Performing a Chi-Square Independence Test In ...

Question

Performing a Chi-Square Independence Test In Exercises 13–28, perform the indicated chi-square independence test by performing the steps below.

a. Identify the claim and state H₀ and Hₐ

b. Determine the degrees of freedom, find the critical value, and identify the rejection region.

c. Find the chi-square test statistic.

d. Decide whether to reject or fail to reject the null hypothesis.

e. Interpret the decision in the context of the original claim.

Attitudes about Safety The contingency table shows the results of a random sample of students by type of school and their attitudes on safety steps taken by the school staff. At α=0.01, can you conclude that attitudes about the safety steps taken by the school staff are related to the type of school? (Adapted from Horatio Alger Association)

Contingency table showing student attitudes on safety steps by school type: public and private, with corresponding counts.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A researcher surveys a random sample of employees from two different companies regarding their satisfaction with workplace wellness programs. The responses are categorized as satisfied, neutral, or dissatisfied. The data are shown in the contingency table below. Company X, Company Y satisfied 2218 neutral, 1520, dissatisfied, 1317. At alpha equals 0.05, you can conclude that satisfaction with wellness programs is related to the company of employment. Perform a G2 test for independence. Awesome. So it appears we're ultimately trying to conclude whether or not this claim that the satisfaction with wellness programs is related to the company of employment. And we're trying to determine whether or not if we can conclude that this claim is true based on the information that is provided to us by the problem itself, and considering we're given a level of significance which is denoted as alpha as 0.05. So that's our value for the level of significance. So now that we know what we're ultimately trying to solve for, let's read off our multiple choice answers to see what our final answer might be. A is yes, there is sufficient evidence to conclude that satisfaction with wellness programs is related to the company. B is no, there is insufficient evidence to conclude that satisfaction with wellness programs is related to the company. C is no, the test statistic is negative, so we cannot conduct the test, and finally, D is insufficient data. Awesome. So, our first step is we need to state the claim and the hypothesis. So our claim, which we'll call C, states that satisfaction with wellness programs is related to the company itself. So, depending on, so what, what does that mean? So it's what it's trying to say is that it's saying that satisfaction with wellness programs is related to a specific company, whether it be Company X does wellness programs better than Company Y, etc. so either or. So we need to determine. If this is a true statement or not, if one company does wellness programs better than the other. Since it's related to the company. So that's our claim. So now we need to state our null hypothesis, which will denote as H0 or H subscript zero, and our null hypothesis states that the satisfaction is independent of company, so I'll put a capital I to mean that it's independent from company. Whereas our alternative hypothesis, H subscript A or HA for short, states that the satisfaction is related to the company, so Related to companies, so I'll call it RC. So once again, H0 or no hypothesis, satisfaction is independent of company, whereas our alternative hypothesis, satisfaction is related to the company, so I'll call it RC for related to company. So now, our 2nd step. we need to determine the degrees of freedom. So as we should recall, DF, the degrees of freedom, is equal to parentheses 2 minus 1, multiplied by parentheses 3 minus 1, which when we plug that into a calculator, you will find that it's equal to 2. And we have our critical value CV. At a significance level of alpha equals 0.05 and a DF degrees of freedom is 2, which will mean that our critical value. Based on this fact, will be that it's equal to 5.991. And once again we can determine the critical value is equal to 5.991 because we're told that alpha, our level of significance, is equal to 0.05, and we're told that the degrees of freedom is equal to 2. So our rejection region for this case will be that T squared. is greater than, so T squared. is greater than 5.991. And once again, this is for our rejection region, which I'll call RR. OK. Moving right along here. Our 3rd step is we need to compute our expected counts. So for company X, so this is our total, so I'll call it T. For total, so we have our company X, which I'll just call X for short. So for Company X we have 22 + 15 + 13. Which is equal to 50, whereas for company Y, We have 18 + 20 + 17, which is equal to 55. And for those that are satisfied, we have 22 + 18, which is equal to 40, and those who are neutral, so once again I'm saying S for satisfied and N for neutral, and for those who are neutral, we have 15 + 20, which is equal to 35, and for dissatisfied, which we'll just simply call D. We have 13, so when you take 13 plus 17, which is equal to 30. Which means that our grand total, which we'll call GT and GT our grand total will be equal to 105. Fantastic. So now, As you should recall, following this line of thought, we need to determine the expected for Company X satisfied, so So now we're doing X satisfied condition, which means we need to take 50 multiplied by 40 divided by 105, which will give us an answer of when we round the two decimal places, 19.05. So for company X, neutral condition, we need to take. 50 Multiplied by 35 divided by 105, which will give us an answer of 16.67. And once again we're rounding all of our answers to two decimal places. So for Company X for the dissatisfied condition, we need to take 50 multiplied by 30 divided by 105, which will give us an answer of 14.29. And for company Y satisfied condition, we need to take 55 multiplied by, so we're taking 55 multiplied by 40. And then we need to divide by 105, which will give us an answer of 20.95. Fantastic. And for the condition where we have company Y is neutral, we need to take 55 multiplied by 35 divided by 105, which will give us an answer up. Once again, this is for the company Y condition neutral, which will give us an answer of 18.33. And finally, our company Y dissatisfied condition. We need to take 55 multiplied by 30, divided by 105, which will give us an answer of. 15.17. Fantastic. So now, moving right along here. Our 4th step is we need to compute G2. And we need to note that when we're computing G2, we need to focus first on the condition where we're focused on Company X is satisfied, so we need to take 22 minus 19.05, the power of 2 divided by 19.05. Which will give us an answer of, when we plug this into our calculator, 0.457 when we round to three decimal places. Fantastic. So now, moving right along here, so for the condition where we have Company X neutral, we need to take 15, so we need to take 15 minus 16.67. And don't forget that we need to take this to the power 2 divided by 16.67, which will give us an answer of 0.167. Once again, rounding to three decimal places. And for the condition where X is dissatisfied, so company X dissatisfied, we have parentheses 13 minus 14.29 di to the power 2, so 13 minus 14.29 to the power 2. Divided by 14.29. Which will give us an answer of drum roll, 0.116. Now focusing on company Y, satisfied condition, we have parentheses 18 minus 20.95 to the power 2 divided by 20.95, which will give us an answer of 0.415. Fantastic. And then for the company Y neutral condition, we have parentheses 20 minus 18.33 to the power 2 divided by 18.33, which will give us an answer of 0.152. Fantastic. So now for our final condition, so for company Y dissatisfied, we need to have parentheses 17 minus 15.71 to the power of 2 divided by 15.71, which will give us an answer of 0.106 when we round to three decimal places. Fantastic. So now, we need to take the sum, so we need to take 0.457. Plus 0.167 plus 0.116 plus 0.415 plus 0.152 plus 0.106, which will give us an answer of 0. So actually, when we plug into our calculator, we will find that this is all equal to 1. 413. So 1.413. So that means that our final decision, we need to state that 1.413. is indeed less than 5.991. So once again, our sum value, which is 1.413, is less than 5.991. So that means for our final interpretation for step 6. For our interpretation, we can state that there is insufficient evidence to conclude that satisfaction with wellness programs is related to the company. So that means going back to look at our multiple choice answers, that will mean that our final answer, without a doubt, will have to be multiple choice answer. Letter B, which once again states that no, there is insufficient evidence to conclude that satisfaction with wellness programs is related to the company. And that's it. That's our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Chi-Square Independence Test

Null and Alternative Hypotheses (H₀ and Hₐ)

Degrees of Freedom and Critical Value

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