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Chapter 10: Problem 16

In Problems \(15-22,(a)\) determine the null and alternative hypotheses, (b) explain what it would mean to make a Type I error, and (c) explain what it would mean to make a Type II error. Historically, the time to order and deliver a pizza at Jimbo's pizza was 48 minutes. Jim, the owner, implements a new system for ordering and delivering pizzas that he believes will reduce the time required to get a pizza to his customers.

Short Answer

Expert verified
The null hypothesis (H_0) is that the mean delivery time is 48 minutes, and the alternative hypothesis (H_a) is that it is less than 48 minutes. A Type I error would be concluding that the new system reduces delivery time when it does not, and a Type II error would be concluding that the new system does not reduce delivery time when it does.

Step by step solution

01

- Determine the Null and Alternative Hypotheses

Jim believes the new system will reduce the time to order and deliver a pizza. Thus, the null hypothesis (H_0) is that the mean time for delivery is equal to 48 minutes, and the alternative hypothesis (H_a) is that the mean time for delivery is less than 48 minutes. \[H_0: \text{The mean delivery time} = 48 \text{ minutes} \] \[H_a: \text{The mean delivery time} < 48 \text{ minutes} \]
02

- Explain Type I Error

A Type I error occurs when we reject the null hypothesis when it is actually true. In this context, this means concluding that the new system reduces the delivery time when in fact, it does not (the actual mean delivery time is still 48 minutes).
03

- Explain Type II Error

A Type II error occurs when we fail to reject the null hypothesis when it is false. Here, this means we conclude that the new system does not reduce the delivery time when in fact, it does (the actual mean delivery time is less than 48 minutes).

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Key Concepts

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

null hypothesis
The null hypothesis, denoted as \(H_0\), is a fundamental aspect of hypothesis testing. In hypothesis testing, the null hypothesis represents a statement of no effect or no difference. It is a default position that indicates no change or no impact. In this problem, the null hypothesis is that the mean delivery time for pizza remains unchanged at 48 minutes. Formally, the null hypothesis is stated as: \[H_0: \text{The mean delivery time} = 48 \text{ minutes} \] The null hypothesis is what we aim to test against using statistical methods.
alternative hypothesis
The alternative hypothesis, symbolized as \(H_a\), is the statement that we want to provide evidence for in hypothesis testing. It represents a new effect or difference that contradicts the null hypothesis. In this scenario, the alternative hypothesis is that Jim's new system will reduce the mean delivery time to less than 48 minutes. Formally, it is stated as: \[H_a: \text{The mean delivery time} < 48 \text{ minutes} \] The alternative hypothesis is essential because it sets the direction of the test and determines the type of statistical test to be used.
Type I error
A Type I error, also known as a 'false positive,' occurs when we incorrectly reject the null hypothesis when it is actually true. In the context of this problem, a Type I error would mean concluding that Jim's new system has successfully reduced the delivery time when, in fact, it has not. It implies that the actual mean delivery time is still 48 minutes, but our test results mistakenly suggest a reduction. The consequence of a Type I error is that we might implement a new system based on falsely optimistic results, leading to no real improvement and potential costs.
Type II error
A Type II error, known as a 'false negative,' happens when we fail to reject the null hypothesis when it is actually false. In this problem, a Type II error would mean concluding that the new system does not reduce the delivery time when it actually does. This occurs if the actual mean delivery time is less than 48 minutes, but our test fails to detect this improvement. The consequence of a Type II error is that we might discard a genuinely effective new system, missing out on potential efficiency gains and better customer service.
mean delivery time
The mean delivery time is an important metric in this problem. It represents the average time it takes for Jimbo's pizza to deliver a pizza to a customer. Historically, this time has been 48 minutes. The goal of implementing a new system is to reduce this mean delivery time. Hence, the mean delivery time forms the basis of both the null and alternative hypotheses. Monitoring and calculating the mean delivery time is crucial for Jim to assess whether the new system is effective and meets customer expectations better than the old system.

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Most popular questions from this chapter

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