A sample standard deviation and sample size are given. Use the one-standard-deviation $\chi ^ { 2 }$-test to conduct the required hvpothesis test.
-$\mathrm { s } = 6 , \mathrm { n } = 10$
$\mathrm { H } _ { 0 } : \sigma = 7 , \mathrm { H } _ { \mathrm { a } } : \sigma \neq 7 , \alpha = 0.05$
A)Test statistic: $X ^ { 2 } = 6.612$, Critical values $= 3.325,16.919$. Do not reject $\mathrm { H } _ { 0 }$.
B) Test statistic: $\mathrm { X } ^ { 2 } = 12.25$, Critical values $= 2.700,19.023$. Do not reject $\mathrm { H } _ { 0 }$.
C) Test statistic: $X ^ { 2 } = 6.612$, Critical values $= 2.700,19.023$. Do not reject $\mathrm { H } _ { 0 }$.
D) Test statistic: $X ^ { 2 } = 12.25$, Critical values $= 3.325,16.919$. 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 required hvpothesis test.
-$\mathrm { s } = 6 , \mathrm { n } = 10$
$\mathrm { H } _ { 0 } : \sigma = 7 , \mathrm { H } _ { \mathrm { a } } : \sigma \neq 7 , \alpha = 0.05$  
A)Test statistic: $X ^ { 2 } = 6.612$, Critical values $= 3.325,16.919$. Do not reject $\mathrm { H } _ { 0 }$.
B) Test statistic: $\mathrm { X } ^ { 2 } = 12.25$, Critical values $= 2.700,19.023$. Do not reject $\mathrm { H } _ { 0 }$.
C) Test statistic: $X ^ { 2 } = 6.612$, Critical values $= 2.700,19.023$. Do not reject $\mathrm { H } _ { 0 }$.
D) Test statistic: $X ^ { 2 } = 12.25$, Critical values $= 3.325,16.919$. Reject $\mathrm { H } _ { 0 }$.

Quizplus · Accepted Answer

The test statistic for a chi-square test is calculated using the formula $X^2 = \frac{(n-1)s^2}{\sigma^2}$, where $s$ is the sample standard deviation, $n$ is the sample size, and $\sigma$ is the population standard deviation under the null hypothesis. Plugging in the given values ($s = 6$, $n = 10$, $\sigma = 7$), we get $X^2 = \frac{(10-1)6^2}{7^2} = 6.612$. The critical values for a chi-square distribution with $n-1 = 9$ degrees of freedom at a significance level of $\alpha = 0.05$ (for a two-tailed test) are approximately $2.700$ and $19.023$. Since the test statistic falls between these critical values, we do not 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 $\chi ^ { 2 }$