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The U.S. Census Bureau reported that in 2015 the proportion of adult Americans, ages 25 and older, who have a bachelor's degree or higher is 0.325.† Consider the population of all adult Americans, ages 25 and over, in 2015 and define p̂ to be the proportion of people in a random sample from this population who have a bachelor's degree or higher.

(a)

Would p̂ based on a random sample of only 10 people from this population have a sampling distribution that is approximately normal? Explain why or why not.

---Select---YesNo , the sampling distribution of p̂ based on a random sample of size 10 residents ---Select---would bewould not be approximately normally distributed because np is ---Select---less thanat least 10.

(b)

What are the mean and standard deviation of the sampling distribution of p̂ if the sample size is 200? (Round your standard deviation to four decimal places.)

mean standard deviation

(c)

Suppose that the sample size is

n = 100

rather than

n = 200.

What are the values for the mean and standard deviation when

n = 100?

(Round your standard deviation to four decimal places.)

mean standard deviation

Does the change in sample size affect the mean and standard deviation of the sampling distribution of p̂? If not, explain why not. (Select all that apply.)

When the sample size decreases, the mean increases. When the sample size decreases, the mean decreases.

When the sample size decreases, the mean stays the same. The sampling distribution is always centered at the population mean, regardless of sample size.

When the sample size decreases, the standard deviation increases.

When the sample size decreases, the standard deviation decreases.

When the sample size decreases, the standard deviation stays the same. The standard deviation of the sampling distribution is always the same as the standard deviation of the population distribution, regardless of sample size.

Answer 1

a)Yes , , the sampling distribution of p̂ based on a random sample of size 10 residents would b approximately normally distributed because np is at least 10.
b)

here mean=     μp=

0.3250

std deviation of proportion=σp=√(p*(1-p)/n)=

0.0331

c)

for n=100:

mean=     μp=

0.3250

std deviation of proportion=σp=√(p*(1-p)/n)=

0.0331

When the sample size decreases, the mean stays the same. The sampling distribution is always centered at the population mean, regardless of sample size

When the sample size decreases, the standard deviation increases.