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Ch. 8 - Hypothesis Testing with Two Samples
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 8 - Hypothesis Testing with Two SamplesProblem 8.4.7
Chapter 8, Problem 8.4.7

Testing the Difference Between Two Proportions In Exercises 7–12, (a) identify the claim and state Ho and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent.


Multiple Sclerosis Drug In a study to determine the effectiveness of using a drug to treat multiple sclerosis, 488 subjects were given the drug and 244 subjects were given a placebo. The numbers of subjects who had 12-week confirmed disability progression were tracked. The results are shown at the left. At α=0.01, can you support the claim that there is a difference in the proportion of subjects who had no 12-week confirmed disability progression? (Adapted from The New England Journal of Medicine)


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Identify the claim and state the null hypothesis (Ho) and the alternative hypothesis (Ha). The claim is that there is a difference in the proportion of subjects who had no 12-week confirmed disability progression between the drug and placebo groups. Ho: p1 = p2 (no difference in proportions), Ha: p1 ≠ p2 (there is a difference in proportions).
Determine the significance level (α = 0.01) and find the critical value(s) for a two-tailed test. Use the standard normal distribution (z-distribution) to find the critical z-values corresponding to α/2 for each tail.
Calculate the sample proportions for each group. For the drug group, p̂1 = 327/488. For the placebo group, p̂2 = 148/244. Then, calculate the pooled proportion (p̂) using the formula: p̂ = (x1 + x2) / (n1 + n2), where x1 and x2 are the counts of successes (no disability progression) and n1 and n2 are the sample sizes.
Compute the standardized test statistic z using the formula: z = (p̂1 - p̂2) / sqrt(p̂(1 - p̂)(1/n1 + 1/n2)). Substitute the values of p̂1, p̂2, p̂, n1, and n2 into the formula to calculate z.
Compare the calculated z-value to the critical z-values to decide whether to reject or fail to reject the null hypothesis. If the z-value falls in the rejection region, reject Ho. Finally, interpret the decision in the context of the original claim: determine whether the data supports the claim that there is a difference in the proportions of subjects with no disability progression between the drug and placebo groups.

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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 two competing hypotheses: the null hypothesis (H0), which states there is no effect or difference, and the alternative hypothesis (Ha), which suggests there is an effect or difference. In this context, the claim is tested by comparing the proportions of subjects with and without disability progression between the drug and placebo groups.
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Step 1: Write Hypotheses

Critical Value and Rejection Region

The critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the significance level (α), which in this case is set at 0.01. The rejection region is the range of values for the test statistic that would lead to rejecting H0. Understanding these concepts is crucial for determining whether the observed difference in proportions is statistically significant.
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Critical Values: t-Distribution

Standardized Test Statistic (z)

The standardized test statistic, often denoted as z, measures how many standard deviations an observed proportion is from the expected proportion under the null hypothesis. It is calculated using the difference between sample proportions, the pooled proportion, and the standard error. This statistic helps in assessing the strength of evidence against the null hypothesis and is essential for making a decision regarding the claim about the drug's effectiveness.
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Step 2: Calculate Test Statistic
Related Practice
Textbook Question

Test the claim about the difference between two population means and at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.

Claim: μ1<μ2, α=0.10, Assume (σ1)^2=(σ2)^2

Sample statistics:

x̅1=0.345, s1=0.305 , n1=11 and x̅2=0.515, s2=0.215, n2=9

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Textbook Question

[APPLET] Teaching Methods

Two teaching methods and their effects on science test scores are being reviewed. A group of students is taught in traditional lab sessions. A second group of students is taught using interactive simulation software. The science test scores for the two groups are shown in the back-to-back stem-and-leaf plot.

At , α=0.01 can you support the claim that the mean science test score is lower for students taught using the traditional lab method than it is for students taught using the interactive simulation software? Assume the population variances are equal.

Textbook Question

Young Adults In a survey of 3500 males ages 20 to 24 whose highest level of education is some college, but no bachelor’s degree, 80.2% were employed. In a survey of 2000 males ages 20 to 24 whose highest level of education is a bachelor’s degree or higher, 86.4% were employed. At α=0.01, can you support the claim that there is a difference in the proportion of those employed between the two groups? (Adapted from National Center for Education Statistics)

Textbook Question

In Exercises 11–14, test the claim about the difference between two population means and at the level of significance . Assume the samples are random and independent, and the populations are normally distributed.

Claim: μ1<μ2; α=0.03

Population statistics:σ1=136 and σ2=215

Sample Statistics: x̅1=5004, n1=144, x̅2=4895, n2=156

Textbook Question

Independent and Dependent Samples In Exercises 5–8, classify the two samples as independent or dependent and justify your answer.

Sample 1: The IQ scores of 60 females

Sample 2: The IQ scores of 60 males

Textbook Question

Testing the Difference Between Two Means In Exercises 15–24, (a) identify the claim and state Ho and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.

ACT English and Reading Scores The mean ACT English score for 120 high school students is 19.9. Assume the population standard deviation is 7.2. The mean ACT reading score for 150 high school students is 21.2. Assume the population standard deviation is 7.1. At α=0.10, can you support the claim that ACT reading scores are higher than ACT English scores? (Source: ACT, Inc.)