in the 2000 census the so called long form received by one of every six households contained 52 quations ranging form your occupation and income all the way to whether you had a bathtub according to the u.s census bureau the mean completion time for the long form is 38 minutes assuming a standard deviation of 5 minutes and a simple random sample of 50 persons who filled out the long form what is the probability that their average time for completion of the form was more than 45 minutes?

Question

in the 2000 census the so called long form received by one of every six households contained 52 quations ranging form your occupation and income all the way to whether you had a bathtub according to the u.s census bureau the mean completion time for the long form is 38 minutes assuming a standard deviation of 5 minutes and a simple random sample of 50 persons who filled out the long form what is the probability that their average time for completion of the form was more than 45 minutes?

Answer 1

We can use the central limit theorem to approximate the sampling distribution of the sample mean completion time, as long as the sample size is sufficiently large. Specifically, we can assume that the sample mean follows a normal distribution with mean μ = 38 minutes and standard deviation σ/√n = 5/√50 = 0.707 minutes, where n = 50 is the sample size.

We want to find the probability that the sample mean is more than 45 minutes, or equivalently, the probability that a standard normal random variable Z = (X̄ - μ) / (σ/√n) is greater than (45 - 38) / 0.707 = 9.90. Using a standard normal table or calculator, we find that this probability is approximately 1.85 x 10^-23.

Therefore, the probability that a simple random sample of 50 persons who filled out the long form had an average completion time of more than 45 minutes is extremely low, less than 1 in a trillion. This suggests that either the sample was unrepresentative or there were some outliers with unusually long completion times.