Question

A random sample of 23 items is drawn from a population whose standard deviation is unknown. The sample mean is x¯ = 840 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 μ with 98% confidence. (Round your answers to 3 decimal places.)
  
The 98% confidence interval is from __to__

(b) Construct an interval estimate of μ with 98% confidence, assuming that s = 30. (Round your answers to 3 decimal places.)
  
The 98% confidence interval is from __to__

(c) Construct an interval estimate of μ with 98% confidence, assuming that s = 60. (Round your answers to 3 decimal places.)
  
The 98% confidence interval is from __to__

(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 = 840
sample standard deviation, s = 15
sample size, n = 23
degrees of freedom, df = n - 1 = 22

Given CI level is 98%, hence α = 1 - 0.98 = 0.02
α/2 = 0.02/2 = 0.01, tc = t(α/2, df) = 2.51

ME = tc * s/sqrt(n)
ME = 2.51 * 15/sqrt(23)
ME = 7.851

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (840 - 2.51 * 15/sqrt(23) , 840 + 2.51 * 15/sqrt(23))
CI = (832.149 , 847.851)

b)

sample mean, xbar = 840
sample standard deviation, s = 30
sample size, n = 23
degrees of freedom, df = n - 1 = 22

Given CI level is 98%, hence α = 1 - 0.98 = 0.02
α/2 = 0.02/2 = 0.01, tc = t(α/2, df) = 2.51

ME = tc * s/sqrt(n)
ME = 2.51 * 30/sqrt(23)
ME = 15.701

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (840 - 2.51 * 30/sqrt(23) , 840 + 2.51 * 30/sqrt(23))
CI = (824.299 , 855.701)

c)
sample mean, xbar = 840
sample standard deviation, s = 60
sample size, n = 23
degrees of freedom, df = n - 1 = 22

Given CI level is 98%, hence α = 1 - 0.98 = 0.02
α/2 = 0.02/2 = 0.01, tc = t(α/2, df) = 2.51

ME = tc * s/sqrt(n)
ME = 2.51 * 60/sqrt(23)
ME = 31.402

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (840 - 2.51 * 60/sqrt(23) , 840 + 2.51 * 60/sqrt(23))
CI = (808.598 , 871.402)

d)

The interval gets wider as s increases.