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

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?

a)

So then the distribution is also normal but with another deviation. And the reason is because"

b)

c)

And using the complment rule, the normal standard deviation or excel we got:

d) Since the distribution for X is normal then the distribution for the sample mean is given by:

And the reason is because"

e)

f)

And using the complment rule, the normal standard deviation or excel we got:

g) The probability for part f is lower since the deviation for the sample size of n =35 is lower compared to the deviation with a sample size of n =10.

Step-by-step explanation:

For this case we assume that we have a random variable X and the distribution of X is given by:

Part a

Since the distribution for X is normal then the distribution for the sample mean is given by:

And for this case n =10 represent the sample size we have:

So then the distribution is also normal but with another deviation. And the reason is because"

Part b

Part c

For this case we can use the z score formula given by:

And we want this probability:

And using the complment rule, the normal standard deviation or excel we got:

Part d

Since the distribution for X is normal then the distribution for the sample mean is given by:

And the reason is because"

Part e

Part f

For this case we can use the z score formula given by:

And we want this probability:

And using the complment rule, the normal standard deviation or excel we got:

Part g

The probability for part f is higher since the deviation for the sample size of n =35 is lower compared to the deviation with a sample size of n =10.

22 dimes and 9 nickels

step-by-step explanation: