To find the mean and standard deviation of the sampling distribution of sample means given a population with a known mean and standard deviation, and a sample size, we can follow these steps:
1. Identify the Population Mean ([tex]\(\mu\)[/tex]) and Population Standard Deviation ([tex]\(\sigma\)[/tex]):
- Given population mean, [tex]\(\mu = 85\)[/tex]
- Given population standard deviation, [tex]\(\sigma = 27\)[/tex]
2. Determine the Sample Size ([tex]\(n\)[/tex]):
- Given sample size, [tex]\(n = 256\)[/tex]
3. Find the Mean of the Sampling Distribution of Sample Means ([tex]\(\mu_x\)[/tex]):
- The mean of the sampling distribution of sample means ([tex]\(\mu_x\)[/tex]) is the same as the population mean ([tex]\(\mu\)[/tex]).
- Therefore, [tex]\(\mu_x = 85\)[/tex]
4. Calculate the Standard Deviation of the Sampling Distribution of Sample Means ([tex]\(\sigma_x\)[/tex]):
- The standard deviation of the sampling distribution of sample means is calculated by dividing the population standard deviation by the square root of the sample size:
[tex]\[
\sigma_x = \frac{\sigma}{\sqrt{n}}
\][/tex]
- Substituting the given values:
[tex]\[
\sigma_x = \frac{27}{\sqrt{256}}
\][/tex]
- Simplify the square root in the denominator:
[tex]\[
\sqrt{256} = 16
\][/tex]
- Hence:
[tex]\[
\sigma_x = \frac{27}{16} = 1.6875
\][/tex]
Therefore, the mean ([tex]\(\mu_{ x }\)[/tex]) of the sampling distribution of sample means is [tex]\(85\)[/tex] and the standard deviation ([tex]\(\sigma_{ x }\)[/tex]) is [tex]\(1.6875\)[/tex].
Answer:
[tex]\(\mu_{ x } = 85 \\
\sigma_{ x } = 1.6875\)[/tex]
