Ch. 7 - Hypothesis Testing with One Sample

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 7 - Hypothesis Testing with One Sample

Problem 7.3.29

Chapter 7, Problem 7.3.29

Faculty Classroom Hours The dean of a university estimates that the mean number of classroom hours per week for full-time faculty is 11.0. As a member of the student council, you want to test this claim. A random sample of the number of classroom hours for eight full-time faculty for one week is shown in the table at the left. At α=0.01, can you reject the dean’s claim?

Verified step by step guidance

Step 1: Formulate the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: μ = 11.0, which states that the mean number of classroom hours is 11.0. The alternative hypothesis is H₁: μ ≠ 11.0, which states that the mean number of classroom hours is not 11.0.

Step 2: Calculate the sample mean (x̄) using the provided data. Add all the classroom hours from the table and divide by the number of observations (n = 8). Use the formula:

x̄ = \frac{\sum x}{n}

Step 3: Calculate the sample standard deviation (s) using the formula:

s = \sqrt{\frac{\sum (x - x̄)^{2}}{n - 1}}

. This measures the variability of the classroom hours.

Step 4: Compute the test statistic (t) using the formula:

t = \frac{x̄ - μ}{\frac{s}{\sqrt{n}}}

, where μ is the hypothesized mean (11.0), x̄ is the sample mean, s is the sample standard deviation, and n is the sample size.

Step 5: Compare the calculated t-value to the critical t-value at α = 0.01 for a two-tailed test with degrees of freedom (df = n - 1). If the absolute value of the calculated t-value exceeds the critical t-value, reject the null hypothesis; 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.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) that represents a default position, and an alternative hypothesis (H1) that represents what we aim to prove. In this case, the null hypothesis would state that the mean classroom hours are equal to 11.0, while the alternative would suggest they are not. The goal is to determine whether the sample data provides sufficient evidence to reject the null hypothesis.

Significance Level (α)

The significance level, denoted as α, is the threshold for determining whether to reject the null hypothesis. It represents the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. In this scenario, α is set at 0.01, indicating a 1% risk of concluding that the mean classroom hours differ from 11.0 when they actually do not. A lower α value means a stricter criterion for evidence against the null hypothesis.

Sample Mean and Standard Deviation

The sample mean is the average of the observed values in a sample, providing an estimate of the population mean. The standard deviation measures the dispersion of the sample data around the mean. In this context, calculating the sample mean of the classroom hours will help assess whether it significantly differs from the dean's claim of 11.0 hours. The standard deviation will also be crucial for determining the variability of the sample and for conducting the hypothesis test.

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Related Practice

Textbook Question

Explain how to find critical values for a t-distribution.

Textbook Question

In Exercises 29–34, find the critical value(s) and rejection region(s) for the type of t-test with level of significance α and sample size n.

Right-tailed test, α=0.02, n=63

Textbook Question

Dive Duration An oceanographer claims that the mean dive duration of a North Atlantic right whale is 11.5 minutes. A random sample of 34 dive durations has a mean of 12.2 minutes and a standard deviation of 2.2 minutes. Is there enough evidence to reject the claim at α=0.10?

Textbook Question

Hypothesis Testing Using Rejection Regions In Exercises 7–12, (a) identify the claim and state H0 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.

Working Students An education researcher claims that 65% of full-time college students work year-round. In a random sample of 105 college students, 66 say they work year-round. At α=0.10, is there enough evidence to reject the researcher’s claim?

Textbook Question

Hypothesis Testing Using a P-Value In Exercises 33–38,

a. identify the claim and state and .

b. find the standardized test statistic z.

c. find the corresponding P-value.

d. decide whether to reject or fail to reject the null hypothesis.

e. interpret the decision in the context of the original claim.

Sprinkler Systems A manufacturer of sprinkler systems designed for fire protection claims that the average activating temperature is at least 135°F. To test this claim, you randomly select a sample of 32 systems and find the mean activation temperature to be 133°F. Assume the population standard deviation is 3.3°F. At alpha=0.10, do you have enough evidence to reject the manufacturer’s claim?

Textbook Question

True or False? In Exercises 5–10, determine whether the statement is true or false. If it is false, rewrite it as a true statement.

A statistical hypothesis is a statement about a sample.