Question

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

a.) for a sample of size 10, state the mean of the sample mean.
b.) for a sample of size 10, state the standard deviation of the sample mean (the "standard error of the mean").
c.) for a sample of size 35, state the mean of the sample mean.
d.) for a sample of size 35, state the standard deviation of the sample mean (the "standard error of the mean").
e.) true or false: as sample size increases, standard deviation of the sample mean (the "standard error of the mean") also increases.

Answer 1

i think its b. hope im right and that this helped

Answer 2

a) By the Central Limit Theorem, 245

b) 6.64

c) By the Central Limit Theorem, 245

d) 3.55

e) False

Explanation:

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem, we have that:

`\mu = 245, \sigma = 21`

a.) for a sample of size 10, state the mean of the sample mean.

By the Central Limit Theorem, 245

b.) for a sample of size 10, state the standard deviation of the sample mean (the "standard error of the mean").

`s = (21)/(√(10)) = 6.64`

c.) for a sample of size 35, state the mean of the sample mean.

By the Central Limit Theorem, 245

d.) for a sample of size 35, state the standard deviation of the sample mean (the "standard error of the mean").

`s = (21)/(√(35)) = 3.55`

e.) true or false: as sample size increases, standard deviation of the sample mean (the "standard error of the mean") also increases.

The standard error of the mean is given by
`s = (\sigma)/(√(n))`. So, as n(sample size) increases, s(standard deviation of the sample mean) decreases. So this is false.