[NOW WORK] Job Satisfaction Is there an association between on...

Question

[NOW WORK] Job Satisfaction Is there an association between one’s level of education and satisfaction with work? A random sample of 5244 employed individuals were asked to disclose their highest level of education and satisfaction with their work/job. The results are shown in the table below. The data are from the General Social Survey.

b. Verify that the requirements for performing a chi-square test of independence are satisfied.

Table showing job satisfaction levels by education with frequencies for very satisfied to very dissatisfied categories.

e. Compare the observed frequencies with the expected frequencies. Which cell contributed most to the test statistic? Was the expected frequency greater than or less than the observed frequency? What does this information tell you?

Accepted Answer

Welcome back everyone. In this problem, a company surveyed 1200 employees about their preferred work shift morning, evening, or night and their department, sales, IT, or HR. The observed frequencies are shown in the table below. Which cell contributed most to the chi square statistic and was the expected frequency greater than or less than the observed frequency for that cell? Now here we're given the shift and the department and their column and the rotta, OK. So how can that help us to first figure out the cell that contributed the most to the casesqua statistic? In other words, how do we measure contribution? Well, recall that the contribution of each cell to the total chi squared statistic. Is defined by the formula that says it's equal to the observed count which we can say is 0. Minus the expected count. Which we can call e squared divided by the expected count. Yeah. Which we could rewrite as O minus ER squared, divided by E. Now so far we have or observed cones, but to use this formula we'll need to figure out our expected frequencies for each cell, and the expected frequency e is going to be equal to the row total for the cell multiplied by the column total for the cell. Divided by the grand total in all, OK? So this is what we need to use to help us figure out the contribution. So let's first try to figure out our expected values for each cell and then use that to calculate the contribution. That's how we would know uh which cell has the maximum contribution. So, let's first create a table. OK, so I'm gonna make a table here. And we know that we have sales, uh, IT. And HR and for sales, the road total is 300 for IT it's 500 and for HR it's 400. For the columns we'll have morning. Whose column total is 540. Evening 4:20 And night. It's 2:40. So now, we just need to calculate those expected values. So let me just create a uh let's make our table look a little bit better here, OK? So no. I think we can move this down a little bit just to make some space, OK. And now, we can go ahead and calculate those expected values. So, for example, for the sale with the sales in the morning, OK, then the expected value is going to be 300 multiplied by 540 divided by our grand total of 1200, OK, which is going to give us a value of 135. So, now we can repeat that idea for the remaining cells, OK? So, for evening. It's gonna be 300 multiplied by 420 divided by 1200, that's 105. And for a night, it's 300 multiplied by 240 divided by 1200, which is 60. So I, I think we kind of have the idea of how we do it because we're just repeating that process for the remaining rows. So in the interest of time, I read the results and let's see if you get the same. But for IT in the morning, 2, it's gonna be 225, IT in the evening, 175, and IT in the night, 100. And for HR in the morning, that's 180, in the evening, 140, and in the night, it will be 80. So these are the expected values, OK, for each cell. I know that we have the observed unexpected values, we can go ahead and calculate the statistic. So to do that, OK, let's write down our cell. We're gonna create a table that is gonna first have the cell. OK. And then it's gonna have our observed value, or expected value, OK. And then we'll have the difference between those values, O minus E, and we'll use that to help us figure out the contribution because remember the contribution is O minus E squared divided by our expected value E, OK? So again, let's just make uh create these rules clearly here, define them clearly. OK. And no, OK. We're going to have sales in the morning, OK, sales in the evening and sales at night. Uh, IT in the morning, IT in the evening and IT at night. And then we'll have HR in the morning. HR in the evening and HR at night. OK? My space is getting a little bit crowded here, but I think we know what we'd have. Let me try to bring it up a little bit. So first, starting with sales in the morning, remember observed value was 180, expected is 135. So O minus E will be 45, and thus the contribution will be 45 2 divided by 135, which is 15. Next, for sales in the evening, that's 60, 105, which gives us -45, thus the contribution will be approximately 19.29. OK. Uh, for sales in the night, that's going to be 60 and 60. So the difference is zeros, contribution 0, I in the morning, 240, 225. So that gives us a difference of 15, which is a contribution of 1. IT in the evening is gonna be 180, 175, 5. For the difference, so that gives us approximately 0.14. OK. I think I need to separate these just to, I need to draw some lines here just to make it clear what the values are, OK, just so we don't get uh confused or mixed up, OK. So now, let me extend this here. Now you guys see I'm, I'm running it really tight here, but we want to make sure we, the, the table helps us to see it a bit more structured, OK. Now I in the night, the observed value is 80, expected value is 100. So the difference is -20 and that gives us a contribution of 4. Finally, H, well, in our final row HR in the morning, that's 120 and then 180. OK. That gives us a difference of -60 here and -60 square divided by 180 equals 20. For a contribution. Then HR in the evening, that's 180, 140. The difference is 40, so the contribution is approximately 11.43. And finally, for HR in the night, that's 180, uh, with a difference of 20, giving us a contribution of 5. So based on what our results are, which cell has the maximum contribution? Well, you'll notice here that it would be HR in the morning, OK, so the HR department in the morning because they have a maximum or they have a contribution of 20, which is the most. Therefore, we can say that the cell that contributed the most to the chi square statistic is the HR morning cell with a contribution of 20. And for this cell, the observed frequency was 120. The expected frequency was 180, thus the observed frequency was less than the expected frequency. So if I just put our answer over on the left in a box. On the right side, we can say HR morning still contributed the most. Yeah. Where The observed frequency. Was less than they expected. Less than the expected frequency. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Chi-Square Test for Independence

Observed vs. Expected Frequencies

Contribution to Chi-Square Statistic

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