Skip to Main Content
Español

Midterm Exam 2 Review (Nonadaptive)
Question 1 of 16 (1 point)|Question Attempt: 1 of Unlimited

1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16

Question 1
A consumer advocacy group is doing a large study on car rental practices. Among other things, the consumer group would like to estimate the mean monthly mileage, μ, of cars rented in the U.S. over the past year. The consumer group plans to choose a random sample of monthly U.S. rental car mileages and then estimate μ using the mean of the sample.
Using the value 750 miles per month as the standard deviation of monthly U.S. rental car mileages from the past year, what is the minimum sample size needed in order for the consumer group to be 95% confident that its estimate is within 150 miles per month of μ?

Carry your intermediate computations to at least three decimal places. Write your answer as a whole number (and make sure that it is the minimum whole number that satisfies the requirements).

(If necessary, consult a list of formulas.)

Check

Save For Later

Submit
© 2024 McGraw Hill LLC. All Rights Reserved.
Terms of Use
|
Privacy Center
|
Accessibility
Clears your work.
Undoes your last action.

To find the minimum sample size needed, we can use the formula for the sample size \( n \) in estimating the mean with a known standard deviation, ensuring a given margin of error at a certain confidence level:

\[
n = \left( \frac{Z_{\alpha/2} \cdot \sigma}{E} \right)^2
\]

Where:
- \( Z_{\alpha/2} \) is the critical value for the desired confidence level (95% in this case).
- \( \sigma \) is the population standard deviation.
- \( E \) is the margin of error.

### Step 1: Identify the values
- Confidence level: 95%. The corresponding critical value \( Z_{\alpha/2} \) for 95% confidence is 1.96.
- Population standard deviation \( \sigma = 750 \) miles per month.
- Margin of error \( E = 150 \) miles per month.

### Step 2: Apply the formula
Substitute the known values into the formula:

\[
n = \left( \frac{1.96 \cdot 750}{150} \right)^2
\]

### Step 3: Calculate the result
First, calculate the fraction inside the parentheses:

\[
\frac{1.96 \cdot 750}{150} = \frac{1470}{150} = 9.8
\]

Now square the result:

\[
n = (9.8)^2 = 96.04
\]

### Step 4: Round up to the nearest whole number
Since the sample size \( n \) must be a whole number and we need the minimum number that satisfies the condition, we round **up**:

\[
n = 97
\]

### Final Answer:
The minimum sample size required is **97**.

Let me know if you want further clarification or details.

---

Here are 5 questions related to this concept:
1. How does increasing the confidence level affect the required sample size?
2. What happens to the sample size if the margin of error decreases?
3. What role does the population standard deviation play in determining sample size?
4. How would the formula change if the population standard deviation were unknown?
5. Can you derive the sample size for a 90% confidence interval instead of 95%?

**Tip**: When calculating sample size, always round up, even if the decimal part is small, to ensure the desired confidence and margin of error are met.

A consumer advocacy group is doing a large study on car rental practices. Among other things, the consumer group would like to estimate the mean monthly mileage, μ, of cars rented in the U.S. over the past year. The consumer group plans to choose a random sample of monthly U.S. rental car mileages and then estimate μ using the mean of the sample. Using the value 750 miles per month as the standard deviation of monthly U.S. rental car mileages from the past year, what is the minimum sample size needed in order for the consumer group to be 95% confident that its estimate is within 150 miles per month of μ?

Learn how to calculate the minimum sample size for a study estimating the mean monthly mileage of rental cars in the U.S. over the past year. This problem involves a 95% confidence interval, a standard deviation of 750 miles per month, and a margin of error of 150 miles. Suitable for college-level statistics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation