Question

A manufacturer would like to determine if the variance in a product dimension exceeds 10.

a) State the Hypothesis to show the variance is greater than 10.

b) Choose a level of alpha.Use a = 0.05 for this problem.

c) To test the hypothesis, the manufacturer takes a sample of 18 parts and measure that product dimension. The data appear in the Dimension worksheet of the HW3 data workbook on Moodle. Collect data and calculate necessary statistics to test the hypothesis.

d) Sketch the sampling distribution. Include the critical value and test statistic.

e) Draw a conclusion and report that in the problem context.

f) What is the p-value for the hypothesis test?

Answer 1

$a.\\ Let~\sigma^2=population~variance~in~a~product~dimension\\ Null~hypothesis,~H_0:\sigma^2=10\\ Alternative~hypothesis,~H_1:\sigma^2>10.\\ b.~Level~of~significance=\alpha=0.05.\\ c.\\ n=sample~size=n=18\\ Test~statistic=\chi^2=\frac{\sum_{i=1}^n(x_i-\bar{x})^2}{10}=\frac{(n-1)s^2}{10}\\ where,~s^2=sample~variance~and~\bar{x}=sample~mean.\\ Under~H_0,~\chi^2\sim \chi^2_{n-1}.\\ Computation~of~s^2:\\ s^2=\frac{\sum_{i=1}^{18}x_i^2-18\bar{x}^2}{17},~\bar{x}=\frac{1}{18}\sum_{i=1}^{18}x_i\\(d)~Critical~value=\chi^2_{0.05,17}= 27.5871\\$

$e.~If~Value~of~\chi^2>Critical~value=\chi^2_{0.05,17}= 27.5871\\ then~we~reject~H_0~at~5\%~level~of~significance~and~conclude~that\\ there~is~enough~evidence~to~conclude~that~variance~in~a~production~dimension\\ exceeds~10.\\ If~Value~of~\chi^2\leq Critical~value=\chi^2_{0.05,17}= 27.5871\\ then~we~fail~to~reject~H_0~at~5\%~level~of~significance~and~conclude~that\\ there~is~not~sufficient~evidence~to~conclude~that~variance~in~a~production~dimension\\ exceeds~10.\\ f.~p-value=P(\chi^2>value~of~\chi^2|\chi^2\sim \chi^2_{17}).\\ Since~data~is~not~provide~here~so~I~mention~the~steps~only.$