Question

Problem 3. If Xi,... . Xio are a random sample from a Normal distribution N(2,32) and 10 - X is the sample mean. (a) What is E(X)? (0.25 point) (b) What is Var(X)? (0.25 point) (c) What is the distribution of X? Please specify the name of the distribution, mean, and variance of the distribution. (0.25 point) d) What is the probability that the random sample mean falls in the interval [1.5,2.5] (0.25 point)

Answer 1

3)

Here, $\bg_white X_{i}\sim N(2,3^{2})$

From Central limit theorem, we know,$\bg_white \overline{X}\sim N(\mu,\frac{\sigma^{2}}{n})\sim N(2,\frac{3^{2}}{10})$

a)

So, $\bg_white E(\overline{X})=2$

b)

Also, $\bg_white Var(\overline{X})=\frac{3^{2}}{10}=0.9$

c)

Using the Central limit theorem, we know,$\bg_white \overline{X}\sim N(\mu,\frac{\sigma^{2}}{n})\sim N(2,\frac{3^{2}}{10})$

i.e. it follows Normal distribution with mean 2 and variance 0.9.

d)

Required probability = $\bg_white P(1.5<\overline{X}<2.5)=P(\frac{1.5-2}{\sqrt{0.9}}<Z<\frac{2.5-2}{\sqrt{0.9}})$

$\bg_white P(-0.53<Z<0.53)=P(Z<0.53)-P(Z<-0.53)$

$\bg_white =P(Z<0.53)-P(Z>0.53)=P(Z<0.53)-[1-P(Z<0.53)]$

$\bg_white =0.70194-[1-0.70194]=0.40388$