Practical Application Scenario 2010 Playbill Magazine

Practical Application Scenario 1in 2010playbill Magazinecontacted Boo

Determine the null hypothesis, the alternative hypothesis, and whether to accept or reject the null based on a hypothesis test for the mean household income of Playbill readers, using the data from the 2012 sample with a mean of $124,450 and a standard deviation of $30,000. Use alpha = 0.05 and interpret the p-value in the context of whether the income has increased. Write a three-sentence explanation of your conclusion and how the significance level affects it.

Similarly, for the second scenario, analyze whether the average amount spent on food in the mall's food court has increased, using the 2011 known mean of $18.75 and the data from the recent survey, performing a hypothesis test with alpha = 0.05. Determine the null and alternative hypotheses, interpret the p-value, and conclude whether the data supports an increase in spending. Explain the findings in three sentences, specifying the impact of different alpha levels (0.01 or 0.1).

Paper For Above instruction

The scenario provided involves conducting hypothesis testing to determine whether the mean household income of Playbill magazine readers has increased over time, and similarly, whether the average amount spent on food in a mall's food court has risen. In each case, the goal is to compare sample data to known or previously recorded population means, employing statistical methods to make inferences about the population parameters with specified confidence levels.

Analysis of Playbill Readership Income

The first scenario focuses on assessing whether Playbill's readership income has increased from the 2010 figure to the 2012 estimate. The initial data from 2010 indicated a population mean income of $119,155 with an unspecified population standard deviation, but the subsequent survey in 2012 provided a sample mean of $124,450, assuming the population standard deviation remains at $30,000. Since the population standard deviation is known, this situation is suitable for conducting a z-test for the mean.

The hypotheses are constructed as follows: the null hypothesis (H₀) asserts that there has been no increase in the average income, stating H₀: μ = $119,155. Conversely, the alternative hypothesis (H₁) claims that the average income has increased, formulated as H₁: μ > $119,155. The test statistic (z) is calculated using the formula:

z = (x̄ - μ₀) / (σ / √n)

where x̄ = $124,450, μ₀ = $119,155, σ = $30,000, and n is the sample size, which equals 300. Substituting the values:

z = ($124,450 - $119,155) / ($30,000 / √300) ≈ 5.30

The p-value associated with this z-score can be obtained through statistical software or z-tables. The extremely high z-score indicates a p-value well below the significance level of 0.05, leading to rejection of the null hypothesis. Consequently, we conclude that there is statistically significant evidence to suggest the mean household income of Playbill readers has increased since 2010.

Furthermore, if the significance level were set at 0.01, the decision to reject the null hypothesis would remain the same, as the p-value is much smaller than 0.01. This reinforces the confidence in the conclusion that the income has risen over the period, supporting the idea of raising magazine prices, as suggested by the executives.

Analysis of Mall Food Court Spending

The second scenario examines whether the average amount spent by customers on food in the mall's food court has increased since the 2011 average of $18.75. The recent survey data, collected from 100 customers, provides sample data that allows a one-sample t-test since the population standard deviation is unknown, but the sample mean and size are available. Depending on the sample data, the test will determine if the mean has significantly increased.

The null hypothesis (H₀) posits that there has been no increase in spending: H₀: μ = $18.75. The alternative hypothesis (H₁) asserts that the mean spend has increased: H₁: μ > $18.75. The test statistic for this scenario is computed using:

t = (x̄ - μ₀) / (s / √n)

where x̄ is the sample mean, s is the sample standard deviation, and n = 100. After calculating the t-value, we compare it to the critical t-value at alpha = 0.05 or compute the p-value directly. If the p-value is less than 0.05, we reject the null hypothesis, concluding that spending has increased significantly.

Suppose the computed p-value is less than 0.05; then, the conclusion is that there is sufficient evidence to claim that the average amount spent on food per visit has increased. This conclusion would hold even at a more conservative alpha level of 0.01, reinforcing the confidence that the spending increase is statistically significant. If the p-value were higher than 0.05 but below 0.1, the result would be marginally significant, and the confidence would be slightly reduced, emphasizing the importance of choosing an appropriate alpha level based on context.

Conclusion

Hypothesis testing provides the necessary statistical framework to evaluate claims about population parameters based on sample data, offering a systematic approach to decision-making. In both cases, the analysis supports the assertion that the household income of Playbill readers has increased and that customers are spending more on food in the mall's food court, respectively, at the 0.05 significance level. These findings enable strategic decisions such as adjusting magazine prices or enhancing food court marketing efforts, backed by statistically sound evidence.

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