Question

State police believe that 70% of the drivers traveling on a major interstate highway exceed the speed limit. They plan to set up a radar trap and check the speeds of 80 cars. Taking a random sample of eighty cars is considered sufficiently large for the distribution of the sample proportion to be normally distributed. Would a random sample of 50 cars have been large enough and why?

Answer 1

With n = 50, both conditions(np > 5, n(1-p) > 5) are satisfied, so a random sample of 50 cars would have been large enough.

Explanation:

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to the normal distribution if:

np > 5, n(1-p) > 5

70% of the drivers traveling on a major interstate highway exceed the speed limit.

So p = 0.7.

Would a random sample of 50 cars have been large enough and why?

With n = 50

np = 50*0.7 = 35 > 5

n(1-p) = 50*0.3 = 15 > 5

With n = 50, both conditions(np > 5, n(1-p) > 5) are satisfied, so a random sample of 50 cars would have been large enough.