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.4

Chapter 7, Problem 7.T.4

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?

Verified step by step guidance

Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The null hypothesis is H₀: p ≤ 0.80 (the proportion of U.S. adults who think mothers should have paid maternity leave is 80% or less). The alternative hypothesis is Hₐ: p > 0.80 (the proportion is greater than 80%).

Step 2: Identify the sample proportion (p̂), the sample size (n), and the significance level (α). Here, p̂ = 0.82, n = 50, and α = 0.05.

Step 3: Calculate the test statistic using the formula for a one-sample z-test for proportions: z = (p̂ - p₀) / √((p₀(1 - p₀)) / n), where p₀ is the hypothesized population proportion (0.80).

Step 4: Determine the critical value for a one-tailed test at α = 0.05. Use a z-table or statistical software to find the z-critical value corresponding to a right-tailed test.

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. Otherwise, fail to reject the null hypothesis. Interpret the result in the context of the research center's claim.

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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 80% or fewer U.S. adults support paid maternity leave, while the alternative hypothesis would assert that more than 80% do. The goal is to determine if the sample data provides sufficient evidence to reject the null hypothesis.

P-Value

The p-value is a measure that helps determine the significance of the results obtained from a hypothesis test. It represents the probability of observing the sample data, or something more extreme, assuming the null hypothesis is true. If the p-value is less than the significance level (α), typically set at 0.05, it indicates strong evidence against the null hypothesis, leading to its rejection in favor of the alternative hypothesis.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. In this context, it can provide insight into the proportion of U.S. adults who support paid maternity leave. By calculating a confidence interval for the sample proportion, researchers can assess whether the interval includes the hypothesized value of 0.80, which would affect the conclusion of the hypothesis test.

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Related Practice

Textbook Question

Writing Hypotheses: Backpack Manufacturer A backpack manufacturer claims that the mean life of its competitor’s backpacks is less than 5 years. You are asked to perform a hypothesis test to test this claim. How would you write the null and alternative hypotheses when

a. you represent the manufacturer and want to support the claim?

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Textbook Question

Interpreting a Decision In Exercises 43–48, determine whether the claim represents the null hypothesis or the alternative hypothesis. If a hypothesis test is performed, how should you interpret a decision that

a. rejects the null hypothesis?

Gas Mileage An automotive manufacturer claims that the standard deviation for the gas mileage of one of the vehicles it manufactures is 3.9 miles per gallon.

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Textbook Question

A nutrition bar manufacturer claims that the standard deviation of the number of grams of carbohydrates in a bar is 1.11 grams. A random sample of 26 bars has a standard deviation of 1.19 grams. At α=0.05, is there enough evidence to reject the manufacturer’s claim? Assume the population is normally distributed.

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Textbook Question

[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.

370

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Textbook Question

Graphical Analysis In Exercises 9–12, state whether each standardized test statistic t allows you to reject the null hypothesis. Explain.

a. t = -1.755

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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?

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