Question

1) Find the critical value t (a/2) needed to construct a confidence interval of the given level with the given sample size. Round the answers to three decimal places

a) 98% sample size 11

critical value-

b) for level 95% and sample size 25

critical value-

c)for level 99% and sample size 14

2) a sample of size n equals 45 has a sample mean X equals 56.9 and sample standard deviation s equals 9.4.

construct a 99% confidence interval for the population mean U. Round the answer to one decimal place

A 99% confidence interval for the population mean is ____ < U < _____

Answer 1

1)

a)

sample size, n = 11
degrees of freedom, df = n - 1 = 10

Given CI level is 98%, hence α = 1 - 0.98 = 0.02
α/2 = 0.02/2 = 0.01, tc = t(α/2, df) = 2.764

b)

sample size, n = 25
degrees of freedom, df = n - 1 = 24

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, tc = t(α/2, df) = 2.064

c)

sample size, n = 14
degrees of freedom, df = n - 1 = 13

Given CI level is 99%, hence α = 1 - 0.99 = 0.01
α/2 = 0.01/2 = 0.005, tc = t(α/2, df) = 3.012

2)

sample mean, xbar = 56.9
sample standard deviation, s = 9.4
sample size, n = 45
degrees of freedom, df = n - 1 = 44

Given CI level is 99%, hence α = 1 - 0.99 = 0.01
α/2 = 0.01/2 = 0.005, tc = t(α/2, df) = 2.692

ME = tc * s/sqrt(n)
ME = 2.692 * 9.4/sqrt(45)
ME = 3.772

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (56.9 - 2.692 * 9.4/sqrt(45) , 56.9 + 2.692 * 9.4/sqrt(45))
CI = (53.1 , 60.7)