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Ch. 7 - Hypothesis Testing with One Sample
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. 7 - Hypothesis Testing with One SampleProblem 7.Q.2
Chapter 7, Problem 7.Q.2

A travel analyst claims the mean daily base price for renting a full-size or less expensive vehicle in Vancouver, British Columbia, is more than \(86. You want to test this claim. In a random sample of 40 full-size or less expensive vehicles available to rent in Vancouver, British Columbia, the mean daily base price is \)93.23. Assume the population standard deviation is \$28.90. At α=0.10, do you have enough evidence to support the analyst’s claim?

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Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: μ ≤ 86, which states that the mean daily base price is less than or equal to \(86. The alternative hypothesis is H₁: μ > 86, which states that the mean daily base price is greater than \)86. This is a one-tailed test.
Step 2: Identify the test statistic to use. Since the population standard deviation (σ) is known, use the z-test for the hypothesis test. The formula for the z-test statistic is: z=x̄−μσn, where x̄ is the sample mean, μ is the hypothesized population mean, σ is the population standard deviation, and n is the sample size.
Step 3: Substitute the given values into the z-test formula. Here, x̄ = 93.23, μ = 86, σ = 28.90, and n = 40. Calculate the standard error of the mean (SE) using the formula: SE=σn. Then, compute the z-test statistic using the formula from Step 2.
Step 4: Determine the critical value for the z-test at a significance level of α = 0.10. For a one-tailed test, find the z-value that corresponds to an area of 0.10 in the upper tail of the standard normal distribution. Use a z-table or statistical software to find this critical value.
Step 5: Compare the calculated z-test statistic to the critical value. If the z-test statistic is greater than the critical value, reject the null hypothesis (H₀) and conclude that there is enough evidence to support the analyst’s claim. 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 daily base price is less than or equal to $86, while the alternative hypothesis would assert that it is greater than $86. The goal is to determine whether there is enough evidence to reject the null hypothesis in favor of the alternative.
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Step 1: Write Hypotheses

P-Value

The p-value is a measure that helps determine the significance of the results obtained from a statistical test. It represents the probability of observing the sample data, or something more extreme, if the null hypothesis is true. In this scenario, if the p-value is less than the significance level (α = 0.10), it indicates strong evidence against the null hypothesis, suggesting that the mean daily base price is indeed greater than $86.
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Step 3: Get P-Value

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 context, constructing a confidence interval for the mean daily base price can provide insight into the range within which the true mean lies. If the entire interval is above $86, it would support the analyst's claim, while an interval that includes $86 would not provide sufficient evidence.
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Related Practice
Textbook Question

A hat company claims that the mean hat size for a male is at least 7.25. A random sample of 12 hat sizes has a mean of 7.15. At α=0.01, can you reject the company’s claim? Assume the population is normally distributed and the population standard deviation is 0.27.

Textbook Question

In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim.

Claim: p ≥0.48, α=0.08. Sample statistics: p_hat = 0.40, n=90

Textbook Question

A nonprofit consumer organization says that less than 25% of the televisions the organization rated in a recent year have an overall score of 70 or more. In a random sample of 35 televisions the organization rated in a recent year, 23% have an overall score of 70 or more. At α=0.05, can you support the organization’s claim?

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Textbook Question

What are the two types of hypotheses used in a hypothesis test? How are they related?

Textbook Question

A government agency reports that the mean amount of earnings for full-time workers ages 18 to 24 with a bachelor’s degree in a recent year is \(52,133. In a random sample of 15 full-time workers ages 18 to 24 with a bachelor’s degree, the mean amount of earnings is \)48,400 and the standard deviation is \$6679. At α=0.05, is there enough evidence to reject the claim? Assume the population is normally distributed.

Textbook Question

[APPLET] A weight loss program claims that program participants have a mean weight loss of at least 10.5 pounds after 1 month. The weight losses after 1 month (in pounds) of a random sample of 40 program participants are listed below. At α=0.01, is there enough evidence to reject the program’s claim?