Questions: A normal population has a mean of 75 and standard deviation of 5. You select random samples of 40. Required: a. Apply the central limit theorem to describe the sampling distribution of the sample mean with n=40. What condition is necessary to apply the central limit theorem? b. What is the standard error of the sampling distribution of sample means? Note: Round your answer to 2 decimal places. c. What is the probability that a sample mean is less than 74 ? Note: Round z-value to 2 decimal places and final answer to 4 decimal places.

A normal population has a mean of 75 and standard deviation of 5. You select random samples of 40.

Required:

a. Apply the central limit theorem to describe the sampling distribution of the sample mean with n=40. What condition is necessary to apply the central limit theorem?

b. What is the standard error of the sampling distribution of sample means?

Note: Round your answer to 2 decimal places.

c. What is the probability that a sample mean is less than 74 ?

Note: Round z-value to 2 decimal places and final answer to 4 decimal places.

Transcript text: A normal population has a mean of $\$ 75$ and standard deviation of $\$ 5$. You select random samples of 40. Required: a. Apply the central limit theorem to describe the sampling distribution of the sample mean with $n=40$. What condition is necessary to apply the central limit theorem? b. What is the standard error of the sampling distribution of sample means? Note: Round your answer to 2 decimal places. c. What is the probability that a sample mean is less than $\$ 74$ ? Note: Round $z$-value to 2 decimal places and final answer to 4 decimal places.

Solution

Solution Steps

Step 1: Central Limit Theorem Application

According to the Central Limit Theorem, the sampling distribution of the sample mean will be approximately normal if the sample size $ n $ is sufficiently large (typically $ n \geq 30 $). In this case, since $ n = 40 $, we can apply the theorem, and the sampling distribution of the sample mean will be normal.

Step 2: Standard Error Calculation

The standard error (SE) of the sampling distribution of the sample mean is calculated using the formula:

\[ SE = \frac{\sigma}{\sqrt{n}} \]

where $ \sigma = 5 $ (the population standard deviation) and $ n = 40 $. Thus,

\[ SE = \frac{5}{\sqrt{40}} \approx 0.79 \]

Step 3: Probability Calculation

To find the probability that the sample mean is less than $ 74 $, we first calculate the Z-score for $ 74 $:

\[ Z = \frac{X - \mu}{SE} = \frac{74 - 75}{0.79} \approx -1.2649 \]

Next, we find the probability:

\[ P(X < 74) = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(-1.2649) - \Phi(-\infty) \approx 0.103 \]