Question

1. in a normally distributed population, the mean is 24 and the standard deviation is 4.8. which one of the following is completely correct with reference to the distribution of sample means of all samples of size 64?

a. its mean is 24 and its standard deviation is 4.8.
b. its mean is 3 and its standard deviation is 4.8.
c. its mean is 24 and its standard deviation is 0.6.
d. its mean is 3 and its standard deviation is 0.6.

Explanation:

🆕 New Concept Discovered: Central Limit Theorem
How sample means behave compared to the population

Step 1: Identify Population Parameters

The population parameters given in the problem are:

• Population mean: \(\mu = 24\)
• Population standard deviation: \(\sigma = 4.8\)
• Sample size: \(n = 64\)

Step 2: Determine Mean of Sample Means

According to the Central Limit Theorem, the mean of the sampling distribution of the sample mean (\(\mu_{\bar{x}}\)) is equal to the population mean (\(\mu\)):

$$ \mu_{\bar{x}} = \mu = 24 $$

Step 3: Calculate Standard Error of the Mean

The standard deviation of the sampling distribution of the sample mean (also called the standard error, \(\sigma_{\bar{x}}\)) is calculated by dividing the population standard deviation by the square root of the sample size:

$$ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} $$

Substitute the given values:

$$ \sigma_{\bar{x}} = \frac{4.8}{\sqrt{64}} = \frac{4.8}{8} = 0.6 $$

Answer:

C. Its mean is 24 and its standard deviation is 0.6.