Question

Find the necessary confidence interval for a population mean μ for the following values. (Round your answers to two decimal places.)

α = 0.05, n = 83, x = 66.2, s² = 2.38

to

Interpret the interval that you have constructed.

There is a 95% chance that an individual sample mean will fall within the interval.There is a 5% chance that an individual sample mean will fall within the interval.    In repeated sampling, 95% of all intervals constructed in this manner will enclose the population mean.95% of all values will fall within the interval.In repeated sampling, 5% of all intervals constructed in this manner will enclose the population mean.

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Answer 1

Solution:

α = 0.05, n = 83, x = 66.2, s² = 2.38

s = 2.38 = 1.54272486205

Note that, Population standard deviation() is unknown. So we use t distribution.

Our aim is to construct 95% confidence interval.

c = 0.95

= 1- c = 1- 0.95 = 0.05

  /2 = 0.05 2 = 0.025

Also, d.f = n - 1 = 83 - 1 = 82

    =    =  _0.025,82 = 1.989

( use t table or t calculator to find this value..)

The margin of error is given by

E =  /2,d.f. * ( / n )

=   1.989 * (1.54272486205 / 83)

=  0.34

Now , confidence interval for mean() is given by:

( - E ) <   <  ( + E)

(66.2 - 0.34)   <   <  (66.2 + 0.34)

65.86 <   < 66.54

Required interval is (65.86 , 66.54 )

Interpret the interval that you have constructed.

Answer:

In repeated sampling, 95% of all intervals constructed in this manner will enclose the population mean.