A large university is interested in learning about the average time it takes students to drive to campus. The university sampled 51 students and asked each to provide the amount of time they spent traveling to campus. The sample results found that the sample mean was $23.243$ minutes and the sample standard deviation was $20.40$ minutes. It is desired to determine if the population standard deviation exceeds 20 minutes. Calculate the test statistic for this test of hypothesis.
A) $\chi ^ { 2 } = 58.11$
B) $\chi ^ { 2 } = 52.02$
C) $\chi ^ { 2 } = 51$
D) $\chi ^ { 2 } = 53.06$

Question

Quizplus · Accepted Answer

The test statistic for testing if the population standard deviation exceeds a certain value is calculated using the chi-square ($\chi^2$) distribution formula: $\chi^2 = \frac{(n-1)s^2}{\sigma^2}$, where $n$ is the sample size, $s$ is the sample standard deviation, and $\sigma^2$ is the hypothesized population variance. Here, $n = 51$, $s = 20.40$ minutes, and $\sigma = 20$ minutes, so $\sigma^2 = 400$. Plugging these values into the formula gives $\chi^2 = \frac{(51-1)(20.40)^2}{400} = \frac{50 \times 416.16}{400} = \frac{20808}{400} = 52.02$.

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