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.

Achievement and School Location The contingency table shows the results of a random sample of students by the location of school and the number of those students achieving basic skill levels in three subjects. At α=0.01, test the hypothesis that the variables are independent. (Adapted from HUD State of the Cities Report)

Contingency table showing student achievement in reading, math, and science by school location: urban and suburban.

Accepted Answer

Hello there. Today we want to 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 collects data on 120 college students, categorizing them by their major, science, arts, business, and their preferred study method. Group solo mixed. The observed counts are as follows. Group solo mixed, science, arts, business, 10, 8, 12. 1814, 1612, 1020. At alpha equals 0.01, test whether the type of major and preferred study method is independent. Awesome. So it appears for this particular problem we're ultimately trying to determine at the significance level of 0.01 we're asked to test whether the type of major and preferred study method is independent. So that is what we're ultimately trying to solve for. So now that we know what we're trying to solve for, let's read off our multiple choice answers to see what our final answer might be. There is sufficient evidence to conclude that the type of major and preferred study method is independent. B is there is insufficient evidence to conclude that the type of major and preferred study method is independent. C is there is sufficient evidence to conclude that the type of major and preferred study method is dependent. And D is insufficient data. Awesome. So our first step in order to solve this particular problem is we need to state our hypotheses. So we need to state the hypothesis for our null hypothesis H0 and our alternative hypothesis HA. So for our null hypothesis H0, we need to note that this is the major and study methods are independent. Whereas for our alternative hypothesis HA, it states that it will be not independent. So now, our second step is we need to compute the row totals for science, arts, and business, which I'll call S for short, for science, A for arts, and B for business for short. So, computing our row totals for science, we need to take 10. Plus 18 plus 12, which is equal to 40, and for arts, we need to take 8. Plus 14 + 10, which is equal to. 32. And finally for business, we need to take 12 + 16 + 20. Which will be equal to 48. Fantastic. So now our 3rd step is we need to compute our column totals, which for the column totals, we need to do it for the group. Solo, so I'll call G for group for short, SO for solo, and M for mixed. So for the group column, we need to take 10 + 8 + 12, which is equal to 30. For solo, we need to take 18 + 14 + 16, which is equal to 48. And finally for mixed, we need to take 12 plus 10 + 20. Which is equal to 42. So that will mean for our 4th step, we need to come up with our grand total, so I'll call it GT for short, for our grand total. So we need to take 40 + 32 + 48, which will be equal to, so once again 40 + 32 plus 48, which will give us a grand total of 120. Fantastic. So now for our 5th step, We need to compute the expected counts for each cell, so for the expected counts for each cell, For the science dash group, so science dash group. We will find that E, the expected count, is equal to 40, multiplied by 30 divided by 120, which is equal to 10, and we need to repeat this methodology for all cells. And when we repeat this methodology for all cells, we will be able to produce the following table, which will include our expected frequencies, so we should be able to, when we repeat. Our Methodology, once again, for all the cells, we will be able to create the following chart, or I should say table. Which as you could see for the expected frequencies, we have our first column which is major, then we have our second column group, third column solo, and then our final column mixed for our rows for science, arts, and business, which going in order from left to right, we have 10, 8.0, 12.0, 16.0, 12.8. 19.2, and then finally we have 14.0, 11.2, and 16.8. Fantastic. So now, our next step, our 6th step, We need to calculate G2, which, as we should recall, T2 is equal to the sum of parentheses minus E to the power of 2 divided by E. And when we calculate or when we use our G squared formula, we'll be able to expand our data table that we've created to produce the following table. Which will look as follows. So, as you could see, We'll be able to produce the following table when we look at, or rather when we use our T squared formula. And as you could see, Looking at our table, we have our observed value, which for our group solo and mixed, and we have our expected value for our group solo mixed, and then finally we have our values for O minus Es square divided by E for our group solo and mixed for our different majors for science, arts, and business, and we've collected all of our data points. So, with that in mind, We now need to compute the following. So now we need to note that T squared is equal to 0.25 plus 0.1125 plus 0.5333 plus 0.2857. Plus 0.1286. Plus 0.6095, which when we add all those values together, we will find that it's equal to 1.9196. Fantastic. So now our next step, and once again we got all the values that we added together from our table that we created using our T squad formula and using our previous data table that we added to this last table where we combined everything into one big table. So, moving that, moving right along here, our seventh step now is we need to determine the degrees of freedom, which, as we should recall, we need to take parentheses 3 minus 1, multiplied by 3 minus 1 in parenthesis, which is simply equal to 4. And our eighth step is we need to determine our critical value at alpha, our level of significance of 0.01. That will mean that our degrees of freedom, DF for short, is equal to 4. So that will mean that our critical value or the value for T squared will be equal to 13.277. All right, we did it. So now, Our ninth and final step, we need to note that since 1.9196 is less than 13, so 13.277, we fail to reject the null hypothesis. I'll put in X next to H0. We fail to reject the null hypothesis and because we fail to reject the null hypothesis. as we should recall, there is sufficient evidence to conclude that the type of major and preferred study method are indeed independent. So that will mean looking at our multiple choice answers. Our final answer, without a doubt, will have to be. Multiple choice answer letter A, which once again is, there is sufficient evidence to conclude that the type of major and preferred study method is independent. 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

Degrees of Freedom and Critical Value

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