Question

Of all freshman at a large college, 16% made the dean’s list in the current

year. As part of a class project, students randomly sample 40 students and check if those students made
the list. They repeat this 1,000 times and build a distribution of sample proportions.
(a) What is this distribution called?
(b) Would you expect the shape of this distribution to be symmetric, right skewed, or left skewed? Explain
your reasoning.
(c) Calculate the variability of this distribution.
(d) What is the formal name of the value you computed in (c)?
(e) Suppose the students decide to sample again, this time collecting 90 students per sample, and they again
collect 1,000 samples. They build a new distribution of sample proportions. How will the variability
of this new distribution compare to the variability of the distribution when each sample contained 40
observations?

Answer 1

Explanation:

(a) This distribution is called the sampling distribution of sample proportions (p-hat).

(b) The distribution will have a bell curve, similar to a normal curve. So it is symmetric.

(c) The variability is:

σ = √(pq/n)

σ = √(0.16 × (1 − 0.16) / 40)

σ = 0.058

(d) The formal name for this is "standard error".

(e) This time, the sample size increases to 90.

σ = √(pq/n)

σ = √(0.16 × (1 − 0.16) / 90)

σ = 0.039

The standard error decreases as sample size increases.