Question

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 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 means of the sample to be 2800 miles and the standart deation to be 850 miles. Based on this information, complete the parts below. (a) What are the null hypothesis H{o} and the alternative hypothesis H{i} that should be used for the test? H{o} ___ H{i} ______ (b) Suppose that the group decides not to reject the null hypothesis. What sort of errar might it be making? Suppose the true mean monthly mileage of cars rented in the US this year is 2700, fit in the blanks to describe a Type I error A Type I error would _________ the hypothesis thatμ is ____ when, in fact, is

Answer 1

(a) Null hypothesis
`\( H_(0) \)`: μ ≤ 2700 and and alternative hypothesis
`\( H_(1) \)`: μ > 2700.

(b) A Type I error would incorrectly reject the hypothesis that μ ≤ 2700 when, in fact, it is true.

Step-by-step explanation:

(a) The null hypothesis
`\( H_(0) \)` represents the status quo or the assumption that there is no significant change. In this case,
`\( H_(0) \)` is
`\( \mu \leq 2700 \)`, suggesting that the mean monthly mileage of cars rented this year is less than or equal to last year's mean of 2700 miles. The alternative hypothesis
`\( H_(1) \)` is
`\( \mu > 2700 \)`, indicating that the mean monthly mileage is greater than 2700 miles.

(b) A Type I error occurs when the null hypothesis
`\( H_(0) \)` is incorrectly rejected when it is actually true. In this context, it would mean wrongly concluding that the mean monthly mileage is greater than 2700 miles when, in reality, it is not. This error is also known as a "false positive." It could lead to unnecessary actions or decisions based on the incorrect belief that there has been a significant increase in mean mileage.

To clarify, a Type I error is akin to a consumer group falsely claiming that there is a significant increase in mean monthly mileage when, according to the null hypothesis, there is no such increase. This emphasizes the importance of careful consideration before rejecting the null hypothesis, especially when potential consequences of a Type I error are significant.