Answer:
![8.637 \leq \sigma^2 \leq 28.93](https://tex.z-dn.net/?f=%208.637%20%5Cleq%20%5Csigma%5E2%20%5Cleq%2028.93)
![2.939 \leq \sigma \leq 5.379](https://tex.z-dn.net/?f=%202.939%20%5Cleq%20%5Csigma%20%5Cleq%205.379)
Step-by-step explanation:
Data given and notation
s represent the sample standard deviation
represent the sample mean
n=23 the sample size
Confidence=95% or 0.95
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population mean or variance lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
The Chi Square distribution is the distribution of the sum of squared standard normal deviates .
Calculating the confidence interval
The confidence interval for the population variance is given by the following formula:
![\frac{(n-1)s^2}{\chi^2_{\alpha/2}} \leq \sigma \leq \frac{(n-1)s^2}{\chi^2_{1-\alpha/2}}](https://tex.z-dn.net/?f=%5Cfrac%7B%28n-1%29s%5E2%7D%7B%5Cchi%5E2_%7B%5Calpha%2F2%7D%7D%20%5Cleq%20%5Csigma%20%5Cleq%20%5Cfrac%7B%28n-1%29s%5E2%7D%7B%5Cchi%5E2_%7B1-%5Calpha%2F2%7D%7D)
The sample standard deviation for this case was s = 3.8
The next step would be calculate the critical values. First we need to calculate the degrees of freedom given by:
![df=n-1=23-1 =22](https://tex.z-dn.net/?f=df%3Dn-1%3D23-1%20%3D22)
Since the Confidence is 0.95 or 95%, the value of
and
, and we can use excel, a calculator or a tabele to find the critical values.
The excel commands would be: "=CHISQ.INV(0.025,22)" "=CHISQ.INV(0.975,22)". so for this case the critical values are:
![\chi^2_{\alpha/2}=36.78](https://tex.z-dn.net/?f=%5Cchi%5E2_%7B%5Calpha%2F2%7D%3D36.78)
![\chi^2_{1- \alpha/2}=10.98](https://tex.z-dn.net/?f=%5Cchi%5E2_%7B1-%20%5Calpha%2F2%7D%3D10.98)
And replacing into the formula for the interval we got:
![8.637 \leq \sigma^2 \leq 28.93](https://tex.z-dn.net/?f=%208.637%20%5Cleq%20%5Csigma%5E2%20%5Cleq%2028.93)
Now we just take square root on both sides of the interval and we got:
![2.939 \leq \sigma \leq 5.379](https://tex.z-dn.net/?f=%202.939%20%5Cleq%20%5Csigma%20%5Cleq%205.379)