Question

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It is estimated that the average test score for a statistics exam will be u = 70, with a variance of o2 = 12. In a random sample of 8 students, the test scores are 65, 73, 74, 68, 70, 74, 69, 71, giving us the sample statistics i = 70.5 and s? = 10. Assume a normal distribution of scores. Answer the following questions to determine if you would agree with the assumption of o2 or not. (a) Use the sampling distribution of S2 to find the value of x2 corresponding to what is seen in the data, and our assumption about the population variance. Round your answer to 3 decimals. (b) Find Xô.975 (df), where df is the degrees of freedom value corresponding to the statistic from Part 7a. (c) Find Xô.025 (df), where df is the degrees of freedom value corresponding to the statistic from Part 7a. (d) Would you agree with our assumption that the population variance is o2 = 12? Why or why not?

Answer 1

a) $\dpi{100} \chi^2 = \frac{(n - 1)s^2}{\sigma^2}$

          $\dpi{100} = \frac{7 * 10}{12}$

          $\dpi{100} = 5.833$

b) $\chi^2_{0.975} = 1.690$

c) $\chi^2_{0.025} = 16.013$

d) Yes, we would agree with our assumption that the population variance is $\sigma^2 = 12$..

Because the test statistic value lies between the critical values(1.690 < 5.833 < 16.013), so we can conclude that the population variance is $\sigma^2 = 12$.