Question

Question 7

Suppose you and a friend both take samples of college students in order to determine if the proportion of college students who prefer Macbooks to PCs is more than 30%. You find that 36% of your sample prefer Macbooks, while your friend’s sample estimates the percentage is 33%. However your test statistic is 1.32 and your friend calculates her test statistic to be 1.76. Which of the following statements is true?

	A.	Your friend's sample must have been smaller than your sample.
	B.	Your sample must have been smaller than your friend's sample.
	C.	Your sample provides stronger evidence against the null hypothesis because 0.36 is further from 0.30
	D.	Your friend's test statistic is calculated incorrectly. It should be smaller than yours.

Answer 1

Q.7) From given information, the null and the alternative hypotheses are,

$H_0: p = 0.30$

$H_a: p > 0.30$

Test statistic is,

$Z = \frac { \hat p - p}{\sqrt \frac { p * (1-p)}{n}}$

Case 1) your sample proportion is 0.36 and test statistic is, Z = 1.32

For test statistic = 1.32 we need sample size is to be,

$n = \frac {Z^2* p * (1-p)}{ (\hat p - p)^2} =\frac { (1.32)^2 * 0.30*(1-0.30) }{(0.36-0.30)^2} = 101.64$

$n \approx 102$

Case 2) your friends sample proportion is 0.33 and test statistic is, Z = 1.76

For test statistic = 1.76 we need sample size is to be,

$n = \frac {Z^2* p * (1-p)}{ (\hat p - p)^2} =\frac { (1.76)^2 * 0.30*(1-0.30) }{(0.36-0.30)^2} = 722.77$

$n \approx 723$

Here, n = 723 > n = 102

Therefore, true statement is,

B) Your sample must have been smaller than your friend's sample.

Note: If you and your friend take same sample size, ( i.e. n = 102) then your friends test statistic should be smaller than yours. And true statement will be D)