Questions: Question 18 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 ( σₓ̄ ) is 0.5 minutes. What is the standard deviation of the population ( σ )? 1. ? 2. 0.5 3. 18 4. 4.5

Question 18 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 ( σₓ̄ ) is 0.5 minutes. What is the standard deviation of the population ( σ )? 1. ? 2. 0.5 3. 18 4. 4.5
Transcript text: Question 18 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 ( $\sigma_{\bar{x}}$ ) is 0.5 minutes. What is the standard deviation of the population ( $\sigma$ )? 1. ? 2. 0.5 3. 18 4. 4.5
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Solution

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Solution Steps

Step 1: Given Information

We are provided with the following information:

  • The mean of the distribution of sample mean wait times is \( \mu = 18 \) minutes.
  • The standard deviation of the sampling distribution is \( \sigma_{\bar{x}} = 0.5 \) minutes.
  • The sample size is \( n = 81 \).
Step 2: Relationship Between Standard Deviations

The relationship between the standard deviation of the sampling distribution (\( \sigma_{\bar{x}} \)) and the population standard deviation (\( \sigma \)) is given by the formula:

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

Step 3: Rearranging the Formula

To find the population standard deviation (\( \sigma \)), we can rearrange the formula:

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

Step 4: Substituting Values

Substituting the known values into the equation:

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

Step 5: Calculating the Population Standard Deviation

Calculating \( \sqrt{81} \):

\[ \sqrt{81} = 9 \]

Now substituting back:

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

Final Answer

The standard deviation of the population (\( \sigma \)) is:

\[ \boxed{\sigma = 4.5} \]

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