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.

Left-tailed test, n=7,α=0.01

Accepted Answer

All right. Hello, everyone. So, this question says, a quality control analyst wants to determine whether the variability in the weight of small snack packets has decreased from what was previously known. She takes a random sample of 7 packets and performs a left-tailed chi square test at the 1% significance level. What is the critical value and rejection region for this left-tailed chi square test? And here we have 4 different answer choices labeled A through D. So the first thing we need to know are the degrees of freedom. And recall that the degrees of freedom are equal to the sample size and subtracted by 1. So here, that's 7 subtracted by 1, which gives you 6 degrees of freedom. Now, notice how this is a left-tailed test. That's aiming to determine whether or not the variability has decreased. Because of this, we're interested in the lower tail of the chi square distribution. So, to find the appropriate critical value, you have to find. The value such that the area to its right. Is 0.99. Or 0.99. Because again, we're interested in the lower tail of the distribution. Furthermore, in this case, the left tail area is alpha, which is equal to 0.01 in this case. So The chi square critical value that we're looking for is chi square of 0.99 and at 6 degrees of freedom. So, using the square distribution table, you would look for the row that corresponds to DF equal to 6, and then look under the column 4 0.99, because again, this corresponds to the right tail area, leading 1% in the left tail. Ultimately, this critical value is equal to 0.872. Which now brings you to the rejection region. Because this is a left-tailed chi square test, you reject the null hypothesis if the test statistic is less than the critical value. And I like to think of this as the test statistic having to be to the left of the critical value in a number line since it's less than. So the rejection region. Is in which chi square is less than 0.872. Again, to the left of the number line. Ultimately, this matches option A in the multiple choice. And If the calculated test statistic is in fact smaller than the critical value, in this context, that would mean that there is significant evidence at the 1% confidence level that the variability in weight has in fact decreased. 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.

Left-tailed test, n=7,α=0.01

Key Concepts

Chi-Square Test

Critical Value

Rejection Region

Watch next