estimating mu when sigma is unknown

Question

If n=20, ¯xx¯(x-bar)=40, and s=11, construct a confidence interval at a 80% confidence level. Assume the data came from a normally distributed population.

Give your answers to one decimal place

Answer 1

z scores for 80% is plus and minus 1.28

You can find 1.28 from a z table or a calculator with statistics functions

the z table is for a normal distribution

standard deviation times z score =

11(1.28) = 14.08

mean plus 11(1.28) =

40 + 14.08 = 54.08

mean minus 11(1.28)=

40 - 14.08 = 25.92

but that's equivalent to a sample size of 1, if you're just dealing with a sample and not the population

the 80% confidence interval is (25.92, 54.08)

that's true if you had the population mean = 40

but if it's just the sample mean then

divide 11(1.28) by the square root of the sample size or sqr(20)

14.08/sqr20 = 14.08/2sqr5 = 7.04/sqr5 = 7.04sqr5/5 = 1.41sqr5 = 1.41(2.2) = 3.1

40+3.1= 43.1

40-3.1= 36.9

(36.9,43.1)

You can find a confidence interval calculator on line and plug in the numbers 40, 11, 20 and 80 for the mean, standard deviation, sample size and confidence level and it spits out (36.9,43.1)