Question

3) A state policeman has a theory that people who drive red cars are more likely to drive too fast. On his day off, he borrows one of the department’s radar guns, parks his car in a rest area, and measures the proportion of red cars and non-red cars that are driving too fast (he decided ahead of time to define "driving too fast" as exceeding the speed limit by more than 5 miles per hour). To produce a random sample, he rolls a die and only includes a car in his sample if he rolls a 5 or a 6. He finds that 18 out of 28 red cars are driving too fast and 75 of 205 other cars are driving too fast. Is this convincing evidence that people who drive red cars are more likely to drive too fast, as the policeman has defined it?

Answer 1

Yes, it is convincing evidence to conclude that the proportion of red cars that drive too fast on this

highway is greater than the proportion of non-red cars that drive too fast.

Explanation:

From the, we wish to first test;

H0; P_r - P_o = 0

And; H0; P_r - P_o > 0

Where; P_r and P_o are the proportion of red cars and other cars, respectively, who are driving too fast.

We will use a significance level of a = 0.05.

Thus;

The procedure is a two-sample z-test for the difference of proportions.

For, Random Conditions: The policemen chose cars randomly by rolling a die.

10%: We can safely assume that the number of cars driving past the rest area is essentially infinite, so the 10% restriction does not apply.

Large counts: The number of successes and failures in the two groups are 18, 10, 75, and 130—all of which are at least 10.

So, P_r = 18/28 = 0.64

P_o = 75/205 = 0.37

P_c = (18 + 75)/(28 + 205) = 0.4

Thus:

z = [(0.64 - 0.37) - 0]/√[[(0.4 x 0.6)/28] + [(0.4 x 0.6)/205]]

z = 2.73

From the one tailed z-score calculator online, I got P value = 0.003167

Thus, the P-value of 0.0032 is less than a = 0.05, so we reject H0. We

have sufficient evidence to conclude that the proportion of red cars that drive too fast on this

highway is greater than the proportion of non-red cars that drive too fast.