Hypothesis Testing

Two-Tailed Tests

Outline

1

Hypothesis Testing

2

Two-Tailed Tests

3

One-Tailed Tests

One-Tailed Tests

Hypothesis Testing

Two-Tailed Tests

Outline

1

Hypothesis Testing

2

Two-Tailed Tests

3

One-Tailed Tests

One-Tailed Tests

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

The Purpose of Statistics Our process until now:

population

describe

approximate

select

sample

parameters

compute

statistics

We gathered and analyzed information about a sample, and inferred information about the population. But the only reason to do all this is to make better decisions!

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

The Context of Statistics Ask questions Select random sample Collect data on sample Compute statistics about sample Infer information about population Use this information to make decisions

We need to learn how to ask suitable questions. We also need to learn how to make good decisions on the basis of statistics.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

The Context of Statistics

Select random sample Collect data on sample Compute statistics about sample Infer information about population

What we’ve done so far

Ask questions

Use this information to make decisions

We need to learn how to ask suitable questions. We also need to learn how to make good decisions on the basis of statistics.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

The Context of Statistics

Select random sample Collect data on sample Compute statistics about sample Infer information about population

What we’ve done so far

Ask questions

Use this information to make decisions

We need to learn how to ask suitable questions. We also need to learn how to make good decisions on the basis of statistics.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

The Context of Statistics

Select random sample Collect data on sample Compute statistics about sample Infer information about population

What we’ve done so far

Ask questions

Use this information to make decisions

We need to learn how to ask suitable questions. We also need to learn how to make good decisions on the basis of statistics.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Asking Good Questions

The context: You need to take some action if the data warrants it. E.g., if your product’s weight is inaccurate, you should fix your production equipment. E.g., if your services are too slow, you should improve them. E.g., if the pollution levels are unacceptably high, you should clean up!

In every case, you will take some action if the data show it’s needed, but you won’t if the data are inconclusive. We formulate this as the null hypothesis and the alternative hypothesis.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

The Two Hypotheses

Definition In a statistical test, the null hypothesis, abbreviated H0 , is the statement that “Nothing is going on,” that “No action is needed because there’s no evidence for it.”

Definition In a statistical test, the alternative hypothesis, abbreviated HA , is the statement that “Something is going on,” that “Action is needed because the evidence shows it.”

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Examples of the Two Hypotheses E.g., if your product’s weight is inaccurate, you should fix your production equipment.

E.g., if your services are too slow, you should improve them.

E.g., if the pollution levels are unacceptably high, you should clean up!

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Examples of the Two Hypotheses E.g., if your product’s weight is inaccurate, you should fix your production equipment. H0 : The product’s weight is accurate. E.g., if your services are too slow, you should improve them.

E.g., if the pollution levels are unacceptably high, you should clean up!

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Examples of the Two Hypotheses E.g., if your product’s weight is inaccurate, you should fix your production equipment. H0 : The product’s weight is accurate. HA : The product’s weight is inaccurate. E.g., if your services are too slow, you should improve them.

E.g., if the pollution levels are unacceptably high, you should clean up!

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Examples of the Two Hypotheses E.g., if your product’s weight is inaccurate, you should fix your production equipment. H0 : The product’s weight is accurate. HA : The product’s weight is inaccurate. E.g., if your services are too slow, you should improve them.

E.g., if the pollution levels are unacceptably high, you should clean up!

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Examples of the Two Hypotheses E.g., if your product’s weight is inaccurate, you should fix your production equipment. H0 : The product’s weight is accurate. HA : The product’s weight is inaccurate. E.g., if your services are too slow, you should improve them. H0 : Your services are acceptable. E.g., if the pollution levels are unacceptably high, you should clean up!

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Examples of the Two Hypotheses E.g., if your product’s weight is inaccurate, you should fix your production equipment. H0 : The product’s weight is accurate. HA : The product’s weight is inaccurate. E.g., if your services are too slow, you should improve them. H0 : Your services are acceptable. HA : Your services are too slow. E.g., if the pollution levels are unacceptably high, you should clean up!

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Examples of the Two Hypotheses E.g., if your product’s weight is inaccurate, you should fix your production equipment. H0 : The product’s weight is accurate. HA : The product’s weight is inaccurate. E.g., if your services are too slow, you should improve them. H0 : Your services are acceptable. HA : Your services are too slow. E.g., if the pollution levels are unacceptably high, you should clean up!

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Examples of the Two Hypotheses E.g., if your product’s weight is inaccurate, you should fix your production equipment. H0 : The product’s weight is accurate. HA : The product’s weight is inaccurate. E.g., if your services are too slow, you should improve them. H0 : Your services are acceptable. HA : Your services are too slow. E.g., if the pollution levels are unacceptably high, you should clean up! H0 : The pollution levels are low enough.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Examples of the Two Hypotheses E.g., if your product’s weight is inaccurate, you should fix your production equipment. H0 : The product’s weight is accurate. HA : The product’s weight is inaccurate. E.g., if your services are too slow, you should improve them. H0 : Your services are acceptable. HA : Your services are too slow. E.g., if the pollution levels are unacceptably high, you should clean up! H0 : The pollution levels are low enough. HA : The pollution levels are too high.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Reality H0 We think

HA

H0 HA

A Type I Error is acting when you didn’t need to. It’s “jumping the gun,” claiming something that’s not true. Its probability is called α. A Type II Error is failing to act when you should have. It’s being cautious, waiting for more evidence. Its probability is called β. A Type I Error is usually worse than Type II, so we worry about α more than β. We call α the level of significance.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Reality We think

H0 HA

H0 Yay!

HA

A Type I Error is acting when you didn’t need to. It’s “jumping the gun,” claiming something that’s not true. Its probability is called α. A Type II Error is failing to act when you should have. It’s being cautious, waiting for more evidence. Its probability is called β. A Type I Error is usually worse than Type II, so we worry about α more than β. We call α the level of significance.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Reality We think

H0 HA

H0 Yay!

HA Yay!

A Type I Error is acting when you didn’t need to. It’s “jumping the gun,” claiming something that’s not true. Its probability is called α. A Type II Error is failing to act when you should have. It’s being cautious, waiting for more evidence. Its probability is called β. A Type I Error is usually worse than Type II, so we worry about α more than β. We call α the level of significance.

Hypothesis Testing

Two-Tailed Tests

We think

H0 HA

One-Tailed Tests

Reality H0 HA Yay! Type I Error Yay!

A Type I Error is acting when you didn’t need to. It’s “jumping the gun,” claiming something that’s not true. Its probability is called α. A Type II Error is failing to act when you should have. It’s being cautious, waiting for more evidence. Its probability is called β. A Type I Error is usually worse than Type II, so we worry about α more than β. We call α the level of significance.

Hypothesis Testing

Two-Tailed Tests

We think

H0 HA

One-Tailed Tests

Reality H0 HA Yay! Type II Error Type I Error Yay!

A Type I Error is acting when you didn’t need to. It’s “jumping the gun,” claiming something that’s not true. Its probability is called α. A Type II Error is failing to act when you should have. It’s being cautious, waiting for more evidence. Its probability is called β. A Type I Error is usually worse than Type II, so we worry about α more than β. We call α the level of significance.

Hypothesis Testing

Two-Tailed Tests

We think

H0 HA

One-Tailed Tests

Reality H0 HA Yay! Type II Error Type I Error Yay!

A Type I Error is acting when you didn’t need to. It’s “jumping the gun,” claiming something that’s not true. Its probability is called α. A Type II Error is failing to act when you should have. It’s being cautious, waiting for more evidence. Its probability is called β. A Type I Error is usually worse than Type II, so we worry about α more than β. We call α the level of significance.

Hypothesis Testing

Two-Tailed Tests

We think

H0 HA

One-Tailed Tests

Reality H0 HA Yay! Type II Error Type I Error Yay!

A Type I Error is acting when you didn’t need to. It’s “jumping the gun,” claiming something that’s not true. Its probability is called α. A Type II Error is failing to act when you should have. It’s being cautious, waiting for more evidence. Its probability is called β. A Type I Error is usually worse than Type II, so we worry about α more than β. We call α the level of significance.

Hypothesis Testing

Two-Tailed Tests

What next? Ask questions Select random sample Collect data on sample Compute statistics about sample Infer information about population Use this information to make decisions

Formulating H0 and HA is the way to frame statistical questions. Next, we need to learn how to test those hypotheses.

One-Tailed Tests

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

What next?

Select random sample Collect data on sample Compute statistics about sample

What we did before

Ask questions

Infer information about population Use this information to make decisions

Formulating H0 and HA is the way to frame statistical questions. Next, we need to learn how to test those hypotheses.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

What next?

Select random sample Collect data on sample Compute statistics about sample

What we did before

Ask questions

Infer information about population Use this information to make decisions

Formulating H0 and HA is the way to frame statistical questions. Next, we need to learn how to test those hypotheses.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

What next?

Select random sample Collect data on sample Compute statistics about sample

What we did before

Ask questions

Infer information about population Use this information to make decisions

Formulating H0 and HA is the way to frame statistical questions. Next, we need to learn how to test those hypotheses.

Hypothesis Testing

Two-Tailed Tests

Outline

1

Hypothesis Testing

2

Two-Tailed Tests

3

One-Tailed Tests

One-Tailed Tests

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

A “two-tailed” test Example You oversee the production of 16 oz. cans of jellied cranberry sauce. You need to make sure the machines are putting the right amount of cranberry sauce into each can; if not, the machines will need to be adjusted. For this situation, The population is all the cans coming out of your factory. You want to be sure the mean µ is 16 ounces; if µ > 16, then you’re losing money, but if µ < 16, you’re cheating your customers.

Thus our hypotheses are H0 : µ = 16 HA : µ 6= 16 The hypothesized population mean is called µ0 , so H0 is “µ = µ0 ” and HA is “µ 6= µ0 .”

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

A “two-tailed” test Example You oversee the production of 16 oz. cans of jellied cranberry sauce. You need to make sure the machines are putting the right amount of cranberry sauce into each can; if not, the machines will need to be adjusted. For this situation, The population is all the cans coming out of your factory. You want to be sure the mean µ is 16 ounces; if µ > 16, then you’re losing money, but if µ < 16, you’re cheating your customers.

Thus our hypotheses are H0 : µ = 16 HA : µ 6= 16 The hypothesized population mean is called µ0 , so H0 is “µ = µ0 ” and HA is “µ 6= µ0 .”

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

A “two-tailed” test Example You oversee the production of 16 oz. cans of jellied cranberry sauce. You need to make sure the machines are putting the right amount of cranberry sauce into each can; if not, the machines will need to be adjusted. For this situation, The population is all the cans coming out of your factory. You want to be sure the mean µ is 16 ounces; if µ > 16, then you’re losing money, but if µ < 16, you’re cheating your customers.

Thus our hypotheses are H0 : µ = 16 HA : µ 6= 16 The hypothesized population mean is called µ0 , so H0 is “µ = µ0 ” and HA is “µ 6= µ0 .”

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

A “two-tailed” test Example You oversee the production of 16 oz. cans of jellied cranberry sauce. You need to make sure the machines are putting the right amount of cranberry sauce into each can; if not, the machines will need to be adjusted. For this situation, The population is all the cans coming out of your factory. You want to be sure the mean µ is 16 ounces; if µ > 16, then you’re losing money, but if µ < 16, you’re cheating your customers.

Thus our hypotheses are H0 : µ = 16 HA : µ 6= 16 The hypothesized population mean is called µ0 , so H0 is “µ = µ0 ” and HA is “µ 6= µ0 .”

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

A “two-tailed” test Example You oversee the production of 16 oz. cans of jellied cranberry sauce. You need to make sure the machines are putting the right amount of cranberry sauce into each can; if not, the machines will need to be adjusted. For this situation, The population is all the cans coming out of your factory. You want to be sure the mean µ is 16 ounces; if µ > 16, then you’re losing money, but if µ < 16, you’re cheating your customers.

Thus our hypotheses are H0 : µ = 16 HA : µ 6= 16 The hypothesized population mean is called µ0 , so H0 is “µ = µ0 ” and HA is “µ 6= µ0 .”

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example You oversee the production of 16 oz. cans of jellied cranberry sauce. You need to make sure the machines are putting the right amount of cranberry sauce into each can; if not, the machines will need to be adjusted.

You want to test where H0 or HA is true. So you take a random sample of 18 cans, weigh them, and compute the sample mean x. 13

14

15

What would you do if. . . x x x x x

= 16.0? = 19.0? = 13.0? = 16.1? = 15.9?

16

17

18

19

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example You oversee the production of 16 oz. cans of jellied cranberry sauce. You need to make sure the machines are putting the right amount of cranberry sauce into each can; if not, the machines will need to be adjusted.

You want to test where H0 or HA is true. So you take a random sample of 18 cans, weigh them, and compute the sample mean x. 13

14

15

What would you do if. . . x x x x x

= 16.0? = 19.0? = 13.0? = 16.1? = 15.9?

16

17

18

19

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example You oversee the production of 16 oz. cans of jellied cranberry sauce. You need to make sure the machines are putting the right amount of cranberry sauce into each can; if not, the machines will need to be adjusted.

You want to test where H0 or HA is true. So you take a random sample of 18 cans, weigh them, and compute the sample mean x. 13

14

15

What would you do if. . . x x x x x

= 16.0? Don’t reject H0 . = 19.0? = 13.0? = 16.1? = 15.9?

16

17

18

19

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example You oversee the production of 16 oz. cans of jellied cranberry sauce. You need to make sure the machines are putting the right amount of cranberry sauce into each can; if not, the machines will need to be adjusted.

You want to test where H0 or HA is true. So you take a random sample of 18 cans, weigh them, and compute the sample mean x. 13

14

15

What would you do if. . . x x x x x

= 16.0? Don’t reject H0 . = 19.0? = 13.0? = 16.1? = 15.9?

16

17

18

19

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example You oversee the production of 16 oz. cans of jellied cranberry sauce. You need to make sure the machines are putting the right amount of cranberry sauce into each can; if not, the machines will need to be adjusted.

You want to test where H0 or HA is true. So you take a random sample of 18 cans, weigh them, and compute the sample mean x. 13

14

15

16

What would you do if. . . x x x x x

= 16.0? Don’t reject H0 . = 19.0? Reject H0 , accept HA . = 13.0? = 16.1? = 15.9?

17

18

19

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example You oversee the production of 16 oz. cans of jellied cranberry sauce. You need to make sure the machines are putting the right amount of cranberry sauce into each can; if not, the machines will need to be adjusted.

You want to test where H0 or HA is true. So you take a random sample of 18 cans, weigh them, and compute the sample mean x. 13

14

15

16

What would you do if. . . x x x x x

= 16.0? Don’t reject H0 . = 19.0? Reject H0 , accept HA . = 13.0? = 16.1? = 15.9?

17

18

19

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example You oversee the production of 16 oz. cans of jellied cranberry sauce. You need to make sure the machines are putting the right amount of cranberry sauce into each can; if not, the machines will need to be adjusted.

You want to test where H0 or HA is true. So you take a random sample of 18 cans, weigh them, and compute the sample mean x. 13

14

15

16

What would you do if. . . x x x x x

= 16.0? Don’t reject H0 . = 19.0? Reject H0 , accept HA . = 13.0? Reject H0 , accept HA . = 16.1? = 15.9?

17

18

19

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example You oversee the production of 16 oz. cans of jellied cranberry sauce. You need to make sure the machines are putting the right amount of cranberry sauce into each can; if not, the machines will need to be adjusted.

You want to test where H0 or HA is true. So you take a random sample of 18 cans, weigh them, and compute the sample mean x. 13

14

15

16

What would you do if. . . x x x x x

= 16.0? Don’t reject H0 . = 19.0? Reject H0 , accept HA . = 13.0? Reject H0 , accept HA . = 16.1? = 15.9?

17

18

19

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example You oversee the production of 16 oz. cans of jellied cranberry sauce. You need to make sure the machines are putting the right amount of cranberry sauce into each can; if not, the machines will need to be adjusted.

You want to test where H0 or HA is true. So you take a random sample of 18 cans, weigh them, and compute the sample mean x. 13

14

15

16

What would you do if. . . x x x x x

= 16.0? = 19.0? = 13.0? = 16.1? = 15.9?

Don’t reject H0 . Reject H0 , accept HA . Reject H0 , accept HA . Don’t reject H0 .

17

18

19

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example You oversee the production of 16 oz. cans of jellied cranberry sauce. You need to make sure the machines are putting the right amount of cranberry sauce into each can; if not, the machines will need to be adjusted.

You want to test where H0 or HA is true. So you take a random sample of 18 cans, weigh them, and compute the sample mean x. 13

14

15

16

What would you do if. . . x x x x x

= 16.0? = 19.0? = 13.0? = 16.1? = 15.9?

Don’t reject H0 . Reject H0 , accept HA . Reject H0 , accept HA . Don’t reject H0 .

17

18

19

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example You oversee the production of 16 oz. cans of jellied cranberry sauce. You need to make sure the machines are putting the right amount of cranberry sauce into each can; if not, the machines will need to be adjusted.

You want to test where H0 or HA is true. So you take a random sample of 18 cans, weigh them, and compute the sample mean x. 13

14

15

16

What would you do if. . . x x x x x

= 16.0? = 19.0? = 13.0? = 16.1? = 15.9?

Don’t reject H0 . Reject H0 , accept HA . Reject H0 , accept HA . Don’t reject H0 . Don’t reject H0 .

17

18

19

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Rejection and Non-Rejection Ranges

µ0

If x falls far away from µ0 , we reject H0 and accept HA . If x falls close to µ0 , however, we don’t reject H0 . The question, then, is where exactly to draw the boundaries between the rejection region and the nonrejection region.

Important! We never actually accept the null hypothesis on the basis of statistical analysis. If the statistics look consistent with H0 , we just “fail to reject H0 .”

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Rejection and Non-Rejection Ranges

µ0 rejection region If x falls far away from µ0 , we reject H0 and accept HA . If x falls close to µ0 , however, we don’t reject H0 . The question, then, is where exactly to draw the boundaries between the rejection region and the nonrejection region.

Important! We never actually accept the null hypothesis on the basis of statistical analysis. If the statistics look consistent with H0 , we just “fail to reject H0 .”

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Rejection and Non-Rejection Ranges nonrejection region

µ0 rejection region If x falls far away from µ0 , we reject H0 and accept HA . If x falls close to µ0 , however, we don’t reject H0 . The question, then, is where exactly to draw the boundaries between the rejection region and the nonrejection region.

Important! We never actually accept the null hypothesis on the basis of statistical analysis. If the statistics look consistent with H0 , we just “fail to reject H0 .”

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Rejection and Non-Rejection Ranges nonrejection region

?

? µ0 rejection region

If x falls far away from µ0 , we reject H0 and accept HA . If x falls close to µ0 , however, we don’t reject H0 . The question, then, is where exactly to draw the boundaries between the rejection region and the nonrejection region.

Important! We never actually accept the null hypothesis on the basis of statistical analysis. If the statistics look consistent with H0 , we just “fail to reject H0 .”

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Rejection and Non-Rejection Ranges nonrejection region

?

? µ0 rejection region

If x falls far away from µ0 , we reject H0 and accept HA . If x falls close to µ0 , however, we don’t reject H0 . The question, then, is where exactly to draw the boundaries between the rejection region and the nonrejection region.

Important! We never actually accept the null hypothesis on the basis of statistical analysis. If the statistics look consistent with H0 , we just “fail to reject H0 .”

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Probabilities Reality H0 HA We think

H0 HA

Suppose the null hypothesis H0 really is true. The probability we decide wrongly is called α, so the probability we arrive at the right conclusion is 1 − α.

Likewise, if HA really is true instead, The probability we decide wrongly is called β, so the probability we arrive at the right conclusion is 1 − β.

If we widen our nonrejection region, we decrease α but increase β. If we narrow our nonrejection region, we decrease β but increase α.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Probabilities Reality H0 HA We think

H0 HA

α

Suppose the null hypothesis H0 really is true. The probability we decide wrongly is called α, so the probability we arrive at the right conclusion is 1 − α.

Likewise, if HA really is true instead, The probability we decide wrongly is called β, so the probability we arrive at the right conclusion is 1 − β.

If we widen our nonrejection region, we decrease α but increase β. If we narrow our nonrejection region, we decrease β but increase α.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Probabilities

We think

H0 HA

Reality H0 HA 1−α α

Suppose the null hypothesis H0 really is true. The probability we decide wrongly is called α, so the probability we arrive at the right conclusion is 1 − α.

Likewise, if HA really is true instead, The probability we decide wrongly is called β, so the probability we arrive at the right conclusion is 1 − β.

If we widen our nonrejection region, we decrease α but increase β. If we narrow our nonrejection region, we decrease β but increase α.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Probabilities

We think

H0 HA

Reality H0 HA 1−α α

Suppose the null hypothesis H0 really is true. The probability we decide wrongly is called α, so the probability we arrive at the right conclusion is 1 − α.

Likewise, if HA really is true instead, The probability we decide wrongly is called β, so the probability we arrive at the right conclusion is 1 − β.

If we widen our nonrejection region, we decrease α but increase β. If we narrow our nonrejection region, we decrease β but increase α.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Probabilities

We think

H0 HA

Reality H0 HA 1−α β α

Suppose the null hypothesis H0 really is true. The probability we decide wrongly is called α, so the probability we arrive at the right conclusion is 1 − α.

Likewise, if HA really is true instead, The probability we decide wrongly is called β, so the probability we arrive at the right conclusion is 1 − β.

If we widen our nonrejection region, we decrease α but increase β. If we narrow our nonrejection region, we decrease β but increase α.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Probabilities

We think

H0 HA

Reality H0 HA 1−α β α 1−β

Suppose the null hypothesis H0 really is true. The probability we decide wrongly is called α, so the probability we arrive at the right conclusion is 1 − α.

Likewise, if HA really is true instead, The probability we decide wrongly is called β, so the probability we arrive at the right conclusion is 1 − β.

If we widen our nonrejection region, we decrease α but increase β. If we narrow our nonrejection region, we decrease β but increase α.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Probabilities

We think

H0 HA

Reality H0 HA 1−α β α 1−β

Suppose the null hypothesis H0 really is true. The probability we decide wrongly is called α, so the probability we arrive at the right conclusion is 1 − α.

Likewise, if HA really is true instead, The probability we decide wrongly is called β, so the probability we arrive at the right conclusion is 1 − β.

If we widen our nonrejection region, we decrease α but increase β. If we narrow our nonrejection region, we decrease β but increase α.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Probabilities

We think

H0 HA

Reality H0 HA 1−α β α 1−β

Suppose the null hypothesis H0 really is true. The probability we decide wrongly is called α, so the probability we arrive at the right conclusion is 1 − α.

Likewise, if HA really is true instead, The probability we decide wrongly is called β, so the probability we arrive at the right conclusion is 1 − β.

If we widen our nonrejection region, we decrease α but increase β. If we narrow our nonrejection region, we decrease β but increase α.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Probabilities

We think

H0 HA

Reality H0 HA 1−α β α 1−β

Suppose the null hypothesis H0 really is true. The probability we decide wrongly is called α, so the probability we arrive at the right conclusion is 1 − α.

Likewise, if HA really is true instead, The probability we decide wrongly is called β, so the probability we arrive at the right conclusion is 1 − β.

If we widen our nonrejection region, we decrease α but increase β. If we narrow our nonrejection region, we decrease β but increase α.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Relating α to the rejection region Suppose H0 is true, so the mean µ really is µ0 . Then x has a t-distribution with mean µ0 and standard error √sn .

µ0

Suppose we set our rejection region and nonrejection region as shown. Then the shaded areas are the probabilities of rejection and nonrejection. We can find these probabilities using a t-table!

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Relating α to the rejection region Suppose H0 is true, so the mean µ really is µ0 . Then x has a t-distribution with mean µ0 and standard error √sn .

µ0

Suppose we set our rejection region and nonrejection region as shown. Then the shaded areas are the probabilities of rejection and nonrejection. We can find these probabilities using a t-table!

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Relating α to the rejection region Suppose H0 is true, so the mean µ really is µ0 . Then x has a t-distribution with mean µ0 and standard error √sn .

µ0

Suppose we set our rejection region and nonrejection region as shown. Then the shaded areas are the probabilities of rejection and nonrejection. We can find these probabilities using a t-table!

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Relating α to the rejection region Suppose H0 is true, so the mean µ really is µ0 . Then x has a t-distribution with mean µ0 and standard error √sn .

1−α

α

µ0

Suppose we set our rejection region and nonrejection region as shown. Then the shaded areas are the probabilities of rejection and nonrejection. We can find these probabilities using a t-table!

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Relating α to the rejection region Suppose H0 is true, so the mean µ really is µ0 . Then x has a t-distribution with mean µ0 and standard error √sn .

1−α

α

µ0

Suppose we set our rejection region and nonrejection region as shown. Then the shaded areas are the probabilities of rejection and nonrejection. We can find these probabilities using a t-table!

Hypothesis Testing

Two-Tailed Tests

Calculating a rejection region from α 1

Let’s say we want a certain α.

2

Draw the appropriate t- or z-curve.

3

Draw the rejection and nonrejection regions, and label them with their probabilities.

4

Then the area of the left tail is α/2.

5

Find the end of the left tail using the t- or z-table.

One-Tailed Tests

Hypothesis Testing

Two-Tailed Tests

Calculating a rejection region from α 1

Let’s say we want a certain α.

2

Draw the appropriate t- or z-curve.

3

Draw the rejection and nonrejection regions, and label them with their probabilities.

4

Then the area of the left tail is α/2.

5

Find the end of the left tail using the t- or z-table.

One-Tailed Tests

Hypothesis Testing

Two-Tailed Tests

Calculating a rejection region from α 1

Let’s say we want a certain α.

2

Draw the appropriate t- or z-curve.

3

Draw the rejection and nonrejection regions, and label them with their probabilities.

4

Then the area of the left tail is α/2.

5

Find the end of the left tail using the t- or z-table.

One-Tailed Tests

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Calculating a rejection region from α 1

Let’s say we want a certain α.

2

Draw the appropriate t- or z-curve.

1−α

α

3

Draw the rejection and nonrejection regions, and label them with their probabilities.

4

Then the area of the left tail is α/2.

5

Find the end of the left tail using the t- or z-table.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Calculating a rejection region from α 1

Let’s say we want a certain α.

2

Draw the appropriate t- or z-curve.

α/2

1−α

α

3

Draw the rejection and nonrejection regions, and label them with their probabilities.

4

Then the area of the left tail is α/2.

5

Find the end of the left tail using the t- or z-table.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Calculating a rejection region from α 1

Let’s say we want a certain α.

2

Draw the appropriate t- or z-curve.

α/2

1−α

α

−t 3

Draw the rejection and nonrejection regions, and label them with their probabilities.

4

Then the area of the left tail is α/2.

5

Find the end of the left tail using the t- or z-table.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Calculating a rejection region from α 1

Let’s say we want a certain α.

2

Draw the appropriate t- or z-curve.

α/2

α

1−α

−t

t

3

Draw the rejection and nonrejection regions, and label them with their probabilities.

4

Then the area of the left tail is α/2.

5

Find the end of the left tail using the t- or z-table.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Using the rejection region calculated from α

α

1−α

−t

t

6

Then we choose our random sample and compute x and s.

7

We calculate the t-score of x, namely

8

x−µ √0 . s/ n

If x’s t-score lies below −t or above t, we reject H0 and accept HA . If x’s t-score lies between −t and t, we do not reject H0 .

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Using the rejection region calculated from α

α

1−α

−t

t

6

Then we choose our random sample and compute x and s.

7

We calculate the t-score of x, namely

8

x−µ √0 . s/ n

If x’s t-score lies below −t or above t, we reject H0 and accept HA . If x’s t-score lies between −t and t, we do not reject H0 .

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Using the rejection region calculated from α

α

1−α

−t

t

6

Then we choose our random sample and compute x and s.

7

We calculate the t-score of x, namely

8

x−µ √0 . s/ n

If x’s t-score lies below −t or above t, we reject H0 and accept HA . If x’s t-score lies between −t and t, we do not reject H0 .

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Using the rejection region calculated from α

α

1−α

−t

t

6

Then we choose our random sample and compute x and s.

7

We calculate the t-score of x, namely

8

x−µ √0 . s/ n

If x’s t-score lies below −t or above t, we reject H0 and accept HA . If x’s t-score lies between −t and t, we do not reject H0 .

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Using the rejection region calculated from α

α

1−α

−t

t

6

Then we choose our random sample and compute x and s.

7

We calculate the t-score of x, namely

8

x−µ √0 . s/ n

If x’s t-score lies below −t or above t, we reject H0 and accept HA . If x’s t-score lies between −t and t, we do not reject H0 .

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Using the rejection region calculated from α

α

1−α

−t

t

6

Then we choose our random sample and compute x and s.

7

We calculate the t-score of x, namely

8

x−µ √0 . s/ n

If x’s t-score lies below −t or above t, we reject H0 and accept HA . If x’s t-score lies between −t and t, we do not reject H0 .

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Remember!

If the sample size n < 30, we use the t-table with n − 1 degrees of freedom. If n ≥ 30, we just use the z-table.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Cranberry sauce Example You oversee the production of 16 oz. cans of jellied cranberry sauce. You need to make sure the machines are putting the right amount of cranberry sauce into each can; if not, the machines will need to be adjusted. You will take a sample of 35 cans, and want to be sure your likelihood of a Type I Error is at most α = 0.04. You find a sample mean of x = 15.8 and standard deviation s = 0.4.

Solution First, let’s figure out our rejection regions.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Cranberry sauce Example You oversee the production of 16 oz. cans of jellied cranberry sauce. You need to make sure the machines are putting the right amount of cranberry sauce into each can; if not, the machines will need to be adjusted. You will take a sample of 35 cans, and want to be sure your likelihood of a Type I Error is at most α = 0.04. You find a sample mean of x = 15.8 and standard deviation s = 0.4.

Solution First, let’s figure out our rejection regions.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Cranberry sauce Example You oversee the production of 16 oz. cans of jellied cranberry sauce. You need to make sure the machines are putting the right amount of cranberry sauce into each can; if not, the machines will need to be adjusted. You will take a sample of 35 cans, and want to be sure your likelihood of a Type I Error is at most α = 0.04. You find a sample mean of x = 15.8 and standard deviation s = 0.4.

Solution First, let’s figure out our rejection regions.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Cranberry sauce Example You oversee the production of 16 oz. cans of jellied cranberry sauce. You need to make sure the machines are putting the right amount of cranberry sauce into each can; if not, the machines will need to be adjusted. You will take a sample of 35 cans, and want to be sure your likelihood of a Type I Error is at most α = 0.04. You find a sample mean of x = 15.8 and standard deviation s = 0.4.

Solution First, let’s figure out our rejection regions.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Cranberry sauce 1

We want α = 0.04.

2

Since n = 35 ≥ 30, we draw the z-curve.

3

We draw the rejection and nonrejection regions, and label them with their probabilities.

4

Then the area of the left tail is 0.02.

5

The z-table tells us the left tail ends at -2.05.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Cranberry sauce 1

We want α = 0.04.

2

Since n = 35 ≥ 30, we draw the z-curve.

3

We draw the rejection and nonrejection regions, and label them with their probabilities.

4

Then the area of the left tail is 0.02.

5

The z-table tells us the left tail ends at -2.05.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Cranberry sauce 1

We want α = 0.04.

2

Since n = 35 ≥ 30, we draw the z-curve.

0.96

0.04

3

We draw the rejection and nonrejection regions, and label them with their probabilities.

4

Then the area of the left tail is 0.02.

5

The z-table tells us the left tail ends at -2.05.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Cranberry sauce 1

We want α = 0.04.

2

Since n = 35 ≥ 30, we draw the z-curve.

0.02

0.96

0.04

3

We draw the rejection and nonrejection regions, and label them with their probabilities.

4

Then the area of the left tail is 0.02.

5

The z-table tells us the left tail ends at -2.05.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Cranberry sauce 1

We want α = 0.04.

2

Since n = 35 ≥ 30, we draw the z-curve.

0.02

0.96

0.04

−2.05 3

We draw the rejection and nonrejection regions, and label them with their probabilities.

4

Then the area of the left tail is 0.02.

5

The z-table tells us the left tail ends at -2.05.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Cranberry sauce

0.96

−2.05 6 7

0.04

2.05

Now our random sample had x = 15.8 and s = 0.4. We calculate the z-score of x, namely x − µ0 15.8 − 16.0 √ = √ = −2.96 s/ n 0.4/ 35

8

Since this z-score lies below −2.05, we reject H0 and accept HA . We decide to recalibrate the machines.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Cranberry sauce

0.96

−2.05 6 7

0.04

2.05

Now our random sample had x = 15.8 and s = 0.4. We calculate the z-score of x, namely x − µ0 15.8 − 16.0 √ = √ = −2.96 s/ n 0.4/ 35

8

Since this z-score lies below −2.05, we reject H0 and accept HA . We decide to recalibrate the machines.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Cranberry sauce

0.96

−2.05 6 7

0.04

2.05

Now our random sample had x = 15.8 and s = 0.4. We calculate the z-score of x, namely x − µ0 15.8 − 16.0 √ = √ = −2.96 s/ n 0.4/ 35

8

Since this z-score lies below −2.05, we reject H0 and accept HA . We decide to recalibrate the machines.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Cranberry sauce

0.96

−2.05 6 7

0.04

2.05

Now our random sample had x = 15.8 and s = 0.4. We calculate the z-score of x, namely x − µ0 15.8 − 16.0 √ = √ = −2.96 s/ n 0.4/ 35

8

Since this z-score lies below −2.05, we reject H0 and accept HA . We decide to recalibrate the machines.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Important Observations

When you are doing statistical testing, you must decide what your hypothesis is before you collect the statistics. decide on your level of significance α before you collect the statistics. Otherwise, you’re more likely to delude yourself, choosing an α weak enough to let you believe what you want to believe.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Why “two-tailed”?

α

It’s called a “two-tailed test” because the probability α of a Type I error is spread out in two tails.

Hypothesis Testing

Two-Tailed Tests

Outline

1

Hypothesis Testing

2

Two-Tailed Tests

3

One-Tailed Tests

One-Tailed Tests

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

One-tailed tests The example we just did was of the form H0 : µ = µ0 . What about examples of the form H0 : µ ≤ µ0 or H0 : µ ≥ µ0 ?

Example You run an Internet Service Provider, and you promise your clients that they will get average speeds of at least 3 Mbps. You’ve had complaints of slow service lately, so you want to run a statistical check. If the average really is below 3 Mbps, you will have to spend money to upgrade your network.

In this case, the population is the download speeds of all your clients at all times, and your sample will consist of the tested speeds of some clients at some times. Your hypotheses are: H0 : µ ≥ 3 HA : µ < 3 (Here µ0 = 3, and H0 is of the form “µ ≥ µ0 .”)

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

One-tailed tests The example we just did was of the form H0 : µ = µ0 . What about examples of the form H0 : µ ≤ µ0 or H0 : µ ≥ µ0 ?

Example You run an Internet Service Provider, and you promise your clients that they will get average speeds of at least 3 Mbps. You’ve had complaints of slow service lately, so you want to run a statistical check. If the average really is below 3 Mbps, you will have to spend money to upgrade your network.

In this case, the population is the download speeds of all your clients at all times, and your sample will consist of the tested speeds of some clients at some times. Your hypotheses are: H0 : µ ≥ 3 HA : µ < 3 (Here µ0 = 3, and H0 is of the form “µ ≥ µ0 .”)

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

One-tailed tests The example we just did was of the form H0 : µ = µ0 . What about examples of the form H0 : µ ≤ µ0 or H0 : µ ≥ µ0 ?

Example You run an Internet Service Provider, and you promise your clients that they will get average speeds of at least 3 Mbps. You’ve had complaints of slow service lately, so you want to run a statistical check. If the average really is below 3 Mbps, you will have to spend money to upgrade your network.

In this case, the population is the download speeds of all your clients at all times, and your sample will consist of the tested speeds of some clients at some times. Your hypotheses are: H0 : µ ≥ 3 HA : µ < 3 (Here µ0 = 3, and H0 is of the form “µ ≥ µ0 .”)

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

One-tailed tests The example we just did was of the form H0 : µ = µ0 . What about examples of the form H0 : µ ≤ µ0 or H0 : µ ≥ µ0 ?

Example You run an Internet Service Provider, and you promise your clients that they will get average speeds of at least 3 Mbps. You’ve had complaints of slow service lately, so you want to run a statistical check. If the average really is below 3 Mbps, you will have to spend money to upgrade your network.

In this case, the population is the download speeds of all your clients at all times, and your sample will consist of the tested speeds of some clients at some times. Your hypotheses are: H0 : µ ≥ 3 HA : µ < 3 (Here µ0 = 3, and H0 is of the form “µ ≥ µ0 .”)

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

One-tailed tests The example we just did was of the form H0 : µ = µ0 . What about examples of the form H0 : µ ≤ µ0 or H0 : µ ≥ µ0 ?

Example You run an Internet Service Provider, and you promise your clients that they will get average speeds of at least 3 Mbps. You’ve had complaints of slow service lately, so you want to run a statistical check. If the average really is below 3 Mbps, you will have to spend money to upgrade your network.

In this case, the population is the download speeds of all your clients at all times, and your sample will consist of the tested speeds of some clients at some times. Your hypotheses are: H0 : µ ≥ 3 HA : µ < 3 (Here µ0 = 3, and H0 is of the form “µ ≥ µ0 .”)

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example You run an Internet Service Provider, and you promise your clients that they will get average speeds of at least 3 Mbps. You’ve had complaints of slow service lately, so you want to run a statistical check. If the average really is below 3 Mbps, you will have to spend money to upgrade your network.

You want to test where H0 or HA is true. So you take a random sample of 15 clients, measure their download speeds, and compute the sample mean x. 1

2

What would you do if. . . x x x x x

= 3.0? = 4.5? = 1.4? = 3.2? = 2.9?

3

4

5

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example You run an Internet Service Provider, and you promise your clients that they will get average speeds of at least 3 Mbps. You’ve had complaints of slow service lately, so you want to run a statistical check. If the average really is below 3 Mbps, you will have to spend money to upgrade your network.

You want to test where H0 or HA is true. So you take a random sample of 15 clients, measure their download speeds, and compute the sample mean x. 1

2

What would you do if. . . x x x x x

= 3.0? = 4.5? = 1.4? = 3.2? = 2.9?

3

4

5

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example You run an Internet Service Provider, and you promise your clients that they will get average speeds of at least 3 Mbps. You’ve had complaints of slow service lately, so you want to run a statistical check. If the average really is below 3 Mbps, you will have to spend money to upgrade your network.

You want to test where H0 or HA is true. So you take a random sample of 15 clients, measure their download speeds, and compute the sample mean x. 1

2

What would you do if. . . x x x x x

= 3.0? Don’t reject H0 . = 4.5? = 1.4? = 3.2? = 2.9?

3

4

5

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example You run an Internet Service Provider, and you promise your clients that they will get average speeds of at least 3 Mbps. You’ve had complaints of slow service lately, so you want to run a statistical check. If the average really is below 3 Mbps, you will have to spend money to upgrade your network.

You want to test where H0 or HA is true. So you take a random sample of 15 clients, measure their download speeds, and compute the sample mean x. 1

2

What would you do if. . . x x x x x

= 3.0? Don’t reject H0 . = 4.5? = 1.4? = 3.2? = 2.9?

3

4

5

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example You run an Internet Service Provider, and you promise your clients that they will get average speeds of at least 3 Mbps. You’ve had complaints of slow service lately, so you want to run a statistical check. If the average really is below 3 Mbps, you will have to spend money to upgrade your network.

You want to test where H0 or HA is true. So you take a random sample of 15 clients, measure their download speeds, and compute the sample mean x. 1

2

What would you do if. . . x x x x x

= 3.0? Don’t reject H0 . = 4.5? Don’t reject H0 . = 1.4? = 3.2? = 2.9?

3

4

5

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example You run an Internet Service Provider, and you promise your clients that they will get average speeds of at least 3 Mbps. You’ve had complaints of slow service lately, so you want to run a statistical check. If the average really is below 3 Mbps, you will have to spend money to upgrade your network.

You want to test where H0 or HA is true. So you take a random sample of 15 clients, measure their download speeds, and compute the sample mean x. 1

2

What would you do if. . . x x x x x

= 3.0? Don’t reject H0 . = 4.5? Don’t reject H0 . = 1.4? = 3.2? = 2.9?

3

4

5

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example You run an Internet Service Provider, and you promise your clients that they will get average speeds of at least 3 Mbps. You’ve had complaints of slow service lately, so you want to run a statistical check. If the average really is below 3 Mbps, you will have to spend money to upgrade your network.

You want to test where H0 or HA is true. So you take a random sample of 15 clients, measure their download speeds, and compute the sample mean x. 1

2

3

What would you do if. . . x x x x x

= 3.0? Don’t reject H0 . = 4.5? Don’t reject H0 . = 1.4? Reject H0 , accept HA . = 3.2? = 2.9?

4

5

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example You run an Internet Service Provider, and you promise your clients that they will get average speeds of at least 3 Mbps. You’ve had complaints of slow service lately, so you want to run a statistical check. If the average really is below 3 Mbps, you will have to spend money to upgrade your network.

You want to test where H0 or HA is true. So you take a random sample of 15 clients, measure their download speeds, and compute the sample mean x. 1

2

3

What would you do if. . . x x x x x

= 3.0? Don’t reject H0 . = 4.5? Don’t reject H0 . = 1.4? Reject H0 , accept HA . = 3.2? = 2.9?

4

5

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example You run an Internet Service Provider, and you promise your clients that they will get average speeds of at least 3 Mbps. You’ve had complaints of slow service lately, so you want to run a statistical check. If the average really is below 3 Mbps, you will have to spend money to upgrade your network.

You want to test where H0 or HA is true. So you take a random sample of 15 clients, measure their download speeds, and compute the sample mean x. 1

2

3

What would you do if. . . x x x x x

= 3.0? = 4.5? = 1.4? = 3.2? = 2.9?

Don’t reject H0 . Don’t reject H0 . Reject H0 , accept HA . Don’t reject H0 .

4

5

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example You run an Internet Service Provider, and you promise your clients that they will get average speeds of at least 3 Mbps. You’ve had complaints of slow service lately, so you want to run a statistical check. If the average really is below 3 Mbps, you will have to spend money to upgrade your network.

You want to test where H0 or HA is true. So you take a random sample of 15 clients, measure their download speeds, and compute the sample mean x. 1

2

3

What would you do if. . . x x x x x

= 3.0? = 4.5? = 1.4? = 3.2? = 2.9?

Don’t reject H0 . Don’t reject H0 . Reject H0 , accept HA . Don’t reject H0 .

4

5

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example You run an Internet Service Provider, and you promise your clients that they will get average speeds of at least 3 Mbps. You’ve had complaints of slow service lately, so you want to run a statistical check. If the average really is below 3 Mbps, you will have to spend money to upgrade your network.

You want to test where H0 or HA is true. So you take a random sample of 15 clients, measure their download speeds, and compute the sample mean x. 1

2

3

What would you do if. . . x x x x x

= 3.0? = 4.5? = 1.4? = 3.2? = 2.9?

Don’t reject H0 . Don’t reject H0 . Reject H0 , accept HA . Don’t reject H0 . Don’t reject H0 .

4

5

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Rejection and Non-Rejection Ranges

µ0

Here there’s just one cutoff between rejecting H0 and not rejecting it. If x falls too far below µ0 , we reject H0 and accept HA . If x falls above µ0 or even close to it, however, we don’t reject H0 . Again, we need to ask where exactly to draw the boundary between the rejection region and the nonrejection region.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Rejection and Non-Rejection Ranges

µ0 rejection region Here there’s just one cutoff between rejecting H0 and not rejecting it. If x falls too far below µ0 , we reject H0 and accept HA . If x falls above µ0 or even close to it, however, we don’t reject H0 . Again, we need to ask where exactly to draw the boundary between the rejection region and the nonrejection region.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Rejection and Non-Rejection Ranges nonrejection region

µ0 rejection region Here there’s just one cutoff between rejecting H0 and not rejecting it. If x falls too far below µ0 , we reject H0 and accept HA . If x falls above µ0 or even close to it, however, we don’t reject H0 . Again, we need to ask where exactly to draw the boundary between the rejection region and the nonrejection region.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Rejection and Non-Rejection Ranges nonrejection region

µ0 rejection region Here there’s just one cutoff between rejecting H0 and not rejecting it. If x falls too far below µ0 , we reject H0 and accept HA . If x falls above µ0 or even close to it, however, we don’t reject H0 . Again, we need to ask where exactly to draw the boundary between the rejection region and the nonrejection region.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Relating α to the rejection region Suppose µ = µ0 . Then x has a t-distribution with mean µ0 and standard error √sn .

µ0

Suppose we set our rejection region and nonrejection region as shown. Then the shaded areas are the probabilities of rejection and nonrejection. We can find these probabilities using a t-table!

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Relating α to the rejection region Suppose µ = µ0 . Then x has a t-distribution with mean µ0 and standard error √sn .

µ0

Suppose we set our rejection region and nonrejection region as shown. Then the shaded areas are the probabilities of rejection and nonrejection. We can find these probabilities using a t-table!

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Relating α to the rejection region Suppose µ = µ0 . Then x has a t-distribution with mean µ0 and standard error √sn .

µ0

Suppose we set our rejection region and nonrejection region as shown. Then the shaded areas are the probabilities of rejection and nonrejection. We can find these probabilities using a t-table!

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Relating α to the rejection region Suppose µ = µ0 . Then x has a t-distribution with mean µ0 and standard error √sn .

1−α α µ0

Suppose we set our rejection region and nonrejection region as shown. Then the shaded areas are the probabilities of rejection and nonrejection. We can find these probabilities using a t-table!

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Relating α to the rejection region Suppose µ = µ0 . Then x has a t-distribution with mean µ0 and standard error √sn .

1−α α µ0

Suppose we set our rejection region and nonrejection region as shown. Then the shaded areas are the probabilities of rejection and nonrejection. We can find these probabilities using a t-table!

Hypothesis Testing

Two-Tailed Tests

Calculating a rejection region from α 1

Let’s say we want a certain level of significance α.

2

Draw the appropriate t- or z-curve.

3

Draw the rejection and nonrejection regions, and label them with their probabilities.

4

Find the end of the left tail using the t- or z-table.

One-Tailed Tests

Hypothesis Testing

Two-Tailed Tests

Calculating a rejection region from α 1

Let’s say we want a certain level of significance α.

2

Draw the appropriate t- or z-curve.

3

Draw the rejection and nonrejection regions, and label them with their probabilities.

4

Find the end of the left tail using the t- or z-table.

One-Tailed Tests

Hypothesis Testing

Two-Tailed Tests

Calculating a rejection region from α 1

Let’s say we want a certain level of significance α.

2

Draw the appropriate t- or z-curve.

3

Draw the rejection and nonrejection regions, and label them with their probabilities.

4

Find the end of the left tail using the t- or z-table.

One-Tailed Tests

Hypothesis Testing

Two-Tailed Tests

Calculating a rejection region from α 1

Let’s say we want a certain level of significance α.

2

Draw the appropriate t- or z-curve.

1−α α

3

Draw the rejection and nonrejection regions, and label them with their probabilities.

4

Find the end of the left tail using the t- or z-table.

One-Tailed Tests

Hypothesis Testing

Two-Tailed Tests

Calculating a rejection region from α 1

Let’s say we want a certain level of significance α.

2

Draw the appropriate t- or z-curve.

1−α α −t 3

Draw the rejection and nonrejection regions, and label them with their probabilities.

4

Find the end of the left tail using the t- or z-table.

One-Tailed Tests

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Using the rejection region calculated from α

1−α α −t

5

Then we choose our random sample and compute x and s.

6

We calculate the t-score of x, namely

7

x−µ √0 . s/ n

If x’s t-score lies in the rejection region, we reject H0 and accept HA . If x’s t-score lies in the nonrejection region, we do not reject H0 .

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Using the rejection region calculated from α

1−α α −t

5

Then we choose our random sample and compute x and s.

6

We calculate the t-score of x, namely

7

x−µ √0 . s/ n

If x’s t-score lies in the rejection region, we reject H0 and accept HA . If x’s t-score lies in the nonrejection region, we do not reject H0 .

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Using the rejection region calculated from α

1−α α −t

5

Then we choose our random sample and compute x and s.

6

We calculate the t-score of x, namely

7

x−µ √0 . s/ n

If x’s t-score lies in the rejection region, we reject H0 and accept HA . If x’s t-score lies in the nonrejection region, we do not reject H0 .

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Using the rejection region calculated from α

1−α α −t

5

Then we choose our random sample and compute x and s.

6

We calculate the t-score of x, namely

7

x−µ √0 . s/ n

If x’s t-score lies in the rejection region, we reject H0 and accept HA . If x’s t-score lies in the nonrejection region, we do not reject H0 .

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Using the rejection region calculated from α

1−α α −t

5

Then we choose our random sample and compute x and s.

6

We calculate the t-score of x, namely

7

x−µ √0 . s/ n

If x’s t-score lies in the rejection region, we reject H0 and accept HA . If x’s t-score lies in the nonrejection region, we do not reject H0 .

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Internet download speeds Example You run an Internet Service Provider, and you promise your clients that they will get average speeds of at least 3 Mbps. You’ve had complaints of slow service lately, so you want to run a statistical check. If the average really is below 3 Mbps, you will have to spend money to upgrade your network. You will take a sample of 15 clients, and you want a level of significance of α = 0.05. You find a sample mean of x = 2.8 and standard deviation s = 0.4.

Solution First, let’s figure out our rejection region.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Internet download speeds Example You run an Internet Service Provider, and you promise your clients that they will get average speeds of at least 3 Mbps. You’ve had complaints of slow service lately, so you want to run a statistical check. If the average really is below 3 Mbps, you will have to spend money to upgrade your network. You will take a sample of 15 clients, and you want a level of significance of α = 0.05. You find a sample mean of x = 2.8 and standard deviation s = 0.4.

Solution First, let’s figure out our rejection region.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Internet download speeds Example You run an Internet Service Provider, and you promise your clients that they will get average speeds of at least 3 Mbps. You’ve had complaints of slow service lately, so you want to run a statistical check. If the average really is below 3 Mbps, you will have to spend money to upgrade your network. You will take a sample of 15 clients, and you want a level of significance of α = 0.05. You find a sample mean of x = 2.8 and standard deviation s = 0.4.

Solution First, let’s figure out our rejection region.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Internet download speeds Example You run an Internet Service Provider, and you promise your clients that they will get average speeds of at least 3 Mbps. You’ve had complaints of slow service lately, so you want to run a statistical check. If the average really is below 3 Mbps, you will have to spend money to upgrade your network. You will take a sample of 15 clients, and you want a level of significance of α = 0.05. You find a sample mean of x = 2.8 and standard deviation s = 0.4.

Solution First, let’s figure out our rejection region.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Internet download speeds 1

We want a level of significance of α = 0.05.

2

Since n = 15, we draw the t-curve with 14 degrees of freedom.

3

We draw the rejection and nonrejection regions, and we label them with their probabilities.

4

Then the t-table for 14 degrees of freedom tells us the tail ends at −1.76.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Internet download speeds 1

We want a level of significance of α = 0.05.

2

Since n = 15, we draw the t-curve with 14 degrees of freedom.

3

We draw the rejection and nonrejection regions, and we label them with their probabilities.

4

Then the t-table for 14 degrees of freedom tells us the tail ends at −1.76.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Internet download speeds 1

We want a level of significance of α = 0.05.

2

Since n = 15, we draw the t-curve with 14 degrees of freedom.

0.95 0.05

3

We draw the rejection and nonrejection regions, and we label them with their probabilities.

4

Then the t-table for 14 degrees of freedom tells us the tail ends at −1.76.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Internet download speeds 1

We want a level of significance of α = 0.05.

2

Since n = 15, we draw the t-curve with 14 degrees of freedom.

0.95 0.05

−1.76 3

We draw the rejection and nonrejection regions, and we label them with their probabilities.

4

Then the t-table for 14 degrees of freedom tells us the tail ends at −1.76.

Hypothesis Testing

Two-Tailed Tests

Example: Internet download speeds

0.95 0.05

−1.76 5 6

Our random sample yielded x = 2.9 and s = 0.4. Thus the t-score of x is x − µ0 2.9 − 3.0 √ = √ = −0.97 s/ n 0.4/ 15

7

Since this lies in the nonrejection region, we do not reject H0 . We do not pay for the network upgrades.

One-Tailed Tests

Hypothesis Testing

Two-Tailed Tests

Example: Internet download speeds

0.95 0.05

−1.76 5 6

Our random sample yielded x = 2.9 and s = 0.4. Thus the t-score of x is x − µ0 2.9 − 3.0 √ = √ = −0.97 s/ n 0.4/ 15

7

Since this lies in the nonrejection region, we do not reject H0 . We do not pay for the network upgrades.

One-Tailed Tests

Hypothesis Testing

Two-Tailed Tests

Example: Internet download speeds

0.95 0.05

−1.76 5 6

Our random sample yielded x = 2.9 and s = 0.4. Thus the t-score of x is x − µ0 2.9 − 3.0 √ = √ = −0.97 s/ n 0.4/ 15

7

Since this lies in the nonrejection region, we do not reject H0 . We do not pay for the network upgrades.

One-Tailed Tests

Hypothesis Testing

Two-Tailed Tests

Example: Internet download speeds

0.95 0.05

−1.76 5 6

Our random sample yielded x = 2.9 and s = 0.4. Thus the t-score of x is x − µ0 2.9 − 3.0 √ = √ = −0.97 s/ n 0.4/ 15

7

Since this lies in the nonrejection region, we do not reject H0 . We do not pay for the network upgrades.

One-Tailed Tests

Hypothesis Testing

Two-Tailed Tests

Three Kinds of Tests If H0 is of the form “µ = µ0 ,” we have a two-tailed test. We must act if x is too small or too large.

If H0 is of the form “µ ≥ µ0 ,” we have a one-tailed test. We must act only if x is too small.

If H0 is of the form “µ ≤ µ0 ,” we have another kind of one-tailed test. Here we act only if x is too large.

One-Tailed Tests

Hypothesis Testing

Two-Tailed Tests

Three Kinds of Tests If H0 is of the form “µ = µ0 ,” we have a two-tailed test. We must act if x is too small or too large.

If H0 is of the form “µ ≥ µ0 ,” we have a one-tailed test. We must act only if x is too small.

If H0 is of the form “µ ≤ µ0 ,” we have another kind of one-tailed test. Here we act only if x is too large.

One-Tailed Tests

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

The same procedure, with just one twist 1 2 3

Let’s say we want a certain level of significance α. Draw the appropriate t- or z-curve. Draw the rejection and nonrejection regions, and label them with their probabilities. 1−α α

4

5 6 7

Find the boundary between the regions using the t- or z-table. Then we choose our random sample and compute x and s. √0 . We calculate the t-score of x, namely x−µ s/ n If x’s t-score lies in the rejection region, we reject H0 and accept HA . If x’s t-score lies in the nonrejection region, we do not reject H0 .

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

The same procedure, with just one twist 1 2 3

Let’s say we want a certain level of significance α. Draw the appropriate t- or z-curve. Draw the rejection and nonrejection regions, and label them with their probabilities. 1−α α

4

5 6 7

Find the boundary between the regions using the t- or z-table. Then we choose our random sample and compute x and s. √0 . We calculate the t-score of x, namely x−µ s/ n If x’s t-score lies in the rejection region, we reject H0 and accept HA . If x’s t-score lies in the nonrejection region, we do not reject H0 .

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

The same procedure, with just one twist 1 2 3

Let’s say we want a certain level of significance α. Draw the appropriate t- or z-curve. Draw the rejection and nonrejection regions, and label them with their probabilities. 1−α α

4

5 6 7

Find the boundary between the regions using the t- or z-table. Then we choose our random sample and compute x and s. √0 . We calculate the t-score of x, namely x−µ s/ n If x’s t-score lies in the rejection region, we reject H0 and accept HA . If x’s t-score lies in the nonrejection region, we do not reject H0 .

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

The same procedure, with just one twist 1 2 3

Let’s say we want a certain level of significance α. Draw the appropriate t- or z-curve. Draw the rejection and nonrejection regions, and label them with their probabilities. 1−α α

4

5 6 7

Find the boundary between the regions using the t- or z-table. Then we choose our random sample and compute x and s. √0 . We calculate the t-score of x, namely x−µ s/ n If x’s t-score lies in the rejection region, we reject H0 and accept HA . If x’s t-score lies in the nonrejection region, we do not reject H0 .

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

The same procedure, with just one twist 1 2 3

Let’s say we want a certain level of significance α. Draw the appropriate t- or z-curve. Draw the rejection and nonrejection regions, and label them with their probabilities. 1−α α

4

5 6 7

Find the boundary between the regions using the t- or z-table. Then we choose our random sample and compute x and s. √0 . We calculate the t-score of x, namely x−µ s/ n If x’s t-score lies in the rejection region, we reject H0 and accept HA . If x’s t-score lies in the nonrejection region, we do not reject H0 .

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

The same procedure, with just one twist 1 2 3

Let’s say we want a certain level of significance α. Draw the appropriate t- or z-curve. Draw the rejection and nonrejection regions, and label them with their probabilities. 1−α α

4

5 6 7

Find the boundary between the regions using the t- or z-table. Then we choose our random sample and compute x and s. √0 . We calculate the t-score of x, namely x−µ s/ n If x’s t-score lies in the rejection region, we reject H0 and accept HA . If x’s t-score lies in the nonrejection region, we do not reject H0 .

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

The same procedure, with just one twist 1 2 3

Let’s say we want a certain level of significance α. Draw the appropriate t- or z-curve. Draw the rejection and nonrejection regions, and label them with their probabilities. 1−α α

4

5 6 7

Find the boundary between the regions using the t- or z-table. Then we choose our random sample and compute x and s. √0 . We calculate the t-score of x, namely x−µ s/ n If x’s t-score lies in the rejection region, we reject H0 and accept HA . If x’s t-score lies in the nonrejection region, we do not reject H0 .

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

The same procedure, with just one twist 1 2 3

Let’s say we want a certain level of significance α. Draw the appropriate t- or z-curve. Draw the rejection and nonrejection regions, and label them with their probabilities. 1−α α

4

5 6 7

Find the boundary between the regions using the t- or z-table. Then we choose our random sample and compute x and s. √0 . We calculate the t-score of x, namely x−µ s/ n If x’s t-score lies in the rejection region, we reject H0 and accept HA . If x’s t-score lies in the nonrejection region, we do not reject H0 .

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

The same procedure, with just one twist 1 2 3

Let’s say we want a certain level of significance α. Draw the appropriate t- or z-curve. Draw the rejection and nonrejection regions, and label them with their probabilities. The tricky bit

1−α α 4

5 6 7

Find the boundary between the regions using the t- or z-table. Then we choose our random sample and compute x and s. √0 . We calculate the t-score of x, namely x−µ s/ n If x’s t-score lies in the rejection region, we reject H0 and accept HA . If x’s t-score lies in the nonrejection region, we do not reject H0 .

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Calculating the boundary for H0 : µ ≤ µ0

1−α α

How do we use the t- or z-table to find the boundary? If it’s a z-curve, it’s easy: just look up 1 − α. If it’s a t-curve, you have two options: either use the full t-table that’s laid out like the z-table, or, if you prefer the smaller t-table, be clever!

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Being clever 1−α α

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Being clever 1−α α

Flip the picture over in your head, 1−α α

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Being clever 1−α α

Flip the picture over in your head, 1−α α

look up this flipped boundary in the t-table, using α,

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Being clever 1−α α

Flip the picture over in your head, 1−α α −t

look up this flipped boundary in the t-table, using α,

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Being clever 1−α α

Flip the picture over in your head, 1−α α −t

look up this flipped boundary in the t-table, using α, and then flip the picture back, changing your boundary from negative to positive. 1−α α t

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Lead paint Example Your office is in a historic old building, and you are concerned that your employees may be getting lead poisoning from the paint. If your employees’ average blood lead level is above 10 µg/dl, you will have to hire contractors to remove the lead paint. You will take a sample of 10 employees, and you want a level of significance of α = 0.10. You find a sample mean of x = 10.6 and standard deviation s = 1.0.

Solution First, let’s figure out our rejection region.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Lead paint Example Your office is in a historic old building, and you are concerned that your employees may be getting lead poisoning from the paint. If your employees’ average blood lead level is above 10 µg/dl, you will have to hire contractors to remove the lead paint. You will take a sample of 10 employees, and you want a level of significance of α = 0.10. You find a sample mean of x = 10.6 and standard deviation s = 1.0.

Solution First, let’s figure out our rejection region.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Lead paint Example Your office is in a historic old building, and you are concerned that your employees may be getting lead poisoning from the paint. If your employees’ average blood lead level is above 10 µg/dl, you will have to hire contractors to remove the lead paint. You will take a sample of 10 employees, and you want a level of significance of α = 0.10. You find a sample mean of x = 10.6 and standard deviation s = 1.0.

Solution First, let’s figure out our rejection region.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Lead paint Example Your office is in a historic old building, and you are concerned that your employees may be getting lead poisoning from the paint. If your employees’ average blood lead level is above 10 µg/dl, you will have to hire contractors to remove the lead paint. You will take a sample of 10 employees, and you want a level of significance of α = 0.10. You find a sample mean of x = 10.6 and standard deviation s = 1.0.

Solution First, let’s figure out our rejection region.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Lead paint 1

We want a level of significance of α = 0.10.

2

Since n = 10, we draw the t-curve with 9 degrees of freedom.

3

We draw the rejection and nonrejection regions, and we label them with their probabilities.

4

Looking in the t-table under 0.10 and 9 degrees of freedom, we find −1.38. Flipping negative to positive, we see that the boundary is at 1.38.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Lead paint 1

We want a level of significance of α = 0.10.

2

Since n = 10, we draw the t-curve with 9 degrees of freedom.

3

We draw the rejection and nonrejection regions, and we label them with their probabilities.

4

Looking in the t-table under 0.10 and 9 degrees of freedom, we find −1.38. Flipping negative to positive, we see that the boundary is at 1.38.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Lead paint 1

We want a level of significance of α = 0.10.

2

Since n = 10, we draw the t-curve with 9 degrees of freedom.

0.90 0.10

3

4

We draw the rejection and nonrejection regions, and we label them with their probabilities. Looking in the t-table under 0.10 and 9 degrees of freedom, we find −1.38. Flipping negative to positive, we see that the boundary is at 1.38.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Lead paint 1

We want a level of significance of α = 0.10.

2

Since n = 10, we draw the t-curve with 9 degrees of freedom.

0.90 0.10

3

4

We draw the rejection and nonrejection regions, and we label them with their probabilities. Looking in the t-table under 0.10 and 9 degrees of freedom, we find −1.38. Flipping negative to positive, we see that the boundary is at 1.38.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Lead paint 1

We want a level of significance of α = 0.10.

2

Since n = 10, we draw the t-curve with 9 degrees of freedom.

0.90 0.10

−1.38 3

4

We draw the rejection and nonrejection regions, and we label them with their probabilities. Looking in the t-table under 0.10 and 9 degrees of freedom, we find −1.38. Flipping negative to positive, we see that the boundary is at 1.38.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Lead paint 1

We want a level of significance of α = 0.10.

2

Since n = 10, we draw the t-curve with 9 degrees of freedom.

0.90 0.10

1.38 3

4

We draw the rejection and nonrejection regions, and we label them with their probabilities. Looking in the t-table under 0.10 and 9 degrees of freedom, we find −1.38. Flipping negative to positive, we see that the boundary is at 1.38.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Lead paint

0.90 0.10

1.38 5 6

Our random sample yielded x = 10.6 and s = 1.0. Thus the t-score of x is x − µ0 10.6 − 10 √ = √ = 1.90 s/ n 1.0/ 10

7

Since this lies in the rejection region, we must reject H0 and accept HA . We decide to hire the lead removal contractors.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Lead paint

0.90 0.10

1.38 5 6

Our random sample yielded x = 10.6 and s = 1.0. Thus the t-score of x is x − µ0 10.6 − 10 √ = √ = 1.90 s/ n 1.0/ 10

7

Since this lies in the rejection region, we must reject H0 and accept HA . We decide to hire the lead removal contractors.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Lead paint

0.90 0.10

1.38 5 6

Our random sample yielded x = 10.6 and s = 1.0. Thus the t-score of x is x − µ0 10.6 − 10 √ = √ = 1.90 s/ n 1.0/ 10

7

Since this lies in the rejection region, we must reject H0 and accept HA . We decide to hire the lead removal contractors.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Example: Lead paint

0.90 0.10

1.38 5 6

Our random sample yielded x = 10.6 and s = 1.0. Thus the t-score of x is x − µ0 10.6 − 10 √ = √ = 1.90 s/ n 1.0/ 10

7

Since this lies in the rejection region, we must reject H0 and accept HA . We decide to hire the lead removal contractors.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Summary When you’re doing a hypothesis test, 1 2

Formulate your hypotheses H0 and HA . DRAW A PICTURE, and decide whether You act if x is too small or too large, you act only if x is too small, or you act only if x is too large.

3

Use α to find the boundary(ies) between rejection and nonrejection.

4

Decide which t-curve you need (or z-curve if n ≥ 30).

5

Convert x into a t-score and see into which region it falls.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Summary When you’re doing a hypothesis test, 1 2

Formulate your hypotheses H0 and HA . DRAW A PICTURE, and decide whether You act if x is too small or too large, you act only if x is too small, or you act only if x is too large.

3

Use α to find the boundary(ies) between rejection and nonrejection.

4

Decide which t-curve you need (or z-curve if n ≥ 30).

5

Convert x into a t-score and see into which region it falls.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Summary When you’re doing a hypothesis test, 1 2

Formulate your hypotheses H0 and HA . DRAW A PICTURE, and decide whether You act if x is too small or too large, you act only if x is too small, or you act only if x is too large.

3

Use α to find the boundary(ies) between rejection and nonrejection.

4

Decide which t-curve you need (or z-curve if n ≥ 30).

5

Convert x into a t-score and see into which region it falls.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Summary When you’re doing a hypothesis test, 1 2

Formulate your hypotheses H0 and HA . DRAW A PICTURE, and decide whether You act if x is too small or too large, you act only if x is too small, or you act only if x is too large.

3

Use α to find the boundary(ies) between rejection and nonrejection.

4

Decide which t-curve you need (or z-curve if n ≥ 30).

5

Convert x into a t-score and see into which region it falls.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Summary When you’re doing a hypothesis test, 1 2

Formulate your hypotheses H0 and HA . DRAW A PICTURE, and decide whether You act if x is too small or too large, you act only if x is too small, or you act only if x is too large.

3

Use α to find the boundary(ies) between rejection and nonrejection.

4

Decide which t-curve you need (or z-curve if n ≥ 30).

5

Convert x into a t-score and see into which region it falls.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Summary When you’re doing a hypothesis test, 1 2

Formulate your hypotheses H0 and HA . DRAW A PICTURE, and decide whether You act if x is too small or too large, you act only if x is too small, or you act only if x is too large.

3

Use α to find the boundary(ies) between rejection and nonrejection.

4

Decide which t-curve you need (or z-curve if n ≥ 30).

5

Convert x into a t-score and see into which region it falls.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Summary When you’re doing a hypothesis test, 1 2

Formulate your hypotheses H0 and HA . DRAW A PICTURE, and decide whether You act if x is too small or too large, you act only if x is too small, or you act only if x is too large.

3

Use α to find the boundary(ies) between rejection and nonrejection.

4

Decide which t-curve you need (or z-curve if n ≥ 30).

5

Convert x into a t-score and see into which region it falls.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Summary When you’re doing a hypothesis test, 1 2

Formulate your hypotheses H0 and HA . DRAW A PICTURE, and decide whether You act if x is too small or too large, you act only if x is too small, or you act only if x is too large.

3

Use α to find the boundary(ies) between rejection and nonrejection.

4

Decide which t-curve you need (or z-curve if n ≥ 30).

5

Convert x into a t-score and see into which region it falls.

Hypothesis Testing

Two-Tailed Tests

One-Tailed Tests

Final Reminders

When you are doing statistical testing for real, you must do the following before you collect the statistics: decide what you’re testing for; decide whether you’re doing a one-tailed or two-tailed test; formulate your hypotheses H0 and HA ; and decide on your level of significance α. Otherwise, you’re just abusing statistics to manipulate yourself or, worse, other people.