Ch. 8 - Hypothesis Testing

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 8 - Hypothesis Testing

Problem 8.CR.4

Chapter 8, Problem 8.CR.4

Hypothesis Test for Lightning Deaths Refer to the sample data given in Cumulative Review Exercise 1 and consider those data to be a random sample of annual lightning deaths from recent years. Use those data with a 0.01 significance level to test the claim that the mean number of annual lightning deaths is less than the mean of 72.6 deaths from the 1980s. If the mean is now lower than in the past, identify one of the several factors that could explain the decline.

Verified step by step guidance

Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The null hypothesis is H₀: μ = 72.6, which states that the mean number of annual lightning deaths is equal to 72.6. The alternative hypothesis is Hₐ: μ < 72.6, which states that the mean number of annual lightning deaths is less than 72.6.

Step 2: Identify the significance level (α). The problem specifies a significance level of 0.01, which means there is a 1% risk of rejecting the null hypothesis when it is actually true.

Step 3: Calculate the test statistic. Use the formula for the t-test statistic:

\frac{x̄ − μ}{\frac{s}{\sqrt{n}}}

, where x̄ is the sample mean, μ is the population mean (72.6), s is the sample standard deviation, and n is the sample size.

Step 4: Determine the critical value. Using a t-distribution table or statistical software, find the critical t-value for a one-tailed test with a significance level of 0.01 and degrees of freedom (df = n - 1).

Step 5: Compare the test statistic to the critical value. If the test statistic is less than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. If the null hypothesis is rejected, consider factors such as improved weather forecasting, better public awareness, or advancements in lightning safety measures as potential explanations for the decline in lightning deaths.

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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), then using sample data to determine whether to reject H0 in favor of H1. In this case, the null hypothesis would state that the mean number of annual lightning deaths is equal to 72.6, while the alternative hypothesis would claim it is less than 72.6.

Significance Level

The significance level, denoted as alpha (α), is the threshold for determining whether the results of a hypothesis test are statistically significant. A significance level of 0.01 indicates that there is a 1% risk of concluding that a difference exists when there is none (Type I error). In this scenario, it means that the test will only reject the null hypothesis if the evidence against it is very strong, reflecting a high standard for evidence.

Mean Comparison

Mean comparison involves evaluating the average values of two or more groups to determine if they differ significantly. In this context, the mean number of annual lightning deaths from the recent sample is compared to the historical mean of 72.6 deaths. If the sample mean is significantly lower, it may suggest a decline in lightning deaths, prompting further investigation into potential factors contributing to this change.

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Textbook Question

Perception and Reality In a presidential election, 308 out of 611 voters surveyed said that they voted for the candidate who won (based on data from ICR Survey Research Group). Use a 0.05 significance level to test the claim that among all voters, the percentage who believe that they voted for the winning candidate is equal to 43%, which is the actual percentage of votes for the winning candidate. What does the result suggest about voter perceptions?

Textbook Question

Robust Explain what is meant by the statements that the t test for a claim about μ is robust, but the (chi)^2 test for a claim about σ is not robust.

Textbook Question

Discarded Plastic Find the test statistic used for the hypothesis test described in Exercise 1.

Textbook Question

Job Search A Gallup poll of 195,600 employees showed that 51% of them were actively searching for new jobs. Use a 0.01 significance level to test the claim that the majority of employees are searching for new jobs

Textbook Question

Lightning Deaths The graph in Cumulative Review Exercise 5 was created by using data consisting of 242 male deaths from lightning strikes and 64 female deaths from lightning strikes. Assume that these data are randomly selected lightning deaths and proceed to test the claim that the proportion of male deaths is greater than . Use a 0.01 significance level. Any explanation for the result?

Textbook Question

Type I Error and Type II Error

a. In general, what is a type I error? In general, what is a type II error?