In a hypothesis test for the population variance, the hypotheses are: $H _ { 0 } : \sigma ^ { 2 } = 25$ . $H _ { 1 } : \sigma ^ { 2 } 25$ . If the sample size is 15 and the test is being carried out at the 5% level of significance, the null hypothesis will be rejected if: A) $\chi ^ { 2 } 24.9958$ . C) $\chi ^ { 2 } 23.6848$ .

Question

In a hypothesis test for the population variance, the hypotheses are: $H _ { 0 } : \sigma ^ { 2 } = 25$ . $H _ { 1 } : \sigma ^ { 2 } > 25$ . If the sample size is 15 and the test is being carried out at the 5% level of significance, the null hypothesis will be rejected if: A) $\chi ^ { 2 } < 6.5706$ . B) $\chi ^ { 2 } > 24.9958$ . C) $\chi ^ { 2 } < 7.2609$ . D) $\chi ^ { 2 } > 23.6848$ .

Quizplus · Accepted Answer

For a hypothesis test for the population variance, when the sample size is n and the null hypothesis is that the population variance is $\sigma ^ { 2 } = \sigma_{0} ^ {2}$, the test statistic follows a chi-square distribution with degrees of freedom equal to n-1.

In this case, we have: 
- Null hypothesis: $H _ { 0 } : \sigma ^ { 2 } = 25$ 
- Alternative hypothesis: $H _ { 1 } : \sigma ^ { 2 } > 25$ 
- Sample size: n = 15 
- Significance level: α = 0.05

To find the critical value of chi-square that would lead to rejecting the null hypothesis, we need to find the upper α-quantile of the chi-square distribution with 14 degrees of freedom. Using a chi-square table or calculator, we find:

$\chi ^ { 2 } _ { 0.05 , 14 } = 23.6848$

Therefore, the null hypothesis will be rejected if the test statistic is greater than 23.6848. Hence, the correct answer is D.

In a Hypothesis Test for the Population Variance, the Hypotheses H0:σ2=25H _ { 0 } : \sigma ^ { 2 } = 25H0​:σ2=25

In a Hypothesis Test for the Population Variance, the Hypotheses $H _ { 0 } : \sigma ^ { 2 } = 25$