Question

A random sample of 23 items is drawn from a population whose standard deviation is unknown. The sample mean is x=820 and the sample standard deviation is s=25. (Round all answers to 3 decimal places)

(a) Construct an interval estimate of u with 99% confidence

(b) Construct an interval estimate of u with 99% confidence, assuming that s=50

(c) Construct an interval estimate of u with 99% confidence, assuming that s=100

Answer 1

a)

99% Confidence Interval
X̅ ± t(α/2, n-1) S/√(n)
t(α/2, n-1) = t(0.01 /2, 23- 1 ) = 2.819
820 ± t(0.01/2, 23 -1) * 25/√(23)
Lower Limit = 820 - t(0.01/2, 23 -1) 25/√(23)
Lower Limit = 805.305
Upper Limit = 820 + t(0.01/2, 23 -1) 25/√(23)
Upper Limit = 834.695
99% Confidence interval is ( 805.305 , 834.695 )

b)

99% Confidence Interval
X̅ ± t(α/2, n-1) S/√(n)
t(α/2, n-1) = t(0.01 /2, 23- 1 ) = 2.819
820 ± t(0.01/2, 23 -1) * 50/√(23)
Lower Limit = 820 - t(0.01/2, 23 -1) 50/√(23)
Lower Limit = 790.610
Upper Limit = 820 + t(0.01/2, 23 -1) 50/√(23)
Upper Limit = 849.390
99% Confidence interval is ( 790.610 , 849.390 )

c)

99% Confidence Interval
X̅ ± t(α/2, n-1) S/√(n)
t(α/2, n-1) = t(0.01 /2, 23- 1 ) = 2.819
820 ± t(0.01/2, 23 -1) * 100/√(23)
Lower Limit = 820 - t(0.01/2, 23 -1) 100/√(23)
Lower Limit = 761.220
Upper Limit = 820 + t(0.01/2, 23 -1) 100/√(23)
Upper Limit = 878.780
99% Confidence interval is ( 761.220 , 878.780 )