A random sample of size 12 from a normal distribution has standard deviation s = 24. Test $H _ { 0 } : \sigma = 41 \text { versus } H _ { 1 } : \sigma 41 . \text { Use the } \alpha = 0.05$ level of significance.
A) Do not reject $H _ { 0 }$
B) Reject $H _ { 0 }$ .

Question

A random sample of size 12 from a normal distribution has standard deviation s = 24. Test $H _ { 0 } : \sigma = 41 \text { versus } H _ { 1 } : \sigma > 41 . \text { Use the } \alpha = 0.05$ 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 formula for the test statistic is $\chi^2 = \frac{(n - 1)s^2}{\sigma^2_0}$, where $n$ is the sample size, $s$ is the sample standard deviation, and $\sigma^2_0$ is the hypothesized population variance. In this case, $n = 12$, $s = 24$, and $\sigma_0 = 41$, so we are testing if the population standard deviation is greater than 41. Given that the sample standard deviation $s = 24$ is less than the hypothesized population standard deviation $\sigma_0 = 41$, it indicates that the data does not provide strong evidence against $H_0$. Without calculating the exact chi-square value and comparing it to the critical value or p-value (since the numerical values and chi-square distribution table are not provided), the logic based on the relationship between the sample standard deviation and the hypothesized population standard deviation suggests that there is not enough evidence to reject $H_0$ at the $\alpha = 0.05$ level of significance.

A Random Sample of Size 12 from a Normal Distribution H0:σ=41 versus H1:σ>41. Use the α=0.05H _ { 0 } : \sigma = 41 \text { versus } H _ { 1 } : \sigma > 41 . \text { Use the } \alpha = 0.05H0​:σ=41 versus H1​:σ>41. Use the α=0.05

A Random Sample of Size 12 from a Normal Distribution $H _ { 0 } : \sigma = 41 \text { versus } H _ { 1 } : \sigma > 41 . \text { Use the } \alpha = 0.05$