Question

A random variable is normally distributed. It has a mean of 245 and a standard deviation of 21. I just need G.

a.)If you take a sample of size 10, can you say what the shape of the distribution for the sample mean is? Why?

b.) For a sample of size 10, state the mean of the sample mean and the standard deviation of the sample mean.

c.) For a sample of size 10, find the probability that the sample mean is more than 241.

d.) If you take a sample of size 35, can you say what the shape of the distribution of the sample mean is? Why?

e.) For a sample of size 35, state the mean of the sample mean and the standard deviation of the sample mean.

f.) For a sample of size 35, find the probability that the sample mean is more than 241.

g.) Compare your answers in part c and f. Why is one smaller than the other?

Answer 1

qa)

as population distribution is normal therefore  shape of the distribution for the sample mean is also normal

b)mean of the sample mean =population mean =245

and  standard deviation of the sample mean =population std deviation/sqrt(n)=21/sqrt(10)=6.6408

c)

for normal distribution z score =(X-μ)/σ

probability =

P(X>241)

=

P(Z>-0.6)=

1-P(Z<-0.6)=

1-0.2743=

0.7257

d)

as population distribution is normal therefore  shape of the distribution for the sample mean is also normal

e)

mean of the sample mean =population mean =245

and  standard deviation of the sample mean =population std deviation/sqrt(n)=21/sqrt(35)=3.5496

f)

probability =

P(X>241)

=

P(Z>-1.13)=

1-P(Z<-1.13)=

1-0.1292=

0.8708

g)

for increasing sample size will reduce standard deviation of sample mean. therefore sample mean will have higher probabiliy to be closer to population mean,hence for part c probability is smaller then part f