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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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.

Recommended video:

Guided course

06:34

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

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

Sample 1: The heights of 37 children

Sample 2: The heights of the same 37 children after 1 year

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 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)

Textbook Question

Claim: μd<0; α=0.10.

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