A recent article in Vitality magazine reported that the mean...

A recent article in Vitality magazine reported that the mean amount of leisure time per week for American men is 40.0 hours. You believe this figure is too large and decide to conduct your own test. In a random sample of 60 men, you find that the mean is 37.8 hours of leisure per week and that the standard deviation of the sample is 12.2 hours. Can you conclude that the information in the article is untrue? Use the .05 significance level. Determine the $p$ -value and explain its meaning.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. It involves formulating a null hypothesis that represents a default position and an alternative hypothesis that represents the claim being tested, followed by using sample data to determine whether the evidence is strong enough to reject the null hypothesis.

Null and Alternative Hypotheses

The null hypothesis is a statement of no effect or no difference, and it serves as the starting assumption for statistical tests. In contrast, the alternative hypothesis represents a claim that contradicts the null, indicating the presence of a statistically significant difference or effect. Clearly defining these hypotheses is essential for any inferential analysis as they frame the test and the conclusions drawn from it.

One-Sample t-Test

A one-sample t-test is used when the population standard deviation is unknown and the sample size is relatively small, making it appropriate to compare the sample mean to a known population mean. This test relies on the t-distribution, which adjusts for extra variability caused by using the sample standard deviation as an estimate of the population standard deviation.

p-Value

The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, under the assumption that the null hypothesis is true. It quantifies the evidence against the null hypothesis; a smaller p-value indicates stronger evidence that the observed data is inconsistent with the null hypothesis, and thus supports the alternative hypothesis.

Significance Level

The significance level, often denoted by ?, is the threshold used to determine whether the p-value indicates a statistically significant result. It represents the probability of rejecting the null hypothesis when it is actually true (a Type I error). In hypothesis testing, comparing the p-value to the significance level helps decide if the null hypothesis should be rejected in favor of the alternative hypothesis.

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