Answer:
Now we just take square root on both sides of the interval and we got:
Step-by-step explanation:
Data given and notation
34,59,61,71,59
We can calculate the sample standard deviation with the following formula:
![s^2= \frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}](https://tex.z-dn.net/?f=%20s%5E2%3D%20%5Cfrac%7B%5Csum_%7Bi%3D1%7D%5En%20%28X_i%20-%5Cbar%20X%29%5E2%7D%7Bn-1%7D)
![s= \sqrt{s^2}](https://tex.z-dn.net/?f=%20s%3D%20%5Csqrt%7Bs%5E2%7D)
s=12.021 represent the sample standard deviation
represent the sample mean
n=5 the sample size
Confidence=99% or 0.99
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^2 \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%5E2%20%5Cleq%20%5Cfrac%7B%28n-1%29s%5E2%7D%7B%5Cchi%5E2_%7B1-%5Calpha%2F2%7D%7D)
The next step would be calculate the critical values. First we need to calculate the degrees of freedom given by:
![df=n-1=5-1=4](https://tex.z-dn.net/?f=df%3Dn-1%3D5-1%3D4)
Since the Confidence is 0.99 or 99%, the value of
and
, and we can use excel, a calculator or a table to find the critical values.
The excel commands would be: "=CHISQ.INV(0.005,4)" "=CHISQ.INV(1-0.005,4)". so for this case the critical values are:
![\chi^2_{\alpha/2}=14.860](https://tex.z-dn.net/?f=%5Cchi%5E2_%7B%5Calpha%2F2%7D%3D14.860)
![\chi^2_{1- \alpha/2}=0.207](https://tex.z-dn.net/?f=%5Cchi%5E2_%7B1-%20%5Calpha%2F2%7D%3D0.207)
And replacing into the formula for the interval we got:
Now we just take square root on both sides of the interval and we got: