You are asked to gather some data on wait times for a popular rollercoaster at the amusement park where you work. You can't track the waiting time of every rider, so you decide to sample 81 people at random each day. The mean of a distribution of sample mean wait times is 18 minutes and the standard deviation of the sampling distribution (sigma subscript x with bar on top end subscript) is 0.5 minutes. 

What is the standard deviation of the population (sigma)?


0.5


4.5


2


18

To determine the standard deviation of the population (\(\sigma\)), we can use the relationship between the standard deviation of the sampling distribution (denoted as \(\sigma_{\bar{x}}\)) and the population standard deviation \(\sigma\).

The formula that relates them is:

\[
\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}
\]

Where:
- \(\sigma_{\bar{x}}\) is the standard deviation of the sampling distribution (0.5 minutes in this case),
- \(\sigma\) is the population standard deviation (what we're solving for),
- \(n\) is the sample size (81 people).

Now, rearrange the formula to solve for \(\sigma\):

\[
\sigma = \sigma_{\bar{x}} \times \sqrt{n}
\]

Substitute the known values:

\[
\sigma = 0.5 \times \sqrt{81}
\]

Since \(\sqrt{81} = 9\), we get:

\[
\sigma = 0.5 \times 9 = 4.5
\]

Thus, the population standard deviation (\(\sigma\)) is **4.5 minutes**.

Would you like more details or have any other questions?

---

Here are 5 related questions to expand on this:

1. How does increasing the sample size affect the standard deviation of the sampling distribution?
2. What is the significance of the standard deviation in a normal distribution?
3. If the sample size were 100 instead of 81, what would the new standard deviation of the sampling distribution be?
4. How would you calculate the margin of error using the standard deviation and sample size?
5. How does the Central Limit Theorem relate to this problem?

**Tip:** When dealing with sampling distributions, remember that larger sample sizes reduce the variability (standard deviation) of the sample means.

You are asked to gather some data on wait times for a popular rollercoaster at the amusement park where you work. You can't track the waiting time of every rider, so you decide to sample 81 people at random each day. The mean of a distribution of sample mean wait times is 18 minutes and the standard deviation of the sampling distribution (sigma subscript x with bar on top end subscript) is 0.5 minutes. What is the standard deviation of the population (sigma)?

Learn how to calculate the population standard deviation (sigma) from the standard deviation of a sampling distribution using a sample size of 81 and a sample mean wait time of 18 minutes. This detailed guide covers key statistical concepts and formulas, including the relationship between sample size and standard deviation. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation