Question

A random sample is selected from a population with mean \(\mu=60\) and standard deviation \(\sigma=3\). Determine the mean and standard deviation of the \(\bar{x}\) sampling distribution for each of the following sample sizes: a. \(n=6\) d. \(n=75\) b. \(n=18\) e. \(n=200\) c. \(n=42\) f. \(n=400\)

Short answer

a. For n=6, the mean is 60 and the standard deviation is 1.22, b. For n=75, the mean is 60 and the standard deviation is 0.35, c. For n=18, the mean is 60 and the standard deviation is 0.71, d. For n=200, the mean is 60 and the standard deviation is 0.21, e. For n=42, the mean is 60 and the standard deviation is 0.46, f. For n=400, the mean is 60 and the standard deviation is 0.15.

Step-by-step solution

Calculate Mean of Sample Distribution

The mean of the sampling distribution of the sample mean equals the population mean, which is \(\mu_{ \bar{x} } = \mu = 60\). This applies to all sample sizes.

Calculate Standard Deviation of Sample Distribution for n=6

Calculate the standard deviation for n=6 using the formula \(\sigma_{\bar{x}} = \sigma / \sqrt{n}\), which gives \(\sigma_{ \bar{x} } = 3 / \sqrt{6} = 1.22\) (rounded to 2 decimal places).

Repeat for n=75

Repeat the same steps for n=75, giving \(\sigma_{ \bar{x} } = 3 / \sqrt{75} = 0.35\) (rounded to 2 decimal places).

Repeat for n=18

Continue for n=18, resulting in \(\sigma_{ \bar{x} } = 3 / \sqrt{18} = 0.71\) (rounded to 2 decimal places).

Repeat for n=200

Calculate for n=200, getting \(\sigma_{ \bar{x} } = 3 / \sqrt{200} = 0.21\) (rounded to 2 decimal places).

Repeat for n=42

Repeat the same steps for n=42, resulting in \(\sigma_{ \bar{x} } = 3 / \sqrt{42} = 0.46\) (rounded to 2 decimal places).

Repeat for n=400

Finally, for n=400, we get \(\sigma_{ \bar{x} } = 3 / \sqrt{400} = 0.15\) (rounded to 2 decimal places).