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

Larson 8th Edition

Ch. 8 - Hypothesis Testing with Two Samples

Problem 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?

Verified step by step guidance

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.

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.

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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Critical Values: t-Distribution

Related Practice

Textbook Question

Take this quiz as you would take a quiz in class. After you are done, check your work against the answers given in the back of the book.For each exercise, perform the steps below.

c. Find the critical value(s) and identify the rejection region(s).

The mean score on a reading assessment test for 49 randomly selected male high school students was 279. Assume the population standard deviation is 41. The mean score on the same test for 50 randomly selected female high school students was 292. Assume the population standard deviation is 39. At α=0.05, can you support the claim that the mean score on the reading assessment test for male high school students is less than the mean score for female high school students? (Adapted from National Center for Education Statistics)

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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 researcher claims that the mean sodium content of sandwiches at Restaurant A is less than the mean sodium content of sandwiches at Restaurant B. The mean sodium content of 22 randomly selected sandwiches at Restaurant A is 670 milligrams. Assume the population standard deviation is 20 milligrams. The mean sodium content of 28 randomly selected sandwiches at Restaurant B is 690 milligrams. Assume the population standard deviation is 30 milligrams. At α=0.05, is there enough evidence to support the claim?

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Textbook Question

Take this quiz as you would take a quiz in class. After you are done, check your work against the answers given in the back of the book.For each exercise, perform the steps below.

a. Identify the claim and state Ho and Ha

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Textbook Question

In Exercises 29 and 30, (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.

A medical research team conducted a study to test the effect of a drug used to treat a type of inflammation. In the study, 68 subjects took the drug and 68 subjects took a placebo. The results are shown below. At α=0.05, can you reject the claim that the proportion of subjects who had at least 24 weeks of accrued remission is the same for the two groups? (Source: The New England Journal of Medicine)

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Textbook Question

Take this quiz as you would take a quiz in class. After you are done, check your work against the answers given in the back of the book.For each exercise, perform the steps below.

a. Identify the claim and state Ho and Ha

b. Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed, and whether to use a z-test or a t-test. Explain your reasoning.

c. Find the critical value(s) and identify the rejection region(s).

d. Find the appropriate standardized test statistic.

e. Decide whether to reject or fail to reject the null hypothesis.

f. Interpret the decision in the context of the original claim.

A music teacher claims that the mean scores on a music assessment test for eighth grade students in public and private schools are equal. The mean score for 13 randomly selected public school students is 146 with a standard deviation of 49, and the mean score for 15 randomly selected private school students is 160 with a standard deviation of 42. At α=0.1, can you reject the teacher’s claim? Assume the populations are normally distributed and the population variances are equal. (Adapted from National Center for Education Statistics)

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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

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