A randomized controlled study was designed to test whether regular drinking of juice is related to the recurrence of urinary tract infections (UTIs) in women. A group of randomly selected female patients with a urinary tract infection were treated with an antibiotic and then randomly assigned to one of 3 groups. One group drank cranberry juice concentrate daily for six months; another group took a juice containing Lactobacillus daily for six months; the last group drank neither cranberry nor Lactobacillus for six months. After six months, the number of women in each group with recurring symptomatic UTI was recorded. Here are the results. Take = 0.05.

Treatment	Recurring UTI	No New UTI
Cranberry Juice	8	42
Lactobacillus Juice	20	30
No Juice	18	32

Based on this table, answer the following questions.

(a) Write the null and alternative hypotheses for the test.

(b) Construct the expected contingency table under the assumption of independence between juice drinking and recurrence of UTI.

(c) From the table, that you just constructed in Part (b), report the expected frequency for number of female patients in the sample who drank Cranberry Juice AND did not exhibit new UTI under the assumption of independence of UTI recurrence and juice drinking.

(d) Is the necessary condition for doing the ² test satisfied here?

(e) Now do the test, compute the p-value and conclude.