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.

Use the contingency table and expected frequencies from Exercise 8. At α=0.05, test the hypothesis that the variables are dependent.

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 hospital administrator wants to know if there is a relationship between patient satisfaction, satisfied, unsatisfied, and type of insurance, private, public, none. The observed frequencies are Satisfied, unsatisfied, private, 30, 10. Public 2520, none, 1520. At alpha equals 0.05 tests the claim that the patient satisfaction and type of insurance are dependent. Awesome. So it appears for this particular prompt we're asked to take a level of significance value which is denoted as alpha of 0.05, and we're asked to use that to help us test the claim that the patient satisfaction and type of insurance are dependent of one another. So now that we know that we're trying to test the specific claim, and that's our final answer 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 there is sufficient evidence to support the claim that patient satisfaction and type of insurance are dependent. there is sufficient evidence to support the claim that patient satisfaction and type of insurance are independent. And see is there is insufficient evidence to support the claim that patient satisfaction and type of insurance are dependent. And D is insufficient data. Awesome. So our first step that we need to take is we need to state the claim and our hypothesis. So, as we should recall, our claim, which you'll call C, our claim is patient satisfaction and type of insurance are dependent. Let us also note that the null hypothesis, which our null hypothesis will denote this as H subscript zero, our null hypothesis states that the variables are independent, and for our alternative hypothesis, which we'll call H subscript A, which we'll call it H0 and HA for short. Our alternative hypothesis states that the variables are dependent. So once again for our null hypothesis, the variables are independent, whereas for alternative hypothesis, our variables are dependent. So I'll make a little note of that. So, once again, for our null hypothesis, the variables are independent, which I'll call I for short, whereas for our alternative hypothesis, our variables are dependent, which I'll call D for short. Awesome. So now our second step is we need to find the degrees of freedom, the critical value, and the rejection region. Awesome. So let us note that for our rows, we have 2 rows and then we have for our columns, we have 3 columns referring to our table that is provided to us for the problem itself. So with that in mind, we can determine our degrees of freedom, which we'll call DF, and our degrees of freedom is equal to parentheses 2 minus 1 multiplied by 3 minus 1 in parentheses. So once again, parentheses 2 minus 1 multiplied by parentheses 3 minus 1, which when we perform that calculation, we will find that it's equal to 2. And now if we recall and use our T square table, we will find that T critical, which I'll call CR for short for critical, so T critical squared will be equal to 5.991, and this once again is at the 0.05 significance level or level of significance. And that will mean that also and that our degrees of freedom is equal to 2. Fantastic. So moving right along here. So that will mean that our rejection region, which we'll call T squad, is greater than 5.991. Fantastic. So now, moving right along here, our third step is we need to calculate the expected frequencies and test statistics. So, now we need to determine our totals, which I'll call T for short, so our totals. So for those that are satisfied, This is equal to 70, and those who are unsatisfied, so I'm gonna call satisfied S and unsatisfied US, unsatisfied is equal to 50. And for private, which I'll call PR and for private is equal to 40, and public, which we'll just call P. is equal to 45. None, which I'll call it N for short, is equal to 35. And our total altogether, which I'll just call T is equal to 120. So We need to recall and note that our equation to solve for the expected for each cell, which we'll call this E subscript I J is equal to the row total, so RT. Multiplied by the column total. All divided by the grand total, which I'll call GT for short, so our grand total. Fantastic. So that will mean For our satisfied private situation or condition I should say, we need to take 70 multiplied by 40 divided by 120, which will give us an answer of when we round to two decial places, 23.33. And for the condition that it's satisfied public. You need to take 70 multiplied by 45 divided by 120, which will give us an answer of 26.25. For the condition that it satisfied none. We need to take 70 multiplied by 35 divided by 120, which gives us an answer of 20.42. And like our previous answers, we're rounding the two decimal places. So for the unsatisfied private condition, We need to take 50, so once again we're taking unsatisfied private conditions, so we need to take 50 multiplied by 40, so 50 multiplied by 40, and then we need to divide it by 120, that's our grand total, and this will be equal to 16.67. And moving right along here, so for the unsatisfied public condition, we need to take 150 multiplied by 45 divided by 120, which is equal to 18.75. And finally, for the unsatisfied non-condition, we need to take 50 multiplied by 35 divided by 120, which will give us an answer of 14.58. And once again we rounded all of our answers to two decimal places. So now we need to recall and use the following equation that T2 is equal to the sum of parentheses minus E to the 2 divided by E. So for our satisfied private condition, we need to take 30 minus 23.33 to the power to divided by 23.33, which will give us an answer of 1.91. Once again, rounding to two decimal places. And for our satisfied public condition, we need to take 25, so we need to take 25 minus 26.25 to the power 2 divided by 26.25, which will give us 0.06 as a final answer. Then we have satisfied none condition, which is 15 minus 20.42 to the power of 2. Divided by drum roll, 20.42, and when we plug that into our calculator, we will find that it is simply equal to 1.44. Fantastic. And for our unsatisfied private condition, we will have 10 minus 16.67 to the power 2 divided by 16.67, which will give us an answer of 2.67. And for our unsatisfied public condition, we will have 20 minus 18.75 to the power 2 divided by 18.75, which will give us an answer of 0.08. And finally, for our unsatisfied none condition, we need to take 20 minus, so 20 minus 14.58, and don't forget we need to take this to the power 2 divided by 14.58. And when we plug that into our calculator, we will find that it's equal to 2.01. So that will mean that when we do our total, we will find that G 2 is equal to 1.91 plus 0.06 plus. 1.44. Plus 2.67 plus 0.08 plus 2.01, which is all equal to 8.17. So now we need to make our final decision and conclusions and our interpretation overall. So we need to note that G squared is equal to 8.17, which is greater than 5.991, so we can reject the null hypothesis, so I'll put an X next to H0. So that will mean because we can reject the Noel hypothesis, we can make the following interpretation. There is sufficient evidence to support the claim that the patient satisfaction and type of insurance are dependent. So that will mean going back to look at our multiple choice answers. Our correct answer without a doubt. Will have to be, when you look again, you will see that our final answer has to be multiple choice answer letter A. Which once again is, there is sufficient evidence to support the claim that patient satisfaction and type of insurance are dependent, and that's it. We've officially solved for this problem. 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 Test of Independence

Null and Alternative Hypotheses

Degrees of Freedom and Critical Value

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