Ch. 9 - Inferences from Two Samples

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 9 - Inferences from Two Samples

Problem 10b

Chapter 9, Problem 10b

Denomination Effect A trial was conducted with 75 women in China given a 100-yuan bill, while another 75 women in China were given 100 yuan in the form of smaller bills (a 50-yuan bill plus two 20-yuan bills plus two 5-yuan bills). Among those given the single bill, 60 spent some or all of the money. Among those given the smaller bills, 68 spent some or all of the money (based on data from “The Denomination Effect,” by Raghubir and Srivastava, Journal of Consumer Research, Vol. 36). We want to use a 0.05 significance level to test the claim that when given a single large bill, a smaller proportion of women in China spend some or all of the money when compared to the proportion of women in China given the same amount in smaller bills.

b. Test the claim by constructing an appropriate confidence interval.

Verified step by step guidance

Step 1: Define the problem and hypotheses. The goal is to test the claim that the proportion of women spending money when given a single large bill is smaller than the proportion of women spending money when given smaller bills. Let p1 represent the proportion of women spending money when given a single large bill, and p2 represent the proportion of women spending money when given smaller bills. The null hypothesis (H0) is p1 = p2, and the alternative hypothesis (H1) is p1 < p2.

Step 2: Calculate the sample proportions. For the group given a single large bill, the sample proportion p̂1 = 60/75. For the group given smaller bills, the sample proportion p̂2 = 68/75. These proportions will be used in the confidence interval calculation.

Step 3: Determine the standard error for the difference in proportions. The formula for the standard error (SE) is:

\sqrt{p̂}

₁(1-p̂₁)n₁+p̂₂(1-p̂₂)n₂, where n1 and n2 are the sample sizes for the two groups.

Step 4: Construct the confidence interval for the difference in proportions. The formula for the confidence interval is:

p̂

₁-p̂₂±z_α/2×SE, where zα/2 is the critical value for the desired confidence level (e.g., 1.96 for 95% confidence).

Step 5: Interpret the confidence interval. If the confidence interval for the difference in proportions does not include 0 and is entirely negative, it supports the claim that the proportion of women spending money when given a single large bill is smaller than the proportion spending money when given smaller bills. Otherwise, the claim is not supported.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample data to determine whether to reject H0 in favor of H1. In this case, the null hypothesis would state that there is no difference in spending behavior between the two groups, while the alternative would suggest that the group with smaller bills spends more.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence, typically 95% or 99%. It provides an estimate of uncertainty around a sample statistic, such as the proportion of women who spent money. Constructing a confidence interval for the difference in proportions will help assess whether the observed difference is statistically significant.

Proportion Comparison

Proportion comparison involves analyzing the differences between two or more proportions to determine if they are statistically different from each other. In this scenario, we compare the proportion of women who spent money from the single large bill group to the proportion from the smaller bills group. This analysis often employs techniques such as the z-test for proportions to evaluate the significance of the observed differences.

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Difference in Proportions: Hypothesis Tests Example 1

Related Practice

Textbook Question

Denomination Effect A trial was conducted with 75 women in China given a 100-yuan bill, while another 75 women in China were given 100 yuan in the form of smaller bills (a 50-yuan bill plus two 20-yuan bills plus two 5-yuan bills). Among those given the single bill, 60 spent some or all of the money. Among those given the smaller bills, 68 spent some or all of the money (based on data from “The Denomination Effect,” by Raghubir and Srivastava, Journal of Consumer Research, Vol. 36). We want to use a 0.05 significance level to test the claim that when given a single large bill, a smaller proportion of women in China spend some or all of the money when compared to the proportion of women in China given the same amount in smaller bills.

c. If the significance level is changed to 0.01, does the conclusion change?

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Textbook Question

In Exercises 17–24, use the indicated Data Sets from Appendix B. The complete data sets can be found at www.TriolaStats.com. Assume that the paired sample data are simple random samples and the differences have a distribution that is approximately normal.

Heights of Presidents Repeat Exercise 12 “Heights of Presidents” using all of the sample data from Data Set 22 “Presidents” in Appendix B.

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Textbook Question

P-VALUE The test statistic of z = 2.14 is obtained when using the data from Exercise 1 and testing the claim that patients treated with dexamethasone and patients given a placebo have the same rate of complete resolution.

a. Find the P-value for the test.

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Textbook Question

a. Test the claim using a hypothesis test.

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Textbook Question

Clinical Trials of OxyContin OxyContin (oxycodone) is a drug used to treat pain, but it is well known for its addictiveness and danger. In a clinical trial, among subjects treated with OxyContin, 52 developed nausea and 175 did not develop nausea. Among other subjects given placebos, 5 developed nausea and 40 did not develop nausea (based on data from Purdue Pharma L.P.). Use a 0.05 significance level to test for a difference between the rates of nausea for those treated with OxyContin and those given a placebo.

a. Use a hypothesis test.

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Textbook Question

F Test Statistic

d. Is the F distribution symmetric, skewed left, or skewed right?

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