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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.3
Chapter 7, Problem 7.Q.3

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.

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Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The null hypothesis (H₀) states that the mean earnings for full-time workers ages 18 to 24 with a bachelor’s degree is \$52,133. The alternative hypothesis (Hₐ) states that the mean earnings are not equal to \$52,133.
Step 2: Determine the appropriate test to use. Since the population standard deviation is not provided and the sample size is small (n = 15), use a t-test for the mean. The test statistic formula is t = (x̄ - μ) / (s / √n), where x̄ is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size.
Step 3: Calculate the degrees of freedom (df) for the t-test. The degrees of freedom for a one-sample t-test is df = n - 1. In this case, df = 15 - 1 = 14.
Step 4: Determine the critical t-value for a two-tailed test at α = 0.05 with df = 14. Use a t-distribution table or statistical software to find the critical t-value. This value will help define the rejection region for the null hypothesis.
Step 5: Compute the test statistic using the formula t = (x̄ - μ) / (s / √n). Compare the calculated t-value to the critical t-value. If the absolute value of the test statistic 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 support. In this case, the null hypothesis would state that the mean earnings of the sample do not differ significantly from the reported mean of $52,133.
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P-value

The p-value is a measure that helps determine the strength of the evidence against the null hypothesis. It represents the probability of obtaining a sample mean as extreme as, or more extreme than, the observed sample mean, assuming the null hypothesis is true. If the p-value is less than the significance level (α), we reject the null hypothesis, indicating that the sample provides sufficient evidence to support the alternative hypothesis.
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Step 3: Get P-Value

Standard Deviation and Sampling Distribution

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In hypothesis testing, the standard deviation of the sample is used to calculate the standard error, which describes how much the sample mean is expected to vary from the population mean. Understanding the sampling distribution is crucial, as it allows us to determine how likely our sample mean is under the null hypothesis, facilitating the calculation of the test statistic and p-value.
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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

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?

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?


Textbook Question

n Exercises 1–6, the statement represents a claim. Write its complement and state which is H0 and which is Ha.


p ≥ 0.64