Question

The U.S. Census Bureau conducts a study to determine the time needed to complete the short form. The Bureau surveys 200 people. The sample mean is 8.2 minutes. There is a known standard deviations of 2.2 minutes. The population distribution is assumed to be normal

A) Construct a 90% confidence interval for the population mean time to complete the forms

B) calculate the error bound

C) If the Census wants to increase its level of confidence and keep the error bound the same by taking another survey, what changes should it make?

D) If the Census did another survey, kept the error bound the same, and surveyed only 50 people instead of 200, what would happen to the level of confidence? Why?

Answer 1

a)

sample mean, xbar = 8.2
sample standard deviation, σ = 2.2
sample size, n = 200

Given CI level is 90%, hence α = 1 - 0.9 = 0.1
α/2 = 0.1/2 = 0.05, Zc = Z(α/2) = 1.64

CI = (xbar - Zc * s/sqrt(n) , xbar + Zc * s/sqrt(n))
CI = (8.2 - 1.64 * 2.2/sqrt(200) , 8.2 + 1.64 * 2.2/sqrt(200))
CI = (7.94 , 8.46)

b)
ME = zc * σ/sqrt(n)
ME = 1.64 * 2.2/sqrt(200)
ME = 0.26

c)
As the level of confidence increases, the interval becomes larger

d)

As the sample size decreases the confidence interval becomes increases