Question

# a random sample of 11 items is drawn from a population whose standard deviation is unknown....

a random sample of 11 items is drawn from a population whose standard deviation is unknown. The sample mean is x= 920 and the sample standard deviation is s = 25. Use Appendix D to find the values of Studengs t.

a) Construct an interval estimate of u with 95% confidence.
b) Construct an interval estimate of u with 95% confidence, assuming tha s=50.
c) Construct an interval estimate of u with 95% confidence, assuming that s= 100

Solution:

Given data is ,

random sample n = 11

sample mean (x) = 920

standard deviation (s) = 25

a) 95% Confidence Interval
X̅ ± t(α/2, n-1) S/√(n)
t(α/2, n-1) = t(0.05 /2, 11- 1 ) = 2.228
920 ± t(0.05/2, 11 -1) * 25/√(11)
Lower Limit = 920 - t(0.05/2, 11 -1) 25/√(11)
Lower Limit = 903.205
Upper Limit = 920 + t(0.05/2, 11 -1) 25/√(11)
Upper Limit = 936.795
95% Confidence interval is ( 903.205 , 936.795 )

b) 95% Confidence Interval is
X̅ ± t(α/2, n-1) S/√(n)
t(α/2, n-1) = t(0.05 /2, 11- 1 ) = 2.228
920 ± t(0.05/2, 11 -1) * 50/√(11)
Lower Limit = 920 - t(0.05/2, 11 -1) 50/√(11)
Lower Limit = 886.410
Upper Limit = 920 + t(0.05/2, 11 -1) 50/√(11)
Upper Limit = 953.590
95% Confidence interval is ( 886.410 , 953.590 )

c) 95% Confidence Interval is
X̅ ± t(α/2, n-1) S/√(n)
t(α/2, n-1) = t(0.05 /2, 11- 1 ) = 2.228
920 ± t(0.05/2, 11 -1) * 100/√(11)
Lower Limit = 920 - t(0.05/2, 11 -1) 100/√(11)
Lower Limit = 852.819
Upper Limit = 920 + t(0.05/2, 11 -1) 100/√(11)
Upper Limit = 987.181
95% Confidence interval is ( 852.819 , 987.181 )

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