Question

Cesar wants to know what the expected weight of a loaf of bread for quality control purposes. Based on a sample of 14 loaves of bread, he finds a sample average of 400 grams with a sample standard deviation of 15 grams.

• What’s the confidence interval at 95%?  What is it at 99.9%?
• What would the confidence intervals (for 95% and 99.9%) be if the sample average was based on 140 loaves instead of 14?

Answer 1

a)

sample mean, xbar = 400
sample standard deviation, s = 15
sample size, n = 14
degrees of freedom, df = n - 1 = 13

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, tc = t(α/2, df) = 2.16

ME = tc * s/sqrt(n)
ME = 2.16 * 15/sqrt(14)
ME = 8.659

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (400 - 2.16 * 15/sqrt(14) , 400 + 2.16 * 15/sqrt(14))
CI = (391.3 , 408.7)

sample mean, xbar = 400
sample standard deviation, s = 15
sample size, n = 14
degrees of freedom, df = n - 1 = 13

Given CI level is 99.9%, hence α = 1 - 0.999 = 0.001
α/2 = 0.001/2 = 0.0005, tc = t(α/2, df) = 4.221

ME = tc * s/sqrt(n)
ME = 4.221 * 15/sqrt(14)
ME = 16.922

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (400 - 4.221 * 15/sqrt(14) , 400 + 4.221 * 15/sqrt(14))
CI = (383.1 , 416.9)

b)

sample mean, xbar = 400
sample standard deviation, s = 15
sample size, n = 140
degrees of freedom, df = n - 1 = 139

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, tc = t(α/2, df) = 1.977

ME = tc * s/sqrt(n)
ME = 1.977 * 15/sqrt(140)
ME = 2.506

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (400 - 1.977 * 15/sqrt(140) , 400 + 1.977 * 15/sqrt(140))
CI = (397.5 , 402.5)

sample mean, xbar = 400
sample standard deviation, s = 15
sample size, n = 140
degrees of freedom, df = n - 1 = 139

Given CI level is 99.9%, hence α = 1 - 0.999 = 0.001
α/2 = 0.001/2 = 0.0005, tc = t(α/2, df) = 3.362

ME = tc * s/sqrt(n)
ME = 3.362 * 15/sqrt(140)
ME = 4.262

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (400 - 3.362 * 15/sqrt(140) , 400 + 3.362 * 15/sqrt(140))
CI = (395.7 , 404.3)