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 7.1.60

Chapter 7, Problem 7.1.60

Graphical Analysis In Exercises 57–60, you are given a null hypothesis and three confidence intervals that represent three samplings. Determine whether each confidence interval indicates that you should reject H0. Explain your reasoning.

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Step 1: Understand the null hypothesis (H0). The null hypothesis states that p ≥ 0.73, where p represents the population proportion. To reject H0, the confidence interval must not include values greater than or equal to 0.73.

Step 2: Analyze confidence interval (a). The interval is 0.73 < p < 0.75. Since this interval starts at 0.73 and includes values greater than 0.73, it does not provide evidence to reject H0.

Step 3: Analyze confidence interval (b). The interval is 0.715 < p < 0.725. This interval does not include 0.73 or any values greater than 0.73, which suggests that H0 can be rejected based on this sampling.

Step 4: Analyze confidence interval (c). The interval is 0.695 < p < 0.745. This interval includes 0.73, so it does not provide evidence to reject H0.

Step 5: Summarize the findings. Confidence interval (b) indicates rejection of H0 because it excludes 0.73 and values greater than 0.73, while intervals (a) and (c) do not provide evidence to reject H0 as they include values greater than or equal to 0.73.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Null Hypothesis (H0)

The null hypothesis (H0) is a statement that there is no effect or no difference, and it serves as the default assumption in statistical testing. In this context, H0 states that the population proportion (p) is greater than or equal to 0.73. To determine whether to reject H0, we compare the confidence intervals from the samples to this hypothesis.

Confidence Interval

A confidence interval is a range of values derived from sample data that is likely to contain the true population parameter. Each confidence interval provides an estimate of where the true proportion (p) lies. If a confidence interval does not include the value specified in the null hypothesis, it suggests that we may reject H0.

Statistical Significance

Statistical significance refers to the likelihood that a result or relationship is caused by something other than mere random chance. In this scenario, if the confidence intervals for the sample proportions (p̂) do not overlap with the null hypothesis value (0.73), it indicates that the results are statistically significant, leading to a potential rejection of H0.

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Textbook Question

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