Question

Education Rate - According to the Department of Education, 32% of adults over the age of 25 in the United States have earned at least a bachelor’s degree. Suppose this is the true proportion of all adults over the age of 25 that have earned at least a bachelor’s degree. A random sample of 200 adults over the age of 25 will be selected and their highest degree will be recorded.
Let X represent the number of adults over the age of 25 in the sample who have earned at least a bachelor’s degree, and
let p̂ represent the proportion of adults over the age of 25 in the sample who have earned at least a bachelor’s degree.

Question 1 Subquestions

1.a Consider all the possible values for the resulting number of adults over the age of 25 in the sample who have earned at least a bachelor’s degree. What is the expected value (or average) of all such values of X?

• 32

• 64

• 0.32

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

1.b Consider all the possible values for the resulting sample proportion p̂. What is the expected value (or average) of all such values of p̂?

• 32

• 64

• 0.32

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

1.c Consider all the possible values for the resulting number of adults over the age of 25 in the sample who have earned at least a bachelor’s degree. What is the standard deviation of all such values of X?

• 0.001088

• 0.03298

• 6.596969

• 43.52

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

1.d Consider all the possible values for the resulting sample proportion p̂. What is the standard deviation of all such values of p̂?

• 0.001088

• 0.03298

• 6.596969

• 43.52

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

1.e What is the exact distribution of the number of adults over the age of 25 in the sample who have earned at least a bachelor’s degree, X?

• Binomial(200, 32)

• Binomial(200, 0.32)

• Normal(64, 6.596969)

• Normal(32, 43.52)

• Normal(0.32, 0.03298)

• Normal(0.32, 0.001088)

• Normal(0,1)

1.f 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 p̂ will be approximately normal?

• Because np = 200(0.32) = 64 and n(1-p) = 200(1 – 0.32) = 136 are both at least 10, the normal model can be used as the approximate distribution of the possible values of the sample proportion p̂.

• Because np̂ = 200(0.32) = 64 and n(1-p̂) = 200(1 – 0.32) = 136 are both at least 10, the normal model can be used as the approximate distribution of the possible values of the sample proportion p̂.

• Because n = 200 is at least 10, the normal model can be used as the approximate distribution of the possible values of the sample proportion p̂.

1.g Sketch (by hand), next to each other, (1) the approximate distribution of X = the number of adults over the age of 25 in the sample who have earned at least a bachelor’s degree, and (2) the approximate distribution of p̂ = the proportion of adults over the age of 25 in the sample who have earned at least a bachelor’s degree.

Be sure that each plot has/includes:
- label for each horizontal axis
- correct shape
- distribution label (distribution name including the values of the parameters) for each distribution
- at least three approprate values, consistent with the corresponding distribution, along each horizontal axis
- a title that includes your full name ("by First Last")

Upload your picture. Be sure that it has correct orientation (not sideways, nor upside down) and that it is a clear image.

What is the approximate probability that a random sample of 200 adults over the age of 25 will result in at least 76 of the adults having earned at least a bachelor’s degree?

• 0.38

• 0.06

• 1.82

• 0.9656

• 0.0344

Answer 1