Midterm Exam 2
Question 8 of 8 (1 point)|Question Attempt: 1 of 1
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Question 8
A consumer advocacy group is doing a large study on car rental practices. Among other things, the consumer group would like to do a statistical test regarding the mean monthly mileage, μ, of cars rented in the U.S. this year. The consumer group has good reason to believe that the mean monthly mileage of cars rented in the U.S. this year is greater than last year's mean, which was 2800 miles.
The group plans to do a statistical test regarding the value of . It chooses a random sample of monthly mileages and computes the mean of the sample to be 2900 miles and the standard deviation to be 800 miles.

Based on this information, complete the parts below.

(a)	What are the null hypothesis H0 and the alternative hypothesis H1 that should be used for the test?

(b)	Suppose that the group decides not to reject the null hypothesis. What sort of error might it be making?
▼(Choose one)
(c)	Suppose the true mean monthly mileage of cars rented in the U.S. this year is 2800. Fill in the blanks to describe a Type I error.
A Type I error would be  ▼(Choose one) the hypothesis that  is ▼(Choose one)  ▼(Choose one) when, in fact,  is ▼(Choose one) .

Let's work through this step by step for each part of the question.

### (a) Null and Alternative Hypotheses

The null hypothesis (\( H_0 \)) typically represents the claim that the consumer advocacy group wants to test against. In this case, they believe the mean monthly mileage of cars rented this year is greater than last year's, which was 2800 miles. Thus, we will test whether the mean monthly mileage this year is greater than 2800 miles.

- **Null hypothesis \( H_0 \):** The mean monthly mileage is equal to or less than last year's, so
  \[
  H_0: \mu \leq 2800
  \]
- **Alternative hypothesis \( H_1 \):** The mean monthly mileage is greater than 2800 miles, so
  \[
  H_1: \mu > 2800
  \]

This is a **one-tailed test** because we are only interested in whether the mean mileage is greater than 2800, not if it is simply different.

### (b) Type of Error If Group Does Not Reject the Null Hypothesis

If the group decides **not to reject the null hypothesis**, it might be making a **Type II error**. A Type II error occurs when the null hypothesis is **not rejected** even though it is **false** in reality. In this context, the null hypothesis claims that the mean mileage is 2800 miles or less, but failing to reject it means the group is accepting that claim, when the true mean might actually be greater than 2800 miles.

- **Answer:** Type II error.

### (c) Type I Error Description

A **Type I error** occurs when the null hypothesis is **rejected** when it is actually **true**.

Given that the true mean is 2800 miles, a Type I error would occur if the group **rejects the null hypothesis** (i.e., believes the mean is greater than 2800 miles) when, in fact, the true mean is 2800 miles.

- **Answer:** A Type I error would be **rejecting** the hypothesis that \( \mu = 2800 \) when, in fact, \( \mu = 2800 \).

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Would you like further clarification on any of the parts? Let me know if you have questions, and I can provide more details.

Here are five related questions to explore this topic further:
1. How would you compute the test statistic for this hypothesis test?
2. What is the significance level (\( \alpha \)) and how does it affect Type I and Type II errors?
3. How would you interpret a p-value in the context of this test?
4. How can increasing the sample size affect the power of a statistical test?
5. What steps should be taken to avoid a Type I error?

**Tip:** In hypothesis testing, a Type I error is generally considered more serious, as it leads to rejecting a true null hypothesis.

A consumer advocacy group is doing a large study on car rental practices. Among other things, the consumer group would like to do a statistical test regarding the mean monthly mileage, μ, of cars rented in the U.S. this year. The consumer group has good reason to believe that the mean monthly mileage of cars rented in the U.S. this year is greater than last year's mean, which was 2800 miles. The group chooses a random sample of monthly mileages, with a sample mean of 2900 miles and a standard deviation of 800 miles. Based on this information, complete the parts below.

(a) What are the null hypothesis H0 and the alternative hypothesis H1 that should be used for the test?

(b) Suppose that the group decides not to reject the null hypothesis. What sort of error might it be making?

(c) Suppose the true mean monthly mileage of cars rented in the U.S. this year is 2800. Fill in the blanks to describe a Type I error.

Explore a detailed statistical hypothesis test regarding car rental practices in the U.S., focusing on the mean monthly mileage. Learn about formulating null and alternative hypotheses, errors in hypothesis testing, and the use of sample mean and standard deviation. Ideal for undergraduate statistics students.

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