In Exercises 7–12, find the critical value(s) and rejection re...

Question

In Exercises 7–12, find the critical value(s) and rejection region(s) for the type of chi-square test with sample size n and level of significance α.

Two-tailed test, n=61,α=0.01

Accepted Answer

All right. Hello, everyone. So, this question says that an engineer is testing whether the variability in battery life spans from a new production line is different, either higher or lower than the variability reported by the manufacturer. A random sample of 61 batteries is tested. The engineer performs a two-tailed chi square test at the 1% significance level. What are the critical values and the rejection region or regions for this two-tailed chi square test? And here on the screen are our 4 different answer choices, labeled A through D. So, first, to find the critical value, we have to know the degrees of freedom. So here, recall that the degrees of freedom or DF corresponds to or is equal to the sample size N subtracted by 1. So here that's 61 subtracted by 1, which gives you 60 degrees of freedom. Now, for the type of test. Notice how this is a two-tailed test, which means that the significance level alpha has to be split equally between both tails. So, alpha divided by 2 equals 0.005 in each tail. From here, you recognize that we need two critical values in this context. Starting off with the right tail critical value. In this case, that's going to be chi square of 0.005 and 60 degrees of freedom. For the left, on the other hand, that's going to be chi square of 0.995 and 60 degrees of freedom. Keep in mind that here, 0.005 and 0.995 represented the area to the right. Of both the right and left tail critical values. And now we can actually determine what they are using the chi square distribution. So, according to the chi square distribution, chi square of 0.005 and 60 degrees of freedom, Is equal to. 91.952. Whereas chi square of 0.995 and 60 degrees of freedom. Is 34.805. Once again, 0.005 and 0.995 represent the areas to the right of each respective critical value. And now on to the rejection region. So, in a two-tailed test, recall that we reject the null hypothesis if the test statistic is less than the left critical value. Or greater than the right critical value. So when writing this out, the rejection regions include When chi square is less than. 35.534. And let me actually take a moment to correct this because the left tilled critical value is in fact 35.534. So as mentioned before. The rejection regions are when chi square is less than 35.534, or when chi square is greater than 91.952. And so there you have it. What this means to say is that if the observed chi square value is either too small or too large, it suggests the variability is significantly different from the claimed value. And based on the information that we just found. This corresponds to. Option B in the multiple choice, making that the correct answer. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

In Exercises 7–12, find the critical value(s) and rejection region(s) for the type of chi-square test with sample size n and level of significance α.

Two-tailed test, n=61,α=0.01

Key Concepts

Chi-Square Test

Critical Value

Rejection Region

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