Question

Do out-of-state motorists violate the speed limit more frequently than in-state motorists? This vital question was addressed by the highway patrol in a large eastern state. A random sample of the speeds of 2,500 selected cars was categorized according to whether the car was registered in the state or in some other state and whether or not the car was violating the speed limit. The data follow.

In state speeding cars: 521

Out of state speeding cars: 328

In state not speeding cars: 1141

Out of state not speeding cars: 510

a.) Do these data provide enough evidence to support the highway patrol's claim at the 5% significance level? Your conclusion must be in terms of the P-Value. Show all necessary work.

b). What type of error is possible and describe this error in terms of the problem?

c). Estimate the difference in the actual percentage of In State and Out of State speed limit violators using a 95% confidence interval. Show all necessary work. Using this interval estimation, is there sufficient evidence to support the highway patrol's claim? Explain Carefully.

d). Carefully interpret the confidence interval estimation.

Use non-parametrics/Mann-Whitney test if the problem calls for it

Answer 1

a)

Let population 1 shows In state cars and population 2 shows out state cars.

Number of In state cars: n1 = 521 +1141 = 1662

Number of out of state cars: n2 = 328 + 510 = 838

Since sample sizes are large and number of successes are greater than 5 so normality can be assumed. That is assumption for z test has been fulfilled.

Conclusion: There data provides enough evidence to support the highway patrol's claim at the 5% significance level.

(b)

In part (a) we reject the null hypothesis so type I error is possible.

Type I error is the probability of rejecting the true null hypothesis. That is we incorrectly conclude that out-of-state motorists violate the speed limit more frequently than in-state motorists while actually out-of-state motorists do not violate the speed limit more frequently than in-state motorists.

(c)

Conclusion: Since confidence interval does not contain zero and all values of confidence intervals are negative so confidence interval provides enough evidence to support the highway patrol's claim at the 5% significance level.

(d)

Interpretation: The confidence interval says that we are 95% confident that true  difference in the actual percentage of In State and Out of State speed limit violators lie between the interval (-0.1177, -0.0381).