Is the water on your airline flight safe to drink? It is not feasible to analyze the water on every flight, so sampling is necessary. In August and September 2004, the Environmental Protection Agency (EPA) found bacterial contamination in water samples from the lavatories and galley water taps on 20 of 158 randomly selected U.S. flights. Alarmed by the data, the EPA ordered sanitation improvements, and then tested water samples again in November and December 2004. In the second sample, bacterial contamination was found in 29 of 169 randomly sampled flights. (a) Use a left- tailed test at α = .05 to check whether the percent of all flights with contaminated water was lower in the first sample. (b) Find the p-value. (c) Discuss the question of significance versus importance in this specific application. (d) Test whether normality may be assumed.
(a) H0: The contamination is the same in both the samples, that is, p1 = p2vs.Ha: The contamination is lower in the first sample, that is, p1 < p2
Hypothesis test for two independent proportions:
= 0.05, Critical z- value for a left-tailed test = -1.645
Decision Rule: Reject H0 if the z- value for the test < -1.645
n1 = 158, p1 = 20/158 = 0.1266, n2 = 169, p2 = 29/169 = 0.1716
p1 – p2 = -0.045
p = (n1p1 + n2p2)/(n1 + n2) = (20 + 29)/(158 + 169) = 0.1498
q = 1 – p = 0.8502
SE = [pq * (n1 + n2)/(n1n2)] = [0.1498 * 0.8502 * 327/(158 * 169)] = 0.0395
z = (p1 – p2)/SE = -0.045/0.0395 = -1.139
Since -1.139 > -1.645, we fail to reject H0
Conclusion: There is no statistical basis to say that the contamination is lower in the first sample
(b) p- value for z = -1.139 is 0.1274
(c) This hypothesis test has indicated that the contamination levels appear to have remained the same before and after the EPA’s sanitation improvement efforts.
(d) Since the sample sizes are greater than 30, normality may be assumed.