A sample standard deviation and sample size are given. Use the one-standard-deviation $\chi ^ { 2 }$ -test to conduct the requiredhypothesis test.
-$\mathrm { s } = 2.4 , \mathrm { n } = 17$
$\mathrm { H } _ { 0 } : \sigma = 3.5 , \mathrm { H } _ { \mathrm { a } } : \sigma \neq 3.5 , \alpha = 0.10$
A)Test statistic: $X ^ { 2 } = 7.523$. Critical values $= 7.962,26.296 .$ Reject $\mathrm { H } _ { 0 }$.
B) Test statistic: $X ^ { 2 } = 9.870$. Critical values $= 7.962,26.296 .$ Do not reject $\mathrm { H } _ { 0 }$.
C) Test statistic: $\mathrm { X } ^ { 2 } = 7.523$. Critical values $= 9.312,23.542$. Reject $\mathrm { H } _ { 0 }$.
D) Test statistic: $\mathrm { X } ^ { 2 } = 9.870$. Critical values $= 9.312,23.542$. Do not reject $\mathrm { H } _ { 0 }$.

Question

A sample standard deviation and sample size are given. Use the one-standard-deviation $\chi ^ { 2 }$ -test to conduct the requiredhypothesis test.
-$\mathrm { s } = 2.4 , \mathrm { n } = 17$
$\mathrm { H } _ { 0 } : \sigma = 3.5 , \mathrm { H } _ { \mathrm { a } } : \sigma \neq 3.5 , \alpha = 0.10$  
A)Test statistic: $X ^ { 2 } = 7.523$. Critical values $= 7.962,26.296 .$ Reject $\mathrm { H } _ { 0 }$.
B) Test statistic: $X ^ { 2 } = 9.870$. Critical values $= 7.962,26.296 .$ Do not reject $\mathrm { H } _ { 0 }$.
C) Test statistic: $\mathrm { X } ^ { 2 } = 7.523$. Critical values $= 9.312,23.542$. Reject $\mathrm { H } _ { 0 }$.
D) Test statistic: $\mathrm { X } ^ { 2 } = 9.870$. Critical values $= 9.312,23.542$. Do not reject $\mathrm { H } _ { 0 }$.

Quizplus · Accepted Answer

The test statistic is calculated as $\frac{(n-1)s^2}{\sigma_0^2}$, which in this case is $\frac{(17-1)2.4^2}{3.5^2} = 7.523$. The critical values for a $\chi^2$-test with 16 degrees of freedom and $\alpha = 0.10$ are 7.962 and 26.296. Since the test statistic (7.523) falls between the critical values, we fail to reject the null hypothesis.

A Sample Standard Deviation and Sample Size Are Given χ2\chi ^ { 2 }χ2

A Sample Standard Deviation and Sample Size Are Given $\chi ^ { 2 }$