Question

Consider the following hypothesis problem.

n = 21, s² = 90, H₀: σ² ≥ 99, Ha: σ² < 99

If the test is to be performed at the 5% level, what are the critical value(s) from the chi-square distribution? (Round your answer(s) to three decimal places. If the test is one-tailed, enter NONE for the unused tail.)

Answer 1

To find the critical value(s) from the chi-square distribution, we need to compare the test statistic to the chi-square distribution with the appropriate degrees of freedom. In this case, the critical value from the chi-square distribution for the given hypothesis test is approximately 31.41.

Step-by-step explanation:

To find the critical value(s) from the chi-square distribution, we need to compare the test statistic to the chi-square distribution with the appropriate degrees of freedom. In this case, the null hypothesis is that the population variance (σ²) is greater than or equal to 99, and the alternative hypothesis is that the population variance is less than 99.

Since it is a one-tailed test, we need to find the critical value for the left tail. The critical value can be found by comparing the test statistic (calculated using the sample variance and sample size) to the chi-square distribution table with the appropriate degrees of freedom.

In this case, the degrees of freedom (df) for the chi-square distribution is n - 1, where n is the sample size. So, df = 21 - 1 = 20. Using a chi-square distribution table or calculator, with df = 20 and a significance level of 0.05, the critical value is approximately 31.41. Therefore, the critical value from the chi-square distribution for the given hypothesis test is 31.41.