Question

To analyze the variation of vitamin supplement tablets, you randomly select and weigh 14 tablets. The results are shown below: 500.000 499.995 500.010 499.997 500.015 499.988 500.000 499.996 500.020 500.002 499.998 499.996 500.003 500.000 a. Assume this sample is taken from a normally distributed population. Construct a 90% level of confidence for the population variance and population standard deviation. b. For quality control, the population standard deviation of the tablets weights should be less than 0.015 milligrams. Does the confidence interval you constructed for the population standard deviation suggest that it is at an acceptable level? Explain your reasoning.

Answer 1

a.

`\begin{gathered} 0.000041\leq\sigma^2\leq0.00015 \\ 0.0064\leq\sigma\leq0.0125 \end{gathered}`

b. The standard deviation is at an acceptable level

Step-by-step explanation:

The confidence interval for the variance can be calculated as:

`\frac{(n-1)s^2}{\chi^2_{(\propto)/(2)}}\leq\sigma^2\leq\frac{(n-1)s^2}{\chi^2_{1-(\propto)/(2)}}`

Where n is the size of the sample, s² is the variance of the sample, (1-∝) is the level of confidence, and χ² is the value of chi-square with n-1 degrees of freedom.

In this case, we get that n is 14 tablets, s² is 0.000071, and 1-∝ = 90%, so ∝ is 10%.

Then, the values of chi-square with 13 (14 -1) degrees of freedom is:

`\begin{gathered} \chi^2_{(\propto)/(2)}=22.36 \\ \chi^2_{1-(\propto)/(2)}=5.89 \end{gathered}`

Therefore, the interval of confidence for the population variance is:

`\begin{gathered} ((14-1)(0.000071))/(22.36)\leq\sigma^2\leq((14-1)(0.000071))/(5.89) \\ 0.000041\leq\sigma^2\leq0.00015 \end{gathered}`

Then, the standard deviation is the square root of the variance, so the interval of confidence for the population standard deviation is:

`\begin{gathered} \sqrt[]{0.000041}\leq\sigma\leq\sqrt[]{0.00015} \\ 0.0064\leq\sigma\leq0.0125 \end{gathered}`

Finally, since the upper limit of the interval for the standard deviation is less than 0.15 milligrams, we can say that the population standard deviation is at an acceptable level with a confidence level of 90%