Question

A random sample of 16 items is drawn from a population whose standard deviation is unknown. The sample mean is = 890 and the sample standard deviation is s= 15. Use Appendix D to find the values of Student's t. (a) Construct an interval estimate of u with 95% confidence. (Round your answers to 3 decimal places.) The 95% confidence interval is from O t o D (b) Construct an interval estimate of u with 95% confidence, assuming that s= 30. (Round your answers to 3 decimal places.) The 95% confidence interval is from O t o D (c) Construct an interval estimate of u with 95% confidence, assuming that s= 60. (Round your answers to 3 decimal places.) The 95% confidence interval is from O t o D (d) Describe how the confidence interval changes as s increases. The interval stays the same as s increases. The interval gets wider as s increases. The interval gets narrower as s increases.

Answer 1

a)

sample mean, xbar = 890
sample standard deviation, s = 15
sample size, n = 16
degrees of freedom, df = n - 1 = 15

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, tc = t(α/2, df) = 2.131

ME = tc * s/sqrt(n)
ME = 2.131 * 15/sqrt(16)
ME = 7.991

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (890 - 2.131 * 15/sqrt(16) , 890 + 2.131 * 15/sqrt(16))
CI = (882.009 , 897.991)

b)

sample mean, xbar = 890
sample standard deviation, s = 30
sample size, n = 16
degrees of freedom, df = n - 1 = 15

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, tc = t(α/2, df) = 2.131

ME = tc * s/sqrt(n)
ME = 2.131 * 30/sqrt(16)
ME = 15.983

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (890 - 2.131 * 30/sqrt(16) , 890 + 2.131 * 30/sqrt(16))
CI = (874.018 , 905.983)

c)

sample mean, xbar = 890
sample standard deviation, s = 60
sample size, n = 16
degrees of freedom, df = n - 1 = 15

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, tc = t(α/2, df) = 2.131

ME = tc * s/sqrt(n)
ME = 2.131 * 60/sqrt(16)
ME = 31.965

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (890 - 2.131 * 60/sqrt(16) , 890 + 2.131 * 60/sqrt(16))
CI = (858.035 , 921.965)

d)

The interval gets wider as s increases