If we know the standard deviation of this population, the confidence interval for the mean can be found as
where "{\\bar X}" - sample mean, "n" - sample size, "\\sigma" - known standard deviation and "{q_{1 - \\frac{\\alpha }{2}}}" is quantile of the standard normal distribution of level "{1 - \\frac{\\alpha }{2}}" . Our significance level is
"\\alpha = 1 - \\frac{C}{{100\\% }} = 1 - \\frac{{60\\% }}{{100\\% }} = 0.4"so we need to find "{q_{1 - \\frac{\\alpha }{2}}} = {q_{1 - \\frac{{0.4}}{2}}} = {q_{0.8}}" quantile. To do so use it's definition
where "\\Phi (x)" is the CDF function of the standard normal distribution. Using the table of this CDF (I used the table from six-sigma-material.com)
We need to find a value of independent variable z such gives "0.8" as result. From the table we see, the value of z lies approximatelty at "z = 0.84" (for it "\\Phi (0.84) = 0.7995"). If we don't need high accuracy we don't use interpolation and just approximately accept
a) Using values we got above with "n=100"
"(\\bar X - {q_{1 - \\frac{\\alpha }{2}}}\\frac{\\sigma }{{\\sqrt n }},\\bar X + {q_{1 - \\frac{\\alpha }{2}}}\\frac{\\sigma }{{\\sqrt n }}) \\approx (10 - 0.84 \\cdot \\frac{5}{{\\sqrt {100} }},10 + 0.84 \\cdot \\frac{5}{{\\sqrt {100} }}) = (9.58,10.42)"b) Do the same with "n=25"
c) Do the same with "n=10"
d)Confidence interval for the mean is increasing with decreasing size of population.
P.S. Note that if we don't know standard deviation for population we need to use other method.
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