Question 388910: A sample of 23 European countries found that the variance of life expectancy was 7.3 years. What is the 95% confidence interval estimate for the variance of life expectancy in Europe?
Answer by haileytucki(390)   (Show Source):
You can put this solution on YOUR website! n = 23
s^2 = 7.3
alpha (due to the 95% C.I.) = 0.05
t- score for 22 degrees of freedom and alpha = 0.05 is t = 2.0739
Standard error, SE = (s^2) * sqrt(2/n) = 7.3 * sqrt(2/23) = 2.1527
See below for how I determined 2.1527:
First, finding the sq.root:
~((2)/(23))
Split the fraction inside the radical into a separate radical expression in the numerator and the denominator. A fraction of roots is equivalent to a root of the fraction.
(~(2))/(~(23))
To rationalize the denominator of a fraction, rewrite the fraction so that the new fraction has the same value as the original and has a rational denominator. The factor to multiply by should be an expression that will eliminate the radical in the denominator. In this case, the expression that will eliminate the radical in the denominator is (~(23))/(~(23)).
(~(2))/(~(23))*(~(23))/(~(23))
Multiply the original expression by a factor of 1 ((~(23))/(~(23))) to eliminate the radical from the denominator.
(~(2)*~(23))/(23)
Simplify the rationalized fraction.
(~((2)(23)))/(23)
Multiply 2 by 23 to get 46.
(~((46)))/(23)
Remove the parentheses around the expression 46.
(~(46))/(23)
The approximate value of ~((2)/(23)) is 0.2948839.
0.2948839
Now, multiplying times 7.3=0.2948839*7.3
Multiply 0.2948839 by 7.3 to get 2.15265247.
2.15265247
Margin of error, E = t * SE = 2.0739 * 2.1517 = 4.4643
LCL = s^2 - E = 7.3 - 4.4643 = 2.8357
UCL = s^2 + E = 7.3 + 4.4643 = 11.7643
Answer= The 95% CI is [2.8357, 11.7643]
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