STATISTICSUse the information below. In this section, you studied the construction of a confidence interval to estimate a population mean when the population is large or infinite. When a population is finite, the formula that determines the standard error of the mean
$$
\sigma _ { \overline { x } }
$$
needs to be adjusted. If N is the size of the population and n is the size of the sample
$$
(\text { where } n \geq 0.05 N)
$$
, then the standard error of the mean is
$$
\sigma _ { \overline { x } } = \frac { \sigma } { \sqrt { n } } \sqrt { \frac { N - n } { N - 1 } }
$$
. The expression
$$
\sqrt { ( N - n ) / ( N - 1 ) }
$$
is called the finite population correction factor. The margin of error is
$$
E = z _ { c } \frac { \sigma } { \sqrt { n } } \sqrt { \frac { N - n } { N - 1 } }
$$
. Use the finite population correction factor to construct each confidence interval for the population mean. (a)
$$
c = 0.99 , \overline { x } = 8.6 , \sigma = 4.9 , N = 200 , n = 25
$$
(b)
$$
c = 0.90 , \overline { x } = 10.9 , \sigma = 2.8 , N = 500 , n = 50
$$
(c)
$$
c = 0.95 , \overline { x } = 40.3 , \sigma = 0.5 , N = 300 , n = 68
$$
(d)
$$
c = 0.80 , \overline { x } = 56.7 , \sigma = 9.8 , N = 400 , n = 36
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