Question

8. A business wants to know whether two versions of an email offer have the same click-through proportion. It sent a random sample of customers an email. It randomly decided which of the two offers went in each email. In the end, it had the following data. 706 out of the 1281 customers that received the first email, clicked on the offer. 614 out of the 1220 customers that received the second email, clicked on the offer.

(a) What is the sample proportion for group 1? (round to 5 digits after the decimal place)

(b) What is the sample proportion for group 2? (round to 5 digits after the decimal place)

(c) What is the difference in the sample proportions between groups 1 and 2? (round to 5 digits after the decimal place)

(d) What is the standard error for the difference in the sample proportions? (Use σp1−p2 and round to 5 digits after the decimal place.)

(e) What is the estimate of the left end of a 95% confidence interval? (Round to 5 digits after the decimal place.)

(f) What is the estimate of the right end of a 95% confidence interval? (Round to 5 digits after the decimal place.)

(g) Do we reject or not reject the null hypothesis that the population click-through proportions are the same at the .05 level of significance? Reject Not reject

(h) Can we interpret the difference in the population proportions as a causal effect? Yes, it has a causal interpretation. No, it does not have a causal interpretation.

Answer 1

a)

p1cap = X1/N1 = 706/1281 = 0.55113

b)

p1cap = X2/N2 = 614/1220 = 0.50328

c)

difference = 0.55113 - 0.50328 = 0.04785

d)

Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.55113 * (1-0.55113)/1281 + 0.50328*(1-0.50328)/1220)
SE = 0.01995

e)

For 0.95 CI, z-value = 1.96
Confidence Interval,
CI = (p1cap - p2cap - z*SE, p1cap - p2cap + z*SE)
CI = (0.55113 - 0.50328 - 1.96*0.01995, 0.55113 - 0.50328 + 1.96*0.01995)
CI = (0.00875 , 0.08695)

left end = 0.00875

f)

right end = 0.08695

g)
Reject

h)

Yes, it has a causal interpretation