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3.5 - The Central Limit Theorem

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Question 1
A population of values has a normal distribution with $\mu=178.8$ and $\sigma=16.3$. You intend to draw a random sample of size $n=115$.
What is the mean of the distribution of sample means?
(Report answer accurate to one decimal place.)
\[
\mu_{\bar{x}}=
\]

What is the standard deviation of the distribution of sample means?
(Report answer accurate to two decimal places.)
\[
\sigma_{x}=
\]
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Step 1 ：The mean of the distribution of sample means is equal to the mean of the population. Given that the mean of the population is 178.8, we can conclude that the mean of the distribution of sample means is also 178.8.

Step 2 ：The standard deviation of the distribution of sample means is calculated by dividing the standard deviation of the population by the square root of the sample size. Given that the standard deviation of the population is 16.3 and the sample size is 115, we can calculate the standard deviation of the distribution of sample means as follows: \( \sigma_{x} = \frac{16.3}{\sqrt{115}} \).

Step 3 ：By calculating the above expression, we find that the standard deviation of the distribution of sample means is approximately 1.52.

Step 4 ：Final Answer: The mean of the distribution of sample means is \( \boxed{178.8} \) and the standard deviation of the distribution of sample means is \( \boxed{1.52} \).