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 7.3.28

Chapter 7, Problem 7.3.28

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?

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Formulate the null hypothesis (H₀) and the alternative hypothesis (Hₐ). H₀: μ = 11.5 (the mean dive duration is 11.5 minutes), Hₐ: μ ≠ 11.5 (the mean dive duration is not 11.5 minutes). This is a two-tailed test.

Determine the test statistic to use. Since the population standard deviation is unknown and the sample size is greater than 30, use the t-test. The formula for the t-test statistic is: t = (x̄ - μ₀) / (s / √n), where x̄ is the sample mean, μ₀ is the hypothesized mean, s is the sample standard deviation, and n is the sample size.

Substitute the given values into the formula. Here, x̄ = 12.2, μ₀ = 11.5, s = 2.2, and n = 34. Compute the standard error (SE) first: SE = s / √n.

Determine the critical t-value for a two-tailed test at α = 0.10 with degrees of freedom (df) = n - 1 = 34 - 1 = 33. Use a t-distribution table or statistical software to find the critical t-value.

Compare the calculated t-test statistic to the critical t-value. If the absolute value of the t-test statistic is greater than 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) and an alternative hypothesis (H1). In this case, the null hypothesis would state that the mean dive duration is 11.5 minutes, while the alternative would suggest it is different. The goal is to determine if the sample data provides sufficient evidence to reject the null hypothesis.

P-value

The p-value is a measure that helps determine the significance of the results in hypothesis testing. It represents the probability of obtaining a sample mean at least as extreme as the one observed, assuming the null hypothesis is true. If the p-value is less than the significance level (α), in this case, 0.10, it indicates strong evidence against the null hypothesis, leading to its rejection.

Confidence Intervals

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence. In this scenario, constructing a confidence interval for the mean dive duration can provide insight into whether the true mean could be 11.5 minutes. If the interval does not include this value, it supports the rejection of the null hypothesis.

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Related Practice

Textbook Question

Explain how to use a t-test to test a hypothesized mean mu when sigma is unknown. What assumptions are necessary?

Textbook Question

Writing In a right-tailed test where P < alpha, does the standardized test statistic lie to the left or the right of the critical value? Explain your reasoning.

Textbook Question

In Exercises 3–8, find the critical value(s) and rejection region(s) for the type of t-test with level of significance alpha and sample size n.

Right-tailed test, α=0.01, n=31

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

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?