Question

The paper "Ladies First?" A Field Study of Discrimination in Coffee Shops" (Applied Economics [2008]: 1-19) describes a study in which researchers observed wait times in coffee shops in Boston. Both wait time and gender of the customer were observed. The mean wait time for a sample of 145 male customers was 85.2 seconds. The mean wait time for a sample of 141 female customers was 113.7 seconds. The sample standard deviations (estimated from graphs in the paper) were 50 seconds for the sample of males and 75 seconds for the sample of females. Suppose that these two samples are representative of the populations of wait times for female coffee shop customers and for male coffee shop customers. Is there convincing evidence that the mean wait time differs for males and females? Test the relevant hypotheses using a significance level of 0.05

Short answer

The solution involves calculating the z-test statistic and comparing it with the critical z-value to see if the null hypothesis can be rejected, thus providing evidence of a difference in mean wait times between males and females.

Step-by-step solution

Formulate the null hypothesis and the alternate hypothesis

The null hypothesis (H0) is that the mean wait time of males(M1) and females(M2) is the same: \( H0: M1 = M2 \).\nThe alternate hypothesis (HA) is that the mean wait times differ: \( HA: M1 \neq M2 \).

Calculate the test statistic and the critical z-score

The test statistic z can be calculated using the formula: \[ Z = \frac{{(M1 - M2) - D0}}{{\sqrt{{\frac{{S1^2}}{{n1}} + \frac{{S2^2}}{{n2}}}}}} \] \nWhere M1 and M2 are the sample means, D0 is the hypothesized difference, S1 and S2 are the standard deviations, and n1 and n2 are the number of observations. Here, M1 is 85.2, M2 is 113.7, D0 is 0, S1 = 50, n1 = 145, S2 = 75, n2 =141. The critical z value at a significance level of 0.05 for a two-tailed test is \(\pm 1.96\).

Conduct the hypothesis test

If the calculated z-score falls within the critical region (-1.96 to 1.96), reject the null hypothesis. Otherwise, we do not have enough evidence to reject the null hypothesis.

Interpret the Result

If the null hypothesis is rejected, there is significant evidence to suggest the mean wait time differs for males and females. If not, there is insufficient evidence to suggest the mean wait times differ.