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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
All textbooksLarson 8th EditionCh. 8 - Hypothesis Testing with Two SamplesProblem 8.CR.14
Chapter 8, Problem 8.CR.14

[APPLET] The annual earnings (in dollars) for 30 randomly selected locksmiths are shown below. Assume the population is normally distributed. (Adapted from Salary.com)


48,69446,85642,91261,67271,11254,861


69,45471,84159,75169,61254,28452,166


66,36048,16465,27235,25061,12765,397


58,92558,91659,01753,07045,19969,941


69,49257,08553,82952,69268,29853,792




A researcher claims that the mean annual earnings for locksmiths is \$55,000. At α=0.05, can you reject the researcher’s claim? Interpret the decision in the context of the original claim.

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Step 1: Formulate the null and alternative hypotheses. The null hypothesis (H₀) states that the mean annual earnings for locksmiths is \(55,000 (μ = 55,000). The alternative hypothesis (H₁) states that the mean annual earnings for locksmiths is not \)55,000 (μ ≠ 55,000).
Step 2: Calculate the sample mean (x̄) and sample standard deviation (s) using the provided data. Use the formulas for mean and standard deviation: x̄ = (Σx) / n and s = sqrt((Σ(x - x̄)²) / (n - 1)), where n is the sample size.
Step 3: Determine the test statistic. Since the population is normally distributed and the sample size is relatively small (n = 30), use the t-test formula: t = (x̄ - μ) / (s / sqrt(n)), where μ is the hypothesized mean, s is the sample standard deviation, and n is the sample size.
Step 4: Find the critical t-value for a two-tailed test at α = 0.05 with degrees of freedom (df = n - 1). Use a t-distribution table or statistical software to find the critical t-value.
Step 5: Compare the calculated t-value to the critical t-value. 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. Interpret the decision in the context of the original claim: If the null hypothesis is rejected, it suggests that the mean annual earnings for locksmiths is significantly different from \$55,000. If the null hypothesis is not rejected, there is insufficient evidence to conclude that the mean annual earnings differ from \$55,000.

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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 context, the null hypothesis would state that the mean annual earnings of locksmiths is $55,000, while the alternative would suggest it is not. The process includes calculating a test statistic and comparing it to a critical value to determine whether to reject H0.
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Step 1: Write Hypotheses

Significance Level (α)

The significance level, denoted as α, is the threshold for determining whether to reject the null hypothesis. In this case, α is set at 0.05, meaning there is a 5% risk of concluding that a difference exists when there is none. If the p-value obtained from the hypothesis test is less than α, we reject the null hypothesis, indicating that the sample provides sufficient evidence against the researcher’s claim.
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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 the context of this question, constructing a confidence interval for the mean earnings of locksmiths can provide insight into whether the true mean could reasonably be $55,000. If the interval does not include this value, it supports rejecting the null hypothesis.
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Related Practice
Textbook Question

In Exercises 3–6, construct the indicated confidence interval for the population mean . Which distribution did you use to create the confidence interval?


c=0.90, x̅=8.21, σ=0.62, n=8

Textbook Question

[APPLET] The annual earnings (in dollars) for 30 randomly selected locksmiths are shown below. Assume the population is normally distributed. (Adapted from Salary.com)


48,69446,85642,91261,67271,11254,861


69,45471,84159,75169,61254,28452,166


66,36048,16465,27235,25061,12765,397


58,92558,91659,01753,07045,19969,941


69,49257,08553,82952,69268,29853,792



Construct a 95% confidence interval for the population mean annual earnings for locksmiths.

Textbook Question

In Exercises 3–6, construct the indicated confidence interval for the population mean . Which distribution did you use to create the confidence interval?


c=0.95, x̅=3.46, s=1.63, n=16

Textbook Question

The mean room rate for two adults for a random sample of 26 three-star hotels in Cincinnati has a sample standard deviation of \$31. Assume the population is normally distributed. (Adapted from Expedia)


Construct a 99% confidence interval for the population variance.

Textbook Question

An education organization claims that the mean SAT scores for male athletes and male non-athletes at a college are different. A random sample of 26 male athletes at the college has a mean SAT score of 1189 and a standard deviation of 218. A random sample of 18 male non-athletes at the college has a mean SAT score of 1376 and a standard deviation of 186. At α=0.05, can you support the organization’s claim? Interpret the decision in the context of the original claim. Assume the populations are normally distributed and the population variances are equal.

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Textbook Question

In a survey of 4860 U.S. adults, 77% said they would date or have already dated someone whose religion was different from theirs. (Source: Pew Research Center)


Construct a 95% confidence interval for the proportion of U.S. adults who say they would date or have already dated someone whose religion was different from theirs.