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Lesson 10.4.2 Hypothesis Tests from Paired Samples
STUDENT NAME
DATE
PART 1 Hypothesis Tests for the Population Mean of Paired Differences Using the sample statistics introduced in Lesson 10.4.1, we can now perform hypothesis tests for the population mean of paired differences. The process for testing claims about paired differences is quite similar to processes seen previously, and is outlined below. 1) Assumptions We assume that the sampling distribution of sample means of paired differences is normal. As before, we consider this to be plausible when the number of paired differences is greater than 30, or the distribution of such differences is normal. 2) Hypotheses The null hypothesis states that the population mean of the paired differences is equal to 0. In other words, it states that there is no mean difference between the paired data sets.
The alternative hypothesis is one of the following inequalities: For a left-tailed test For a right-tailed test For a two-tailed test
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Lesson 10.4.2 Hypothesis Tests from Paired Samples
3) Test Statistic Assuming the null hypothesis is true, we consider the sampling distribution of means of sample differences. This sampling distribution will have the following parameters: Mean: Estimated Standard Error: Because the sample standard deviation of differences is being used to estimate the corresponding population standard deviation, the test statistic varies according to the T-distribution. This statistic is the same statistic used in our first encounter with a T-test, but we have given the variables different names.
4) P-value Assuming the null hypothesis is true, the P-value is the probability of randomly observing a sample mean difference at least as extreme as the one observed. We will use technology or tables to determine P-values.
For a left-tailed test, the P-value is the area to the left of the test statistic.
For a right-tailed test, the P-value is the area to the right of the test statistic.
For a two-tailed test, the P-value is twice the area of the tail to the right of a positive test statistic or to the left of a negative test statistic.
5) Make a Decision Small P-values (less than or equal to α, the level of significance) are sufficient justification for the rejection of the null hypothesis in favor of the alternative. Rejecting the null hypothesis indicates that the mean difference in the data pairs is significantly different from zero. 6) Conclusion in Context
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Lesson 10.4.2 Hypothesis Tests from Paired Samples
Present a conclusion that directly addresses the fundamental research question. Remember that rejecting the null hypothesis means that the data are in support of the alternative. If we fail to reject the null hypothesis, we do not say that we support it – only that we were unable to reject it.
PART 2 Paired Sample Hypothesis Tests Using T-distributions Use the guidelines above to answer the following questions.
Assessing a Treatment A team of psychologists recommends that people who experience anxiety attacks should receive regular counseling. They state that the counseling sessions provide a therapeutic experience and they promote this type of therapy rather than pharmaceutical treatments. To test this approach, you randomly select 6 people who experience anxiety attacks and are interested in this treatment. You administer an anxiety test before and after the counseling sessions. Higher scores on the test indicate increased anxiety. At the 5% level of significance, can you affirm that the counseling sessions reduce anxiety? Assume the population of test score differences is normally distributed.
1
Person
1
2
3
4
5
6
Pre Counseling
66
71
80
78
85
90
Post Counseling
64
65
79
81
81
91
Assumptions A Are the criteria for approximate normality met for the sampling distribution of differences?
B
Do the sample data come in pairs? Explain.
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Lesson 10.4.2 Hypothesis Tests from Paired Samples
2
Hypotheses A State the null and alternative hypotheses.
B
3
Is this a right-, left-, or two-tailed test?
Test Statistic A Determine the mean and standard deviation of the sample differences, and n, the number of data pairs in the sample.
B
Is the mean of sample differences consistent with the alternative hypothesis?
C
Calculate the test statistic for the observed mean of sample differences.
D
Sketch the T-distribution and identify the position of the observed test statistic.
E
Is the test statistic unusual?
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Lesson 10.4.2 Hypothesis Tests from Paired Samples
4
P-value: Use technology or tables to determine the P-value.
5
Make a Decision A Using the P-value above and the level of significance, what can you conclude about the null hypothesis?
B
6
What can you conclude about the alternative hypothesis?
Conclusion in Context: Interpret your decision in the context of the problem.
Comparing Grocery Prices You suspect there is a difference in prices between two grocery stores (Store A and Store B) in your area. To investigate, you identify the prices of 6 identical items at Stores A and B. Since prices change over time, you gather all data on a single day. At the 5% level of significance, is there enough evidence to conclude that a mean difference exists between the prices of grocery items in the two stores? Assume the population of price differences is normally distributed. All prices are in dollars. Product
Eggs
Cheese
Napkins
Bread
Juice
Milk
Store A
2.59
3.29
3.59
2.99
3.49
2.89
Store B
2.99
3.79
3.99
3.19
3.99
2.69
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Lesson 10.4.2 Hypothesis Tests from Paired Samples
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Lesson 10.4.2 Hypothesis Tests from Paired Samples
1
Assumptions A Are the criteria for approximate normality met for the sampling distribution of differences?
B
2
Hypotheses A State the null and alternative hypotheses.
B
3
Do the sample data come in pairs? Explain.
Is this a right-, left-, or two-tailed test?
Test Statistic A Determine the mean and sample standard deviation of the sample differences, and n, the number of data pairs in the sample.
B
Is the mean of sample differences consistent with your alternative hypothesis?
C
Calculate the test statistic for your observed mean of sample differences.
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Lesson 10.4.2 Hypothesis Tests from Paired Samples
D
Sketch the T-distribution and identify the position of the observed test statistic.
E
Is the test statistic unusual?
4
P-value: Use technology or tables to find the P-value.
5
Make a Decision A Using the P-value above and the level of significance, what can you conclude about the null hypothesis?
B
6
What can you conclude about the alternative hypothesis?
Conclusion in Context: Interpret your decision in the context of the problem?
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Lesson 10.4.2 Hypothesis Tests from Paired Samples
TAKE IT HOME Infectious Agents Viruses are infectious agents that often cause disease in plants. Different viruses have different potency levels, and this fact can be used to detect whether a new virus is infecting plants in the field. In a potency comparison experiment, two viruses were placed on a tobacco leaf of 10 randomly selected plants. The viruses were randomly assigned to one-half of each of the leaves. The table below presents the potency of the viruses, as measured by the number of lesions appearing on the leaf half. At the 5% level of significance, test whether virus X is less potent than virus Y. Assume the population of differences is normally distributed.
1
Leaf #
1
2
3
4
5
6
7
8
9
10
Virus X
9
8
3
4
8
4
17
3
14
20
Virus Y
19
8
13
5
16
8
17
6
19
17
Assumptions A Are the criteria for approximate normality met for the sampling distribution of differences?
B
2
Do the sample data come in pairs? Explain.
Hypotheses A State the null and alternative hypotheses.
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Lesson 10.4.2 Hypothesis Tests from Paired Samples
B
3
4
Is this a right-, left-, or two-tailed test?
Test Statistic A Determine the mean of sample differences, and sample standard deviation of the sample differences, and the number of pairs in the sample.
B
Is the mean of sample differences consistent with your alternative hypothesis?
C
Calculate the test statistic for your observed mean of sample differences.
D
Sketch the T-distribution and identify the position of the observed test statistic.
E
Is the test statistic unusual?
P-value: Use technology or tables to find the P-value.
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Lesson 10.4.2 Hypothesis Tests from Paired Samples
5
Make a Decision A Using the P-value above and the level of significance, what can you conclude about the null hypothesis?
B
6
What can you conclude about the alternative hypothesis?
Conclusion in Context: Interpret your decision in the context of the problem.
+++++ This lesson is part of STATWAY™, A Pathway Through College Statistics, which is a product of a Carnegie Networked Improvement Community that seeks to advance student success. Version 1.0, A Pathway Through Statistics, Statway™ was created by the Charles A. Dana Center at the University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching. This version 1.5 and all subsequent versions, result from the continuous improvement efforts of the Carnegie Networked Improvement Community. The network brings together community college faculty and staff, designers, researchers and developers. It is an open-resource research and development community that seeks to harvest the wisdom of its diverse participants in systematic and disciplined inquiries to improve developmental mathematics instruction. For more information on the Statway Networked Improvement Community, please visit carnegiefoundation.org. For the most recent version of instructional materials, visit Statway.org/kernel. +++++ STATWAY™ and the Carnegie Foundation logo are trademarks of the Carnegie Foundation for the Advancement of Teaching. A Pathway Through College Statistics may be used as provided in the CC BY license, but neither the Statway trademark nor the Carnegie Foundation logo may be used without the prior written consent of the Carnegie Foundation.
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Lesson 10.4.2 Hypothesis Tests from Paired Samples
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