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Pizza Hut, after test-marketing a new product called the Bigfoot Pizza, concluded that introduction of the Bigfoot nationwide would increase its sales by more than \(14 \%\) (USA Today, April 2, 1993). This conclusion was based on recording sales information for a random sample of Pizza Hut restaurants selected for the marketing trial. With \(\mu\) denoting the mean percentage increase in sales for all Pizza Hut restaurants, consider using the sample data to decide between \(H_{0}: \mu=14\) and \(H_{a}: \mu>14\). a. Is Pizza Hut's conclusion consistent with a decision to reject \(H_{0}\) or to fail to reject \(H_{0}\) ? b. If Pizza Hut is incorrect in its conclusion, is the company making a Type I or a Type II error?

Step by step solution

01

Understanding Pizza Hut's Conclusion

By stating that they expect an increase of more than 14%, they are essentially siding with the alternative hypothesis \(H_{a}: \mu>14\). This indicates their rejection of the null hypothesis \(H_{0}: \mu=14\).

02

Categorizing the Possible Error

Since they have rejected the null hypothesis (in the anticipation of sales increase being more than 14%), if Pizza Hut is wrong - in essence, if the actual increase turns out to be 14% or less, they would be making a Type I error. A Type I error occurs when the null hypothesis is true, but it is rejected based on the sample data.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis

In hypothesis testing in statistics, the null hypothesis (\(H_0\)) serves as a starting point for evaluation. It is a statement used for statistical testing that assumes no effect or no difference, essentially suggesting that any observed variations are due to chance rather than a specific intervention or action. For instance, in the case of Pizza Hut's Bigfoot Pizza, the null hypothesis posits that the mean percentage increase in sales is exactly 14%, symbolized as \(H_{0}: \mu=14\). This assertion provides a benchmark for comparing the effectiveness of the new product introduction.

Alternative Hypothesis

The alternative hypothesis (\(H_a\) or \(H_1\)) presents a contrasting claim to the null hypothesis and typically reflects the research or experiment hypothesis that there is a significant effect or difference. For the Pizza Hut example, the alternative hypothesis is that the new Bigfoot Pizza will increase sales by more than 14% across all restaurants, denoted as \(H_{a}: \mu>14\). If the sample data provides enough evidence, the null hypothesis can be rejected in favor of the alternative hypothesis.

Type I Error

A Type I error occurs when the null hypothesis is incorrectly rejected when it is in fact true. The consequence of such an error is believing that a difference or effect exists when it does not. In statistical terms, it's a 'false positive.' For our Pizza Hut scenario, if the company concludes that sales will increase by more than 14% based on their sample and that's not truly reflective of all their restaurants, they are committing a Type I error—their optimism is not justified by the larger reality.

Mean Percentage Increase

The mean percentage increase is a measure of the average change, expressed in percentages, over a given period. When Pizza Hut claims a 14% increase in sales, it is saying, on average, there is a 14% rise in the sales figures. In hypothesis testing, the mean percentage increase is pitted against the claim in the null hypothesis (\(H_0\)) to understand if the observed sample data shows a significant increase or not.

Statistical Significance

Statistical significance refers to the likelihood that the result obtained in a study or experiment is not due to random chance. It helps researchers determine whether to reject the null hypothesis. The level of statistical significance is often set at a particular threshold (e.g., a p-value of 0.05) before conducting the test. For Pizza Hut, achieving statistical significance would mean that the evidence from their sample is strong enough to conclude that the sales increase is indeed greater than 14% and not just a random variation in their data.