Question

According to a recent​ survey, the population distribution of number of years of education for​ self-employed individuals in a certain region has a mean of 14.714.7 and a standard deviation of 3.23.2.(a) Find the mean and standard error of the sampling distribution of x^^\_ for a random sample of size n = 100.

mu.gifx^^\_ =
sigma.gifx^^\_ =

(b) Repeat (a) for n = 400.

mu.gifx^^\_ =
sigma.gifx^^\_ =

(c) Which of the following most closely describes the effect of increasing n?

a.The mean of the sampling distribution stays the same, but the standard error decreases.
b.The mean of the sampling distribution stays the same, but the standard error increases.
c.Both the mean and the standard error of the sampling distribution decrease.
d.Both the mean and the standard error of the sampling distribution increase.

Answer 1

a)
`n=100,\,\mu_s = 14.7, \,\sigma_s=0.32`

b)
`n=400,\,\mu_s = 14.7, \,\sigma_s=0.16`

c) "a.The mean of the sampling distribution stays the same, but the standard error decreases."

Explanation:

a) We have a population with mean of 14.7 and standard deviation of 3.2.

We have to calculate the parameters of the sampling distribution (mean and standard deviation), for a sample size of n=100.

The mean for the sampling distribution is the same of the population:

`\mu_s=\mu=14.7`

The standard deviation of the sampling distribution is related to the population standard deviation by the inverse of the square of the sample size:

`\sigma_s=\sigma/√(n)=3.2/√(100)=3.2/10=0.32`

b) For a sample size of n=400, the mean is the same (14.7), but the standard deviation becomes:

`\sigma_s=\sigma/√(n)=3.2/√(400)=3.2/20=0.16`

c) "a.The mean of the sampling distribution stays the same, but the standard error decreases."

As we calculated, the standard error decreases as the sample size increases. This is because the variability of the variable is reduced as the sample size is bigger.

The mean stays the same independently of the sample size.