Question

suppose a random sample of n = 100 measurements is selected from a population with mean μ and standard deviation σ. for each of the following values of μ and σ, give the values of μ_x̄ and σ_x̄. a. μ = 10, σ = 5 μ_x̄ = □ σ_x̄ = □ (type an integer or a decimal.) b. μ = 100, σ = 100 c. μ = 20, σ = 50 d. μ = 10, σ = 220

Explanation:

Step1: Recall the formula for the mean of the sampling - distribution of the sample mean

The mean of the sampling - distribution of the sample mean $\mu_{\bar{x}}$ is equal to the population mean $\mu$. That is, $\mu_{\bar{x}}=\mu$.

Step2: Recall the formula for the standard deviation of the sampling - distribution of the sample mean

The standard deviation of the sampling - distribution of the sample mean (also known as the standard error) is given by $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size. Here, $n = 100$.

For part a:

$\mu = 10$, $\sigma = 5$

Step3: Calculate $\mu_{\bar{x}}$

Since $\mu_{\bar{x}}=\mu$, then $\mu_{\bar{x}}=10$.

Step4: Calculate $\sigma_{\bar{x}}$

$\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}=\frac{5}{\sqrt{100}}=\frac{5}{10}=0.5$

For part b:

$\mu = 100$, $\sigma = 100$

Step5: Calculate $\mu_{\bar{x}}$

Since $\mu_{\bar{x}}=\mu$, then $\mu_{\bar{x}}=100$.

Step6: Calculate $\sigma_{\bar{x}}$

$\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}=\frac{100}{\sqrt{100}}=\frac{100}{10}=10$

For part c:

$\mu = 20$, $\sigma = 50$

Step7: Calculate $\mu_{\bar{x}}$

Since $\mu_{\bar{x}}=\mu$, then $\mu_{\bar{x}}=20$.

Step8: Calculate $\sigma_{\bar{x}}$

$\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}=\frac{50}{\sqrt{100}}=\frac{50}{10}=5$

For part d:

$\mu = 10$, $\sigma = 220$

Step9: Calculate $\mu_{\bar{x}}$

Since $\mu_{\bar{x}}=\mu$, then $\mu_{\bar{x}}=10$.

Step10: Calculate $\sigma_{\bar{x}}$

$\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}=\frac{220}{\sqrt{100}}=\frac{220}{10}=22$

Answer:

a. $\mu_{\bar{x}} = 10$, $\sigma_{\bar{x}}=0.5$
b. $\mu_{\bar{x}} = 100$, $\sigma_{\bar{x}}=10$
c. $\mu_{\bar{x}} = 20$, $\sigma_{\bar{x}}=5$
d. $\mu_{\bar{x}} = 10$, $\sigma_{\bar{x}}=22$