10.30 In Dallas, some fire trucks were painted yellow (instead of red) to heighten their visibility. During a test period, the fleet of red fire trucks made 153,348 runs and had 20 accidents, while the fleet of yellow fire trucks made 135,035 runs and had 4 accidents. At α = .01, did the yellow fire trucks have a significantly lower accident rate? (a) State the hypotheses. (b) State the decision rule and sketch it. (c) Find the sample proportions and z test statistic. (d) Make a decision. (e) Find the p-value and interpret it. (f ) If statistically significant, do you think the difference is large enough to be important? If so, to whom, and why? (g) Is the normality assumption fulfilled? Explain.
Source: The Wall Street Journal, June 26, 1995, p. B1.
Accident Rate for Dallas Fire Trucks
Statistic Red Fire Trucks Yellow Fire Trucks
Number of accidents x1 = 20 accidents x2 = 4 accidents
Number of fire runs n1 = 153,348 runs n2 = 135,035 runs
(a) H0: p1 = p2 vs. Ha: p1 > p2(b) Reject the null hypothesis if z > z0.01 = 2.3263
(c) Sample proportions and z test statistic:
p1 / p2 / pc
0.0001304 / 0.0000286 / 8.27323E-05 / p (as decimal)
0.0001304 / 0.0000286 / 8.27323E-05 / p (as fraction)
19.9965792 / 3.862001 / 23.8585802 / X
153348 / 135035 / 288383 / n
0.0001018 / Difference
0 / hypothesized difference
3.39424E-05 / std. error
2.999201734 / Z
(d) Since z > z0.01 (2.999 > 2.3263), we reject H0
(e) p-value = 0.0014 Þ The probability that z can be as extreme as the test statistic value is 0.0014.
(f) Yes (because the p-value is very low compared to alpha = 0.01). This means the yellow fire trucks have a significantly lower accident rate than the red fire trucks.
(g) Since the sample sizes are very large, the normality assumptions are fulfilled.