Question: A food research laboratory conducts a series of taste tests to compare two sauce recipes, A and B. Customers are presented with the sauces in random order. Of the 125 customers who tried sauce A first, 74 expressed a preference for sauce A. Of the 132 customers who tried sauce B first, 52 expressed a preference for sauce A.

(a) Looking at the data in total, calculate a 95% confidence interval for the overall proportion of customers who prefer sauce A. Does this confidence interval contain 0.5? What conclusion could you draw from this.

(b) Test the hypothesis that the proportion of customers who prefer sauce A is the same for those who try sauce A first, as for those who try sauce B first. State your null and alternative hypotheses clearly, your test statistic, p-value, and conclusion.