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 boll-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 US 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 $156$tractors with flip-up shields, $15$were removed. We wish to perform a test of $H_0:p_b=p_f$versus $H_a:p_b>p_f$, where $p_b$and $p_f$are the proportions of all the tractors with bolt-on and flip-up shields removed, respectively. Which of the following is not a condition for performing the significance test ?

(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

We are given random samples of old tractors with both types of shields, from a study by the US National Safety Council.

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

Explanation

Random: I'm satisfied because the samples were supplied to me at random.

Normal: Satisfied because there are at least 10 successes $(35,15)$and failures

$(183-35=148,136-15=121)$

Because the sample sizes are fewer than $10\%$of the population size, the independent is satisfied.

With the exception of (a), all requirements have been met. Because the number of successes is not close to half the sample size, (a) cannot be satisfied because the distribution is neither symmetric nor normal.