Question

A study of tipping behaviours examined the relationship between the colour of the shirt worn by the server and whether or not the customer left a tip.

Group 1: In a random sample of 65 customers served by a server wearing a red shirt, 42 left a tip.

Group 2: In a random sample of 352 customers served by a server wearing a non-red shirt, 179 left a tip.

At the 5% level of significance, is there a difference in tip leaving between red-shirted servers and non-red-shirted servers?

Note: Be sure to use at least four decimal places in all intermediate calculations.

Answer 1

p1cap = X1/N1 = 42/65 = 0.6462
p1cap = X2/N2 = 179/352 = 0.5085
pcap = (X1 + X2)/(N1 + N2) = (42+179)/(65+352) = 0.53

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p1 = p2
Alternate Hypothesis, Ha: p1 ≠ p2

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.6462-0.5085)/sqrt(0.53*(1-0.53)*(1/65 + 1/352))
z = 2.04

P-value Approach
P-value = 0.0414
As P-value < 0.05, reject the null hypothesis.

The two proportions are not equal, this means there is significant difference between two proportions