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The Pizza Shack has been experimenting with different recipes for their pizza crust, thinking they might replace their current recipe. They are planning to sample pizza made with the new crust. Before sampling, a strategy is needed so that after the tasting results are in, Pizza Shack will know how to interpret their customers' preferences. The decision is not being taken lightly because there is much to be gained or lost depending on whether or not the decision is a popular one. A one-tailed hypothesis test of \(p=P(\text { prefer new crust })=0.50\) is being planned. a. If \(H_{a}: p>0.50\) is used, explain the meaning of the four possible outcomes and their resulting actions. b. If \(H_{a}: p<0.50\) is used, explain the meaning of the four possible outcomes and their resulting actions. c. Which alternative hypothesis do you recommend be used, \(p>0.5\) or \(p<0.5 ?\) Explain.

Step by step solution

01

Explain the Outcomes for \(H_a : p > 0.50\)

This alternative hypothesis is that more than half of the customers prefer the new crust. The four possible outcomes are: 1. Reject \(H_0\) when \(H_0\) is true: In this case, the pizzeria would mistakenly think that the new crust is preferred when it's not. This would be a type I error. 2. Fail to reject \(H_0\) when \(H_0\) is true: The pizzeria would correctly identify that the new crust is not preferred. 3. Reject \(H_0\) when \(H_0\) is false: The pizzeria would correctly identify that the new crust is preferred. 4. Fail to reject \(H_0\) when \(H_0\) is false: In this case, the pizzeria would mistakenly think that the new crust is not preferred when it is. This would be a type II error.

02

Explain the Outcomes for \(H_a : p < 0.50\)

This alternative hypothesis states that less than half of the customers prefer the new crust recipe. The four possible outcomes are: 1. Reject \(H_0\) when \(H_0\) is true: The pizzeria would mistakenly think the new crust isn't preferred when it is. This would be a type I error. 2. Fail to reject \(H_0\) when \(H_0\) is true: The pizzeria would correctly identify that the new crust is preferred. 3. Reject \(H_0\) when \(H_0\) is false: The pizzeria would correctly identify that the new crust isn't preferred. 4. Fail to reject \(H_0\) when \(H_0\) is false: The pizzeria would mistakenly think the new crust is preferred when it's not. This would be a type II error.

03

Recommend an alternative hypothesis

The pizzeria might be more interested in knowing if the new crust is significantly better than the old one. Therefore, it would be more beneficial to use \(H_a: p > 0.50\), meaning that more than half of the customers prefer the new crust. If the test result fails to reject the null hypothesis, they can retain the old crust. But if the null hypothesis is rejected, it suggests that the new crust might be more popular and could lead to a potential increase in customers and profits.

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Key Concepts

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

Type I Error

In the context of hypothesis testing, a Type I error occurs when we reject the null hypothesis when it is actually true. Imagine Pizza Shack conducting a taste test to see if their new crust is preferred by more than half of their customers. Here, the null hypothesis (\(H_0\)) would state that exactly, or less than, half of the customers prefer the new crust. If the test results lead Pizza Shack to believe that the new crust is preferred (rejecting \(H_0\)), but in reality, it’s not preferred, they have committed a Type I error.

• This is like making a false positive conclusion, where Pizza Shack might mistakenly introduce a less popular crust based on incorrect data interpretation.
• Such an error could result in loss of customer satisfaction and loyalty, potentially affecting sales negatively.

Addressing Type I errors involves setting a significance level (α), such as 0.05, which indicates a 5% risk that Pizza Shack would mistakenly reject the null hypothesis.

Type II Error

Type II errors occur in hypothesis testing when we fail to reject the null hypothesis even though it is actually false. In the Pizza Shack scenario, if the alternative hypothesis suggests more than half the customers do prefer the new crust (\(H_a: p > 0.50\)), failing to reject \(H_0\) when it's false means missing the opportunity to adopt a crust that is actually preferred.

• This is known as a false negative, where the data analysis fails to detect a true preference for the new crust.
• The consequences can include missed opportunities for customer satisfaction improvements and potential sales increases.

The probability of a Type II error is denoted by \(\beta\), and a good test design minimizes this probability while balancing the risk of Type I errors.

Alternative Hypothesis

The alternative hypothesis represents what Pizza Shack is trying to prove and stands opposite the null hypothesis. For their taste test, the alternative hypotheses could be:

• \(H_a: p > 0.50\) - More than half of the customers prefer the new crust.
• \(H_a: p < 0.50\) - Less than half of the customers prefer the new crust.

Choosing the correct alternative hypothesis depends on Pizza Shack's objectives. If they want to confirm a superior preference for the new crust, \(H_a: p > 0.50\) would be appropriate. By structuring their test this way, Pizza Shack will gather data showing if a significant portion of customers favors the new recipe, which could help make informed business decisions. Selecting the alternative hypothesis well is crucial for the integrity and applicability of the test outcomes.

Null Hypothesis

The null hypothesis (\(H_0\)) serves as the default or baseline assumption for the hypothesis test. In Pizza Shack's scenario, the null hypothesis might state that no more than half of the customers prefer the new crust (\(H_0: p \leq 0.50\)).

• It is essentially a claim that the new crust is not significantly better than what is currently offered.
• The purpose of the hypothesis test is to evaluate whether there is enough evidence to reject this default assumption.

Maintaining a clear null hypothesis helps Pizza Shack to interpret their test results objectively. It creates a benchmark against which the preference for the new crust is tested. Rejecting the null hypothesis suggests a statistically significant preference exists, guiding Pizza Shack to potentially adopt the new recipe based on valid statistical analysis.