https://file.aiquickdraw.com/mathbot/file/1729042846394bslu2uxn.jpg

The problem asks about the distribution of Harley-Davidson motorcycles in a sample, providing the following:

- **Proportion (mean)**: Harley-Davidson motorcycles make up **14%** of all motorcycles, so the mean is given as **0.14**.
- The sample size is **520 motorcycle owners**.

Next, you are asked to calculate the **standard deviation** of the sample proportion. The formula for the standard deviation of a sample proportion is:

\[
\sigma = \sqrt{\frac{p(1-p)}{n}}
\]

Where:
- \( p \) is the proportion of Harley-Davidson owners (0.14),
- \( 1 - p \) is the complement proportion (0.86),
- \( n \) is the sample size (520).

Let's calculate:

\[
\sigma = \sqrt{\frac{0.14 \times 0.86}{520}} 
\]
\[
\sigma = \sqrt{\frac{0.1204}{520}} 
\]
\[
\sigma = \sqrt{0.00023153846} 
\]
\[
\sigma \approx 0.0152
\]

So, the standard deviation is approximately **0.0152** (to four decimal places).

Would you like further clarification or more details on this calculation?

Here are five related questions you might find useful:
1. How does increasing the sample size affect the standard deviation in this context?
2. What would the standard deviation be for a different proportion (e.g., 20%)?
3. Why is the formula for standard deviation of a proportion different from that of a population?
4. What are other ways to reduce the variability of the sample proportion?
5. How would you interpret the standard deviation in this scenario?

**Tip:** When calculating standard deviation for proportions, remember it depends on both the sample size and the variability (difference between the proportion and its complement).

Harley-Davidson motorcycles make up 14% of all motorcycles registered in the United States. You plan to interview an SRS of 520 motorcycle owners. (a) What is the approximate distribution of the proportion of your sample who own Harleys? Calculate the mean. (b) Calculate the standard deviation.

Learn how to calculate the mean and standard deviation of a sample proportion when 14% of motorcycles are Harley-Davidsons. The sample size is 520 motorcycle owners, and the standard deviation formula for proportions is applied step-by-step. Suitable for Grades 10-12 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