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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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1
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

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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.

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Textbook Question

In Exercises 7–10, the statement represents a claim. Write its complement and state which is Ho and which is Ha.


σ=0.63

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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

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Textbook Question

Take this quiz as you would take a quiz in class. After you are done, check your work against the answers given in the back of the book.For each exercise, perform the steps below.


a. Identify the claim and state Ho and Ha


b. Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed, and whether to use a z-test or a t-test. Explain your reasoning.


c. Find the critical value(s) and identify the rejection region(s).


d. Find the appropriate standardized test statistic.


e. Decide whether to reject or fail to reject the null hypothesis.


f. Interpret the decision in the context of the original claim.



[APPLET] The table shows the credit scores for 12 randomly selected adults who are considered high-risk borrowers before and two years after they attend a personal finance seminar. At α=0.01, is there enough evidence to support the claim that the personal finance seminar helps adults increase their credit scores? Assume the populations are normally distributed.

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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.

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