Question

A college student is curious about undergraduate ownership of iPhones. She thinks that women might be more likely to own iPhones than men. To study this question, she gets permission from the human subjects committee to take a survey of students, then, with help from the Registrar, gets a list of SIDs of all enrolled undergraduate students. She shuffles the SIDs into a random order, then selects the first 34 SIDs from the shuffled list. The registrar sends email to those selected students asking whether they own an iPhone. There are 22,225 undergraduates enrolled at the college in the semester in question, of whom 12,052 are female and 10,173 are male. All 34 of the students in the sample survey respond. It turns out that 18 of the students are female and 16 are male. Fourteen of the female students and twelve of the male students in the sample own iPhones.

Suppose that gender has nothing to do with ownership of iPhones. Then, given that the number of female students in the sample is 18 and that the total number of owners of iPhones in the sample is 26, the number of female owners of iPhones is like the number of tickets labeled 1 in a simple random sample of size 18 from a box that contains 34 tickets of which 26 are labeled 1 and the rest are labeled 0. 38.

1) In this scenario, if the null hypothesis is true, the expected number of iPhone owners among female students is closest to:

(a) 13.3518 (b) 12.9388 (c) 15.0035 (d) 15.4165 (e) 13.7647 10 39. I

2) In this scenario, if the null hypothesis is true, the standard error of the number of iPhone owners among female students is closest to:

(a) 1.3283 (b) 1.1403 (c) 1.2531 (d) 1.2907 (e) 1.1027 40.

3) In this scenario, the observed number of female owners differs from its expected value under the null hypothesis by (pick the closest answer):

(a) 0.3765 (b) 0.3059 (c) 0.1294 (d) 0.2 (e) 0.2353 41.

4) In this scenario, if the null hypothesis is true, the probability that the number of female owners differs from its expected value by as much or more than observed is closest to:

(a) 10% (b) 30% (c) 50% (d) 70% (e) 90%

Answer 1

Solution

Out of 34 students, 26 owned iPhones. So, proportion of iPhone owners is 26/38 = 0.7647.

Part (1)

If gender has nothing to do with ownership of iPhones, proportion of female iPhone owners must be 0.7647 and hence, if the null hypothesis is true, the expected number of iPhone owners among female students is:

Number of female students x 0.7647 = 13.7647 Option (e) Answer 1

Part (2)

If the null hypothesis is true, the number of iPhone owners among female students is distributed as B(18, 0.7647)

Since standard error of binomial = √{np(1 - p)}

= √(0.7647 x 0.2353 x 18)

= 1.7997 Answer 2

Part (3)

Difference between the observed number of female owners and its expected value under the null hypothesis

= 14 – 13.7647

= 0.2353 Option (e) Answer 3

Part (d)

The required percentage =100 x 0.2353/13.7647

= 17% Answer 4

DONE