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.9

Chapter 8, Problem 8.R.9

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?

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 mean sodium content of sandwiches at Restaurant A is less than the mean sodium content of sandwiches at Restaurant B. This translates to Ha: μA < μB (alternative hypothesis). The null hypothesis is the opposite, Ho: μA ≥ μB.

Step 2: Determine the critical value(s) and rejection region(s). Since the claim involves a 'less than' comparison, this is a one-tailed test. Using the significance level α = 0.05, find the critical z-value from the standard normal distribution table. The rejection region will be z < critical value.

Step 3: Calculate the standardized test statistic z. Use the formula for comparing two population means:

z = \frac{(x_{A} - x_{B})}{\sqrt{{σ_{A}}^{2} n_{A} + {σ_{B}}^{2} n_{B}}}

. Plug in the values: μA = 670, μB = 690, σA = 20, σB = 30, nA = 22, nB = 28.

Step 4: Compare the calculated z-value to the critical value. If the calculated z-value falls in the rejection region (z < critical value), reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

Step 5: Interpret the decision in the context of the original claim. If the null hypothesis is rejected, there is enough evidence to support the claim that the mean sodium content of sandwiches at Restaurant A is less than that of Restaurant B. If the null hypothesis is not rejected, there is not enough evidence to support the claim.

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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 population parameters based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents a statement of no effect or no difference, and the alternative hypothesis (Ha), which represents the claim being tested. In this case, the researcher claims that the mean sodium content at Restaurant A is less than that at Restaurant B.

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 indicates the probability of making a Type I error. The rejection region is the range of values for the test statistic that leads to the rejection of H0. For a one-tailed test at α=0.05, the critical value helps identify whether the observed test statistic falls into this region.

Standardized Test Statistic (z)

The standardized test statistic, often denoted as z, measures how many standard deviations an observed sample mean is from the hypothesized population mean under the null hypothesis. It is calculated using the formula z = (X̄ - μ) / (σ/√n), where X̄ is the sample mean, μ is the population mean under H0, σ is the population standard deviation, and n is the sample size. This statistic is crucial for determining whether to reject or fail to reject the null hypothesis based on the comparison with the critical value.

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Step 2: Calculate Test Statistic

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 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

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

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. (Adapted from National Center for Education Statistics)

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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

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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