A random sample of size 16 from a normal distribution has standard deviation s=32 . Test $H _ { 0 } : \sigma = 54 \text { versus } H _ { 1 } : \sigma 54$ Use the $\alpha = 0.01$ level of significance.
A) Do not reject $H _ { 0 }$
B) Reject $H _ { 0 }$

Question

A random sample of size 16 from a normal distribution has standard deviation  s=32 . Test $H _ { 0 } : \sigma = 54 \text { versus } H _ { 1 } : \sigma > 54$ Use the $\alpha = 0.01$ level of significance.
A) Do not reject $H _ { 0 }$
B) Reject $H _ { 0 }$

Quizplus · Accepted Answer

The test statistic for testing the variance or standard deviation of a normal distribution is based on the chi-square distribution. The test statistic is calculated as $\chi^2 = \frac{(n-1)s^2}{\sigma^2_0}$, where $n$ is the sample size, $s$ is the sample standard deviation, and $\sigma^2_0$ is the hypothesized population variance. In this case, $n=16$, $s=32$, and $\sigma_0=54$, so we are testing if the population standard deviation is greater than 54. The calculated $\chi^2$ value would be less than the critical $\chi^2$ value at the 0.01 level of significance for the right-tailed test, indicating insufficient evidence to reject $H_0$. This is because the sample standard deviation is less than the hypothesized population standard deviation, which does not support the alternative hypothesis that the population standard deviation is greater than 54.

A Random Sample of Size 16 from a Normal Distribution H0:σ=54 versus H1:σ>54H _ { 0 } : \sigma = 54 \text { versus } H _ { 1 } : \sigma > 54H0​:σ=54 versus H1​:σ>54

A Random Sample of Size 16 from a Normal Distribution $H _ { 0 } : \sigma = 54 \text { versus } H _ { 1 } : \sigma > 54$