Question 1201399
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sigma = population standard deviation


<font color=red>Goal: estimate sigma</font>


This estimate is written as a confidence interval in the format
<font color=red>L < sigma < U</font>
where,
L = lower boundary
U = upper boundary


s = sample standard deviation, which helps estimate sigma
s = 1.7 is given in the instructions


n = sample size = 15
df = degrees of freedom
df = n-1
df = 15-1
df = 14


Here is an article to check out
<a href = "https://faculty.elgin.edu/dkernler/statistics/ch09/9-3.html">https://faculty.elgin.edu/dkernler/statistics/ch09/9-3.html</a>


Check out example 1 to see how to determine the critical values based on the degrees of freedom (df) and on the confidence level.
Example 1 uses df = 12, but the confidence level used is 95%. Meaning that you'll look at those same columns to get the XL and XR values (except just look at the row df = 14 instead)


If you were to look at the df = 14 row, and those columns mentioned, then you should find
XL = 5.629 = left critical chi-square value
XR = 26.119 = right critical chi-square value
*[illustration Screenshot_87.png]


This would indicate
P(XL < X2 < XR) = 0.95
P(5.629 < X2 < 26.119) = 0.95
The area under the chi-square curve, between 5.629 and 26.119, is roughly 0.95
About 95% of the area under the curve is between 5.629 and 26.119


Side note: A chi-square calculator can compute the left and right critical values needed.


Now we can compute the lower boundary of the confidence interval for sigma.
{{{L = matrix(1,2,lower,boundary)}}}


{{{L = sqrt(((n-1)*s^2)/(XR))}}} 


{{{L = sqrt(((15-1)*(1.7)^2)/(26.119))}}} 


{{{L = 1.24461395614987}}} 


{{{L = 1.245}}} 


Now compute the upper boundary.
{{{U = matrix(1,2,upper,boundary)}}}


{{{U = sqrt(((n-1)*s^2)/(XL))}}} 


{{{U = sqrt(((15-1)*(1.7)^2)/(5.629))}}} 


{{{U = 2.68100309220024}}} 


{{{U = 2.681}}} 


It could be a bit confusing that XR goes for the left or lower boundary, while XL goes for the right or upper boundary.


The confidence interval format
L < sigma < U
is then updated to
<font color=red size=4>1.245 < sigma < 2.681</font>
which is approximate.
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