Question

The power takeoff driveline on tractors used in agriculture is a potentially serious hazard to operators of farm equipment. The driveline is covered by a shield in new tractors, but for a variety of reasons, the shield is often missing on older tractors. Two types of shields are the bolt-on and the flip-up. It was believed that the bolt-on shield was perceived as a nuisance by the operators and deliberately removed, but the flip-up shield is easily lifted for inspection and maintenance and may be left in place. In a study initiated by the U.S. National Safety Council, random samples of older tractors with both types of shields were taken to see what proportion of shields were removed. Of $183$tractors designed to have bolt-on shields, 35 had been removed. Of the 136 tractors with flip-up shields, 15 were removed. We wish to perform a test of $H_0:p_b-p_i$f versus $H_a:pb\ne p_f$ where pb and pf are the proportions of all tractors with the bolt-on and flip-up shields removed, respectively. Which of the following conditions for performing the appropriate significance test is definitely not satisfied in this case?

(a) Both populations are Normally distributed.

(b) The data come from two independent samples.

(c) Both samples were chosen at random.

(d) The counts of successes and failures are large enough to use Normal calculations.

(e) Both populations are at least 10 times the corresponding sample sizes

Short answer

The correct answer is (a) Both populations are Normally distributed.

Step-by-step solution

Given Information

The power takeoff driveline on tractors used in agriculture is a potentially serious hazard to operators of farm equipment.

Explanation

Conditions for performing a two-sample $z$-test: Random, Normal and Independent.

Random: Satisfied, because the samples have been given to be random samples.

Normal: Satisfied, because the number of successes $(35,15)$and failures localid="1650367462435" $183-35=148,$$136-15=121$are at least $10$.

Independent: Satisfied, because the sample sizes are less than $10\%$of the population size.

Thus all except for (a) has been satisfied. (a) cannot be satisfied, because the number of successes is not close to half the sample size and thus the distribution is not symmetric nor normal.