Continuous Random Variables Lecture 22 Section 7.5.4 Robb T. Koether Hampden-Sydney College

Mon, Oct 3, 2011

Robb T. Koether (Hampden-Sydney College)

Continuous Random Variables

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Outline

1

Homework Review

2

Hypothesis Testing

3

Random Variables The Uniform Distribution A Non-uniform Distribution

4

Assignment

5

Answers to Even-numbered Exercises

Robb T. Koether (Hampden-Sydney College)

Continuous Random Variables

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Outline

1

Homework Review

2

Hypothesis Testing

3

Random Variables The Uniform Distribution A Non-uniform Distribution

4

Assignment

5

Answers to Even-numbered Exercises

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Continuous Random Variables

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Homework Review

Exercise 6.25, page 380. Machine A makes parts whose lengths are approximately normally distributed with a mean of 4.6 mm and a standard deviation of 0.1 mm. Machine B makes parts whose lengths are approximately normally distributed with a mean of 4.9 mm and a standard deviation of 0.1 mm. Suppose that you have a box of parts which you believe are from Machine A, but you’re not sure. You decide to test the hypotheses H0 : The parts are from Machine A versus H1 : The parts are from Machine B, by randomly selecting one part from the box and measuring it.

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Homework Review

Exercise 6.25, page 380. (a) Draw the distributions for the lengths of parts under H0 and under H1 . For both sketches, label the x-axis from 4.2 to 5.2 by 0.1. Be sure to include all important features.

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Continuous Random Variables

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Homework Review

Exercise 6.25, page 380. (a) Draw the distributions for the lengths of parts under H0 and under H1 . For both sketches, label the x-axis from 4.2 to 5.2 by 0.1. Be sure to include all important features. 4

3

2

1

4.4

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4.6

4.8

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5.0

5.2

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Homework Review

Exercise 6.25, page 380. (b) Suppose that you get a length of 4.8 mm. (i) In your sketch for part (a), shade in the region that corresponds to this p-value and clearly label the region as such.

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Continuous Random Variables

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Homework Review

Exercise 6.25, page 380. (b) Suppose that you get a length of 4.8 mm. (i) In your sketch for part (a), shade in the region that corresponds to this p-value and clearly label the region as such. 4

3

2

1

4.4

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4.6

4.8

Continuous Random Variables

5.0

5.2

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Homework Review

Exercise 6.25, page 380. (b)

(ii) Compute the p-value for your test.

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Continuous Random Variables

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Homework Review

Exercise 6.25, page 380. (b)

(ii) Compute the p-value for your test.

The p-value is normalcdf(4.8,E99,4.6,0.1) = 0.0228.

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Continuous Random Variables

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Homework Review

Exercise 6.25, page 380. (b)

(ii) Compute the p-value for your test.

The p-value is normalcdf(4.8,E99,4.6,0.1) = 0.0228. (c) What is your decision at the 0.01 level?

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Continuous Random Variables

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Homework Review

Exercise 6.25, page 380. (b)

(ii) Compute the p-value for your test.

The p-value is normalcdf(4.8,E99,4.6,0.1) = 0.0228. (c) What is your decision at the 0.01 level? The decision at the 0.01 level is to accept H0 .

Robb T. Koether (Hampden-Sydney College)

Continuous Random Variables

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Outline

1

Homework Review

2

Hypothesis Testing

3

Random Variables The Uniform Distribution A Non-uniform Distribution

4

Assignment

5

Answers to Even-numbered Exercises

Robb T. Koether (Hampden-Sydney College)

Continuous Random Variables

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Bag A vs. Bag B

Suppose we have two bags, Bag A and Bag B. Each bag contains millions of vouchers. In Bag A, the values of the vouchers have distribution N(50, 10). In Bag B, the values of the vouchers have distribution N(80, 15).

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Continuous Random Variables

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Bag A vs. Bag B

0.04

0.03

0.02

0.01

40

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60

80

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100

120

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Bag A vs. Bag B

We select one voucher at random from one bag. Decision Rule: If its value is less than or equal to $65, then we will decide that it was from Bag A. What is the value of α? What is the value of β?

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Continuous Random Variables

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Bag A vs. Bag B 0.04

0.03

0.02

0.01

40

Robb T. Koether (Hampden-Sydney College)

60

80

Continuous Random Variables

100

120

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Bag A vs. Bag B 0.04

0.03

0.02

0.01

40

60

80

100

120

α = normalcdf(65,E99,50,10) = 0.0668

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Continuous Random Variables

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Bag A vs. Bag B 0.04

0.03

0.02

0.01

40

60

80

100

120

β = normalcdf(-E99,65,80,15) = 0.1587

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Continuous Random Variables

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Bag A vs. Bag B 0.04

0.03

0.02

0.01

40

60

80

100

120

If the means are very close together, then α and β will be large.

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Continuous Random Variables

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Bag A vs. Bag B 0.04

0.03

0.02

0.01

40

60

80

100

120

α = normalcdf(65,E99,60,10) = 0.3085

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Continuous Random Variables

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Bag A vs. Bag B 0.04

0.03

0.02

0.01

40

60

80

100

120

β = normalcdf(-E99,65,70,15) = 0.3694

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Continuous Random Variables

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Bag A vs. Bag B 0.04

0.03

0.02

0.01

40

60

80

100

120

140

If the means are far apart, then α and β will both be very small.

Robb T. Koether (Hampden-Sydney College)

Continuous Random Variables

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Bag A vs. Bag B 0.04

0.03

0.02

0.01

40

60

80

100

120

140

α = normalcdf(65,E99,40,10) = 0.0062

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Continuous Random Variables

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Bag A vs. Bag B 0.04

0.03

0.02

0.01

40

60

80

100

120

140

β = normalcdf(-E99,65,100,15) = 0.0098

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Continuous Random Variables

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Outline

1

Homework Review

2

Hypothesis Testing

3

Random Variables The Uniform Distribution A Non-uniform Distribution

4

Assignment

5

Answers to Even-numbered Exercises

Robb T. Koether (Hampden-Sydney College)

Continuous Random Variables

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Random Variables

Definition (Random variable) A random variable is a variable whose value is determined by the outcome of a random process.

Definition (Discrete random variable) A discrete random variable is a random variable whose set of possible values is a discrete set.

Definition (Continuous random variable) A continuous random variable is a random variable whose set of possible values is a continuous set.

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Continuous Random Variables

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Continuous Probability Density Functions

Definition (Continuous Probability Density Function) A continuous probability distribution function, or pdf, for a random variable X is a continuous function with the property that the area below the graph of the function between any two points a and b equals the probability that a ≤ X ≤ b. Remember, AREA = PROPORTION = PROBABILITY

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Continuous Random Variables

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Outline

1

Homework Review

2

Hypothesis Testing

3

Random Variables The Uniform Distribution A Non-uniform Distribution

4

Assignment

5

Answers to Even-numbered Exercises

Robb T. Koether (Hampden-Sydney College)

Continuous Random Variables

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Example

The TI-83 will return a random number between 0 and 1 if we enter rand and press ENTER. These numbers have a uniform distribution from 0 to 1. Let X be the random number whose value is determined by the rand function.

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Continuous Random Variables

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Example

f(x) 1

x 0

1

What is the probability that the random number is at least 0.3?

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Continuous Random Variables

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Example

f(x) 1

x 0

0.3

1

What is the probability that the random number is at least 0.3?

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Continuous Random Variables

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Example

f(x) 1

x 0

0.3

1

What is the probability that the random number is at least 0.3?

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Continuous Random Variables

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Example

f(x) 1

x 0

0.3

1

What is the probability that the random number is at least 0.3?

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Continuous Random Variables

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Example

f(x) 1

Area = 0.7 x 0

0.3

1

What is the probability that the random number is at least 0.3?

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Continuous Random Variables

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Example

f(x) 1

x 0

0.25

0.75

1

What is the probability that the random number is between 0.25 and 0.75?

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Continuous Random Variables

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Example

f(x) 1

x 0

0.25

0.75

1

What is the probability that the random number is between 0.25 and 0.75?

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Continuous Random Variables

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Example

f(x) 1

x 0

0.25

0.75

1

What is the probability that the random number is between 0.25 and 0.75?

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Continuous Random Variables

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Example

f(x) 1

Area = 0.5 x 0

0.25

0.75

1

What is the probability that the random number is between 0.25 and 0.75?

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Continuous Random Variables

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Outline

1

Homework Review

2

Hypothesis Testing

3

Random Variables The Uniform Distribution A Non-uniform Distribution

4

Assignment

5

Answers to Even-numbered Exercises

Robb T. Koether (Hampden-Sydney College)

Continuous Random Variables

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A Non-Uniform Distribution

f(x) ?

x 5

6

7

8

9

10

What is the height of this distribution?

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Continuous Random Variables

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A Non-Uniform Distribution

f(x) h

Area = (1/2)bh x 5

6

7

8

9

10

What is the height of this distribution?

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Continuous Random Variables

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A Non-Uniform Distribution

f(x) 0.4

x 5

6

7

8

9

10

What is the height of this distribution?

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Continuous Random Variables

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A Non-Uniform Distribution

f(x) 0.4

x 5

6

7

8

9

10

What is the probability that 6 ≤ X ≤ 8?

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Continuous Random Variables

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A Non-Uniform Distribution

f(x) 0.4

x 5

6

7

8

9

10

What is the probability that 6 ≤ X ≤ 8?

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Continuous Random Variables

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A Non-Uniform Distribution

f(x) 0.4 0.24 0.08 x 5

6

7

8

9

10

What is the probability that 6 ≤ X ≤ 8?

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Continuous Random Variables

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A Non-Uniform Distribution

f(x) 0.4 0.24 0.08

Area = 0.32

5

6

7

x 8

9

10

What is the probability that 6 ≤ X ≤ 8?

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Continuous Random Variables

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A Non-Uniform Distribution

f(x) 1/(b - a)

x a

b

The uniform distribution from a to b is denoted U(a, b).

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Continuous Random Variables

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Outline

1

Homework Review

2

Hypothesis Testing

3

Random Variables The Uniform Distribution A Non-uniform Distribution

4

Assignment

5

Answers to Even-numbered Exercises

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Continuous Random Variables

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Assignment

Homework Read Section 7.5.4, pages 478 - 481. Exercises 64, 65, 66a-c, 68, 69, page 481. Review Exercises 105, 106, page 489.

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Continuous Random Variables

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Outline

1

Homework Review

2

Hypothesis Testing

3

Random Variables The Uniform Distribution A Non-uniform Distribution

4

Assignment

5

Answers to Even-numbered Exercises

Robb T. Koether (Hampden-Sydney College)

Continuous Random Variables

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Answers to Even-numbered Exercises

Page 481, Exercises 64, 66, 68 7.64 (a) 1/15

0

5

10

15

(b) 7.5. (c) 31 . (d) (i) To the right. (ii) 0.1333. (iii) Accept H0 .

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Continuous Random Variables

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Answers to Even-numbered Exercises

Page 481, Exercises 64, 66, 68 7.66 (a) 1/15

30

35

40

45

(b) 37.5 days. (c) 0.20. 7.68 (a) 0.0227. (b) 0.0082.

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Continuous Random Variables

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Answers to Even-numbered Exercises

Page 489, Exercise 106 7.106 (a) 1/5

0

1

2

3

4

5

(b) 0.40.

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Continuous Random Variables

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