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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.R.24
Chapter 8, Problem 8.R.24

In Exercises 23 and 24, (a) identify the claim and state Ho and Ha , (b) find the critical value(s) and identify the rejection region(s), (c) calculate d̄ and sd, (d) find the standardized test statistic t, (e) decide whether to reject or fail to reject the null hypothesis, and (f) interpret the decision in the context of the original claim. Assume the samples are random and dependent, and the populations are normally distributed.


A physical fitness instructor claims that a weight loss supplement will help users lose weight after two weeks. The table shows the weights (in pounds) of 9 adults before using the supplement and two weeks after using the supplement. At α=0.10, is there enough evidence to support the physical fitness instructor’s claim?


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Step 1: Identify the claim and state the null hypothesis (Ho) and alternative hypothesis (Ha). The claim is that the weight loss supplement helps users lose weight after two weeks. Ho: μd = 0 (there is no difference in weight before and after using the supplement). Ha: μd < 0 (the weight after using the supplement is less than the weight before).
Step 2: Find the critical value(s) and identify the rejection region(s). Since α = 0.10 and the test is one-tailed (left-tailed), use a t-distribution table with degrees of freedom (df = n - 1 = 9 - 1 = 8) to find the critical value. The rejection region is t < critical value.
Step 3: Calculate the mean difference (d̄) and the standard deviation of the differences (sd). Compute the differences for each user (Weight before - Weight after), then calculate d̄ = Σd / n and sd = sqrt[Σ(d - d̄)^2 / (n - 1)].
Step 4: Find the standardized test statistic t using the formula t = (d̄ - 0) / (sd / sqrt(n)), where n is the number of paired observations.
Step 5: Decide whether to reject or fail to reject the null hypothesis. Compare the calculated t-value with the critical value. If t < critical value, reject Ho; otherwise, fail to reject Ho. Interpret the decision in the context of the original claim: determine whether there is enough evidence to support the instructor's claim that the supplement helps users lose weight.

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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 represents no effect or no difference, and the alternative hypothesis (Ha), which indicates the presence of an effect or difference. In this context, the instructor's claim about the weight loss supplement serves as the alternative hypothesis.
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Step 1: Write Hypotheses

Dependent Samples

Dependent samples refer to pairs of observations that are related or matched in some way, often used in before-and-after studies. In this case, the weights of the same individuals before and after using the supplement are compared, making the samples dependent. This relationship necessitates the use of specific statistical tests, such as the paired t-test, to analyze the differences in weights.
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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 0.10. The rejection region is the range of values for the test statistic that would lead to rejecting H0. Understanding these concepts is crucial for interpreting the results of the statistical test and making informed decisions based on the data.
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Related Practice
Textbook Question

In Exercises 11–16, test the claim about the difference between two population means μ1 and μ2 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= 520, s1= 25, n1= 7 and x̅2= 500, s2= 55, n2= 6

Textbook Question

In Exercises 25–28, determine whether a normal sampling distribution can be used. If it can be used, test the claim about the difference between two population proportions p1 and p2 at the level of significance α. Assume the samples are random and independent.


Claim: p1<p2; α=0.05


Sample statistics: x1 = 86, n1=900 and x2 = 107, n2 = 1200

Textbook Question

In Exercises 19–22, test the claim about the mean of the differences for a population of paired data at the level of significance α. Assume the samples are random and dependent, and the populations are normally distributed.


Claim: μd≠0; α=0.05.


Sample statistics: d̄=17.5, sd=4.05, n=37



Textbook Question

In Exercises 5–8, test the claim about the difference between two population means μ1 and μ2 at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.


Claim: μ1>μ2; α=0.05


Population statistics: σ1= 0.30 and σ2= 0.23


Sample statistics: x̅1 = 1.28, n1 = 96, and x̅2= 1.34, n2= 85

Textbook Question

"In Exercises 9 and 10, (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.


A career counselor claims that the mean annual salaries of entry level paralegals in Dayton, Ohio, and Coventry, Rhode Island, are the same. The mean annual salary of 40 randomly selected entry level paralegals in Dayton is \$58,180. Assume the population standard deviation is \$10,990. The mean annual salary of 35 randomly selected entry level paralegals in Coventry is \$61,120. Assume the population standard deviation is \$11,850. At α=0.10, is there enough evidence to reject the counselor’s claim? (Adapted from Salary.com)"

Textbook Question

In Exercises 19–22, test the claim about the mean of the differences for a population of paired data at the level of significance α. Assume the samples are random and dependent, and the populations are normally distributed.


Claim: μd<0; α=0.10.


Sample statistics: d̄=3.2, sd=5.68, n=25