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Midterm Exam 2
Question 8 of 8 (1 point)
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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 less than last year's mean, which was 2700 miles.
The group plans to do a statistical test regarding the value of μ. It chooses a random sample of monthly mileages and computes the mean of the sample to be 2565 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?
H0:
H1:
(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 2700. Fill in the blanks to describe a Type I error.
A Type I error would be  ▼(Choose one) the hypothesis that μ is ▼(Choose one)  ▼(Choose one) when, in fact, μ is ▼(Choose one) .

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Let's work through this step by step:

### (a) **What are the null hypothesis \( H_0 \) and the alternative hypothesis \( H_1 \)?**

- The **null hypothesis \( H_0 \)** represents the assumption that there is no difference from the previous year's mean (or that the mean is equal to last year's mean, 2700 miles). 

- The **alternative hypothesis \( H_1 \)** represents the claim the consumer group is investigating, which is that the mean monthly mileage is less than last year.

Thus, we have:

\[
H_0: \mu = 2700
\]
\[
H_1: \mu < 2700
\]

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

If the group fails to reject the null hypothesis, but in reality, the true mean is less than 2700, then the group is making a **Type II error**. A Type II error occurs when the null hypothesis is not rejected, even though it is false.

Answer: **Type II error.**

### (c) **Suppose the true mean monthly mileage of cars rented in the U.S. this year is 2700. Fill in the blanks to describe a Type I error.**

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

- A Type I error would be **rejecting** the hypothesis that μ is **2700** when, in fact, μ is **2700**.

Answer: A Type I error would be **rejecting** the hypothesis that **μ is 2700** when, in fact, **μ is 2700**.

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### Do you want any further details or have any additional questions?

Here are 5 related questions to expand on this topic:
1. How do you calculate the p-value for a hypothesis test?
2. What is the difference between a one-tailed and a two-tailed hypothesis test?
3. What factors influence the likelihood of making a Type I error?
4. How do you calculate the power of a statistical test?
5. What is the significance level \( \alpha \), and how does it relate to Type I errors?

#### Tip: 
The **p-value** in hypothesis testing helps you decide whether to reject the null hypothesis. If the p-value is less than the significance level (usually 0.05), you reject the null hypothesis.

A consumer advocacy group is doing a large study on car rental practices. They want to perform a statistical test regarding the mean monthly mileage (μ) of cars rented in the U.S. this year. They believe the mean monthly mileage is less than last year's 2700 miles. A sample mean is 2565 miles with a standard deviation of 800 miles. What are the null and alternative hypotheses? What type of error occurs if the null hypothesis is not rejected or incorrectly rejected?

Explore a statistical test of the mean monthly mileage of U.S. rental cars. Learn about null and alternative hypotheses, Type I and Type II errors, and how a sample mean of 2565 miles is compared to a population mean of 2700 miles with a standard deviation of 800 miles.

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