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?
(A) The sample proportion for the U of W has less sampling variability than that for the U of M.

(B) The sample proportion for the U of W has more sampling variability that that for the U of M.

(C) The sample proportion for the U of W has approximately the same sampling variability as that for the 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.

Could you explain why answer is (B)

Answer 1

Solution

Let N₁, n₁ and p_1cap be respectively the population size, sample size and sample proportion for U of W.

Similarly, N₂, n₂ and p_2cap represent for U of M.

Back-up Theory

The sampling variability of sample proportion, p_cap = p_cap(1- p_2cap)/n …………………………................................…… (1)

Now to work out the solution,

Theoretically, given, ‘within each university, approximately 20% of undergraduate students favour this decision’ => p_1cap = p_2cap ……. ………………………………………………………………….................................................…….. (2)

Also, given, ‘at each university, a simple random sample of 3% of the undergraduate students is selected’ => n₁ < n₂ since N₁ < N₂…………………………........................................................……………………………………… (3)

(1), (2) and (3) =>sampling variability of sample proportion, p_1cap > sampling variability of sample proportion, p_2cap.

Thus, Option B ANSWER

[Going beyond to make it easier to grasp,

University	N	n	pcap	Variability
W	9000	9000 x 0.03 = 270	0.2	(0.2 x 0.8/270) = 0.0006
M	27000	27000 x 0.03 = 810	0.2	(0.2 x 0.8/810) = 0.0002
DONE