Question

The number of undergraduate students at the University of Winnipeg is approximately
9,000, while the University of Manitoba has approximately 27,000 undergraduate students.
Suppose that, at each university, a simple random sample of 3% of the undergraduate
students is selected and the following question is asked: “Do you approve of the
provincial government’s decision to lift the tuition freeze?” Suppose that, within each
university, approximately 20% of undergraduate students favour this decision. What can
be said about the sampling variability associated with the two sample proportions

Answer 1

What can
be said about the sampling variability associated with the two sample proportions?
(A) The sample proportion from U of W has less sampling variability than that from U
of M.
(B) The sample proportion from U of W has more sampling variability that that from
U of M.
(C) The sample proportion from U of W has approximately the same sampling variability
as that from U of M.
(D) It is impossible to make any statements about the sampling variability of the two
sample proportions without taking many samples.
(E) It is impossible to make any statements about the sampling variability of the two
sample proportions because the population sizes are different.

Answer 2

The answer is B. but I want to know why. can everyone explain it to me?

Answer 3

For a given population , let p be a sample proportion of size n, \(\large \pi \) be the population proportion.
It is true that for the sampling distribution of the sample proportion:
$$
\Large \rm \mu_p= \pi \\~\\
\Large\rm \sigma_ p=\sqrt{\frac{ \pi (1-\pi)}{n}}
$$
The standard deviation of the sample proportion gives us a measure of the variability of the sample proportion. So we are going to compare the two standard deviations.I will use M and W to stand for Manitoba and Winnepeg

$$
\Large\rm {
\sigma_{p_W}=\sqrt{\frac{ .20 (1-.20)}{.03\cdot 27000}}=
\sqrt{\frac{ .20 (1-.20)}{810}}= 0.014
\\
\\~\\~\\~\\\
\sigma_{p_M}=\sqrt{\frac{ .20 (1-.20)}{.03 \cdot 9000}}
= \sqrt{\frac{ .20 (1-.20)}{270}}= 0.024
}
$$

Answer 4

We see that the bigger the size of the sample, the less the variability.
Or conversely, the smaller the size of the sample the greater the variability.

Answer 5

Because of the sample size n in the denominator.
Note that we got the sample size n here by multiplying 3% by the respective population as given in the directions.