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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.20
Chapter 8, Problem 8.R.20

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

Verified step by step guidance
1
Step 1: Identify the null and alternative hypotheses. The null hypothesis (H₀) states that μd = 0, while the alternative hypothesis (H₁) states that μd < 0. This is a one-tailed test since the claim is μd < 0.
Step 2: Calculate the test statistic using the formula for the t-test for paired data: t = (d̄ - μd) / (sd / √n). Here, d̄ is the sample mean of the differences, sd is the standard deviation of the differences, n is the sample size, and μd is the hypothesized mean difference (0 under H₀).
Step 3: Determine the degrees of freedom (df) for the t-test. For paired data, df = n - 1. In this case, df = 25 - 1 = 24.
Step 4: Find the critical t-value for a one-tailed test at the significance level α = 0.10 with df = 24. Use a t-distribution table or statistical software to find this value.
Step 5: Compare the calculated t-statistic to the critical t-value. If the t-statistic is less than the critical t-value, reject the null hypothesis (H₀). Otherwise, fail to reject the null hypothesis.

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Key Concepts

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

Paired Data

Paired data refers to two sets of related observations, often collected from the same subjects under different conditions. This type of data is used in statistical tests to determine if there is a significant difference between the two conditions. In this context, the differences between paired observations are analyzed to test claims about their mean.
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Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample statistics to determine whether to reject H0. In this case, the claim is that the mean of the differences (μd) is less than zero, which will be tested against the significance level α.
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Step 1: Write Hypotheses

Level of Significance (α)

The level of significance, denoted as α, is the threshold for determining whether to reject the null hypothesis in hypothesis testing. It represents the probability of making a Type I error, which is rejecting a true null hypothesis. In this scenario, α is set at 0.10, indicating a 10% risk of concluding that a difference exists when there is none.
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Related Practice
Textbook Question

In Exercises 1–4, classify the two samples as independent or dependent and justify your answer.


Sample 1: The retail prices of 20 motorcycles

Sample 2: The retail prices of 20 minivans

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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

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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

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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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

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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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