A Recent Article In Vitality Magazine Reported That The Mean

A Recent Article In Vitality Magazine Reported That The Mean Amount Of

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, with a standard deviation of 12.2 hours. Can you conclude that the information in the article is untrue? Use the 0.05 significance level. Determine the p-value and explain its meaning.

Paper For Above instruction

The question at hand involves testing the validity of a reported population mean based on sample data through hypothesis testing. Specifically, Vitality magazine claims that the average leisure time for American men is 40 hours per week. To evaluate this claim, a researcher has collected a sample of 60 men, revealing a sample mean of 37.8 hours with a standard deviation of 12.2 hours. The core purpose is to determine whether this observed difference provides sufficient statistical evidence to challenge the magazine's assertion, with the significance level set at 0.05. Additionally, the analysis requires calculating the p-value and interpreting its implications concerning the null hypothesis.

Understanding the Context and Formulating Hypotheses:

The initial step involves establishing the null hypothesis (H₀), which states that the true population mean leisure time is 40 hours, aligning with the magazine's report:

H₀: μ = 40 hours.

The alternative hypothesis (H₁) contends that the true mean leisure time is less than 40 hours, reflecting the researcher's suspicion that the figure might be inflated:

H₁: μ

This forms a one-tailed, left-sided t-test owing to the sample size being less than 30 or the population standard deviation being unknown, thus requiring the use of the t-distribution.

Calculating the Test Statistic:

The sample mean (37.8 hours), sample standard deviation (12.2 hours), and sample size (n = 60) are used to compute the test statistic. Since the standard deviation from a sample is given, and the population standard deviation remains unknown, the t-statistic is appropriate. The formula is:

t = (x̄ - μ₀) / (s/√n)

Plugging in the values:

t = (37.8 - 40) / (12.2/√60)

t = (-2.2) / (12.2/7.746)

t = (-2.2) / (1.574)

t ≈ -1.396

This negative t-value indicates that the sample mean is less than the hypothesized mean, aligning with the alternative hypothesis direction.

Determining the p-value:

The p-value represents the probability of observing a test statistic as extreme or more extreme than the calculated value under the null hypothesis.

Using degrees of freedom df = n - 1 = 59, and a t-distribution table or calculator, the p-value for t ≈ -1.396 in a one-tailed test is approximately 0.084.

Comparing p-value to Significance Level:

Since the p-value (0.084) exceeds the significance level of 0.05, we fail to reject the null hypothesis. There is insufficient evidence to conclude that the true mean leisure time is less than 40 hours per week.

Interpreting the Results:

The p-value of approximately 0.084 indicates that if the true mean leisure time were indeed 40 hours, there is an 8.4% chance of observing a sample mean as low as 37.8 hours due to sampling variability alone. Because this probability is greater than 5%, the observed data is consistent with the population mean of 40 hours, and we cannot reject the claim made by the magazine based on this sample at the 0.05 significance level.

Conclusion:

Statistically, there is not enough evidence to refute the magazine's report that American men typically have 40 hours of leisure time per week. Although the sample mean appears lower, the variation within the data and the small p-value show that this difference could be due to random sampling fluctuation. Therefore, it would be premature to conclude that the magazine's figure is untrue solely based on this analysis. Further research with larger samples or additional data might be necessary to draw more definitive conclusions.

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