Answer:
Option D - ![[0.12](https://tex.z-dn.net/?f=%5B0.12%3C%5Csigma%3C0.24%5D)
Step-by-step explanation:
Given : The manufacturer measures 19 randomly selected dowels and finds the standard deviation of the sample to be s=0.16.
To find : The 95% confidence interval for the population standard deviation sigma?
Solution :
Number of sample n=19
The degree of freedom is Df=n-1=19-1=18
The standard deviation of the sample is s=0.16
Applying chi-square table to find critical value,
Upper critical value of
is ![UC=\chi(\frac{0.05}{2},18) = 31.5264](https://tex.z-dn.net/?f=UC%3D%5Cchi%28%5Cfrac%7B0.05%7D%7B2%7D%2C18%29%20%3D%2031.5264)
Lower critical value of
is
![LC=\chi(1-\frac{0.05}{2},18) = 8.2307](https://tex.z-dn.net/?f=LC%3D%5Cchi%281-%5Cfrac%7B0.05%7D%7B2%7D%2C18%29%20%3D%208.2307)
Lower limit of the 95% confidence interval for the population variance
![L=\frac{(df)\times (s^2)}{UC}](https://tex.z-dn.net/?f=L%3D%5Cfrac%7B%28df%29%5Ctimes%20%28s%5E2%29%7D%7BUC%7D)
![L=\frac{18\times (0.16^2)}{31.5264}](https://tex.z-dn.net/?f=L%3D%5Cfrac%7B18%5Ctimes%20%280.16%5E2%29%7D%7B31.5264%7D)
![L=\frac{18\times0.0256}{31.5264}](https://tex.z-dn.net/?f=L%3D%5Cfrac%7B18%5Ctimes0.0256%7D%7B31.5264%7D)
![L=\frac{0.4608}{31.5264}](https://tex.z-dn.net/?f=L%3D%5Cfrac%7B0.4608%7D%7B31.5264%7D)
![L=0.0146](https://tex.z-dn.net/?f=L%3D0.0146)
Upper limit of the 95% confidence interval for the population variance
![U=\frac{(df)\times(s^2)}{LC}](https://tex.z-dn.net/?f=U%3D%5Cfrac%7B%28df%29%5Ctimes%28s%5E2%29%7D%7BLC%7D)
![U=\frac{18\times (0.16^2)}{8.2307}](https://tex.z-dn.net/?f=U%3D%5Cfrac%7B18%5Ctimes%20%280.16%5E2%29%7D%7B8.2307%7D)
![U=\frac{18\times0.0256}{8.2307}](https://tex.z-dn.net/?f=U%3D%5Cfrac%7B18%5Ctimes0.0256%7D%7B8.2307%7D)
![U=\frac{0.4608}{8.2307}](https://tex.z-dn.net/?f=U%3D%5Cfrac%7B0.4608%7D%7B8.2307%7D)
![U=0.0559](https://tex.z-dn.net/?f=U%3D0.0559)
So, The 95% confidence interval for the population variance is [0.0146, 0.0560]
Now, The 95% confidence interval for the population standard deviation is
![[\sqrt{0.0146}](https://tex.z-dn.net/?f=%5B%5Csqrt%7B0.0146%7D%3C%5Csigma%3C%5Csqrt%7B0.0560%7D%5D)
![[0.1208](https://tex.z-dn.net/?f=%5B0.1208%3C%5Csigma%3C0.2366%5D)
or ![[0.12](https://tex.z-dn.net/?f=%5B0.12%3C%5Csigma%3C0.24%5D)
Therefore, Option D is correct.
The 95% confidence interval for the population standard deviation is ![[0.12](https://tex.z-dn.net/?f=%5B0.12%3C%5Csigma%3C0.24%5D)