Question

According to the Department of Education:

• 27% of adults over the age of 65 in the United States have earned at least a bachelor's degree
• 36% of adults 25 to 34 years old in the United States have earned at least a bachelor's degree

Suppose these proportions are true for the respective populations.   

A researcher is planning on the following samples:
A random sample of 200 adults over the age of 65 will be selected and their highest degree will be recorded.
A random sample of 200 adults 25 to 34 years old will be selected and their highest degree will be recorded.

Let p̂₁represent the proportion of adults over the age of 65 in the sample who have earned at least a bachelor’s degree.
Let p̂₂ represent the proportion of adults 25 to 34 years old in the sample who have earned at least a bachelor’s degree.

Question 4 Subquestions

4.a

0.5 point(s)

Consider all the possible values for the resulting differences in sample proportions p̂₁- p̂₂ . What is the expected value of all such values of the differences of p̂₁- p̂₂?

• -0.09

• 0.09

• -18

• 18

• unknown since we have not taken actual random samples yet

4.b

0.5 point(s)

Consider all the possible values for the resulting differences in sample proportions p̂₁- p̂₂ . What is the standard deviation of all such values of the differences of p̂₁- p̂₂?  

• 0.04828

• 0.0462

• 9.2466

• unknown since we have not yet taken an actual random samples

4.c

0.5 point(s)

Which of the following is the appropriate condition (with correct notation) regarding the sample size that would confirm that the sampling distribution of the possible values of p̂₁- p̂₂  will be approximately normal?

• Because n₁= 200 and n₂= 200 are both least 10, the normal model can be used as the approximate distribution of the possible values of the difference in sample proportions p̂₁- p̂₂.

• Because n₁p̂₁= 200(0.27) = 54, n₁(1- p̂₁ )= 200(1 – 0.27) = 146, n₂p̂₂ = 200(0.36) = 72, and n₂(1- p̂₂ )= 200(1 – 0.36) = 128 are all at least 10, the normal model can be used as the approximate distribution of the possible values of the difference in sample proportions p̂₁- p̂₂.

• Because n₁p₁= 200(0.27) = 54, n₁(1- p₁ )= 200(1 – 0.27) = 146, n₂p₂ = 200(0.36) = 72, and n₂(1- p₂ )= 200(1 – 0.36) = 128 are all at least 10, the normal model can be used as the approximate distribution of the possible values of the difference in sample proportions p̂₁- p̂₂.

4.d

0.5 point(s)

What is the approximate probability that a random sample of 200 adults over the age of 65 and a random sample of adults 25 to 34 years old will result in a difference in sample proportion, p̂₁- p̂₂ , of at most -0.01?

• 0.0154

• 0.0418

• 0.9582

• 0.9846

4.e

0.5 point(s)

The researcher has decided to triple the sample size for both groups:

A random sample of 600 adults over the age of 65 will be selected and their highest degree will be recorded.
A random sample of 600 adults 25 to 34 years old will be selected and their highest degree will be recorded.  

The increase in sample sizes, from 200 to 600, will _________________ the expected value of all possible values of the differences in sample proportions, p̂₁- p̂₂.

• increase

• decrease

• not affect

4.f

0.5 point(s)

The researcher has decided to triple the sample size for both groups:

A random sample of 600 adults over the age of 65 will be selected and their highest degree will be recorded.
A random sample of 600 adults 25 to 34 years old will be selected and their highest degree will be recorded.  

The increase in sample sizes, from 200 to 600, will _________________ the value of the standard deviation of all possible values of the differences in sample proportions, p̂₁- p̂₂.

• increase

• decrease

• not affect

Answer 1