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.T.7

Chapter 7, Problem 7.T.7

[APPLET] A researcher claims that the mean age of the residents of a small town is more than 38 years. The ages (in years) of a random sample of 30 residents are listed below. At α=0.10, is there enough evidence to support the researcher’s claim? Assume the population standard deviation is 9 years.

Verified step by step guidance

Step 1: Formulate the null and alternative hypotheses. The null hypothesis (H₀) is that the mean age of the residents is 38 years (μ = 38). The alternative hypothesis (H₁) is that the mean age of the residents is greater than 38 years (μ > 38).

Step 2: Calculate the sample mean (x̄). Add all the ages provided in the sample and divide by the total number of residents (n = 30). Use the formula:

x̄ = \frac{\sum x}{n}

Step 3: Compute the test statistic using the z-test formula for a population mean. The formula is:

z = \frac{x̄ - μ}{\frac{σ}{\sqrt{n}}}

, where μ = 38, σ = 9, and n = 30.

Step 4: Determine the critical value for α = 0.10 in a one-tailed z-test. Look up the z-value corresponding to a significance level of 0.10 in a z-table. This critical value will help decide whether to reject or fail to reject the null hypothesis.

Step 5: Compare the calculated z-test statistic to the critical value. If the test statistic is greater than the critical value, reject the null hypothesis and conclude that there is enough evidence to support the researcher’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 states that the mean age is 38 years or less, while the alternative hypothesis posits that it is greater than 38 years. The goal is to determine if there is enough evidence to reject the null hypothesis at a specified significance level.

Significance Level (α)

The significance level, denoted as α, is the probability of rejecting the null hypothesis when it is actually true, also known as a Type I error. In this scenario, α is set at 0.10, meaning there is a 10% risk of concluding that the mean age is greater than 38 years when it is not. This threshold helps researchers decide how strong the evidence must be to support the alternative hypothesis.

Z-Test for Means

A Z-test for means is a statistical test used to determine if there is a significant difference between the sample mean and a known population mean when the population standard deviation is known. In this case, the sample of 30 residents' ages will be analyzed using the Z-test to compare the sample mean against the hypothesized mean of 38 years, utilizing the provided population standard deviation of 9 years to calculate the Z-score.

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Related Practice

Textbook Question

In Exercises 11 and 12, find the P-value for the hypothesis test with the standardized test statistic z. Decide whether to reject H0 for the level of significance α.

Two-tailed test, z = 2.57, α = 0.10

Textbook Question

You want your test to support a positive claim about your college, not just fail to reject one. Should you state your claim so that the null hypothesis contains the claim or the alternate hypothesis contains the claim? Explain.

Textbook Question

When you reject a true claim with a level of significance that is virtually zero, what can you infer about the randomness of your sampling process?

Textbook Question

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

σ > 1.9

Textbook Question

A research center claims that more than 80% of U.S. adults think that mothers should have paid maternity leave. In a random sample of 50 U.S. adults, 82% think that mothers should have paid maternity leave. At α=0.05, is there enough evidence to support the center’s claim?

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Textbook Question

A travel analyst says that the mean price of a meal for a family of 4 in a resort restaurant is at most \$100. A random sample of 33 meal prices for families of 4 has a mean of \$110 and a standard deviation of \$19. At α=0.01, is there enough evidence to reject the analyst’s claim?