Question

Suppose that we will take a random sample of size n from a population having mean u and standard deviation o. For each of

the following situations, find the mean, variance, and standard deviation of the sampling distribution of the sample mean X:

a. μ = 10, o=2, n = 25

b. μ=500, o=.5, n = 100

C. μ=3, o =., n=4

d. μ= 100, o=1, n = 1,600

What will I do here if the mean and stdv is already given? TT sorry I am confused

Answer 1

a.

$$\mu_{\bar{X}}=\mu=10$$

$$Var(\bar{X})=\sigma_{\bar{X}}^2=\dfrac{\sigma^2}{n}=\dfrac{2^2}{25}=0.16$$

$$\sigma_{\bar{X}}=\dfrac{\sigma}{\sqrt{n}}=\dfrac{2}{\sqrt{25}}=0.4$$

b.

$$\mu_{\bar{X}}=\mu=500$$

$$Var(\bar{X})=\sigma_{\bar{X}}^2=\dfrac{\sigma^2}{n}=\dfrac{0.5^2}{100}=0.0025$$

$$\sigma_{\bar{X}}=\dfrac{\sigma}{\sqrt{n}}=\dfrac{0.5}{\sqrt{100}}=0.05$$

c.

$$\mu_{\bar{X}}=\mu=3$$

$$Var(\bar{X})=\sigma_{\bar{X}}^2=\dfrac{\sigma^2}{n}=\dfrac{0.1^2}{4}=0.0025$$

$$\sigma_{\bar{X}}=\dfrac{\sigma}{\sqrt{n}}=\dfrac{0.1}{\sqrt{4}}=0.05$$

d.

$$\mu_{\bar{X}}=\mu=100$$

$$Var(\bar{X})=\sigma_{\bar{X}}^2=\dfrac{\sigma^2}{n}=\dfrac{1^2}{1600}=0.000625$$

$$\sigma_{\bar{X}}=\dfrac{\sigma}{\sqrt{n}}=\dfrac{1}{\sqrt{1600}}=0.025$$