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The results of Computerworld's Annual Job Satisfaction Survey showed that 28$\%$ ofinformation systems (IS) managers are very satisfied with their job, 46$\%$ are somewhatsatisfied, 12$\%$ are neither satisfied nor dissatisfied, 10$\%$ are somewhat dissatisfied, and 4$\%$are very dissatisfied. Suppose that a sample of 500 computer programmers yielded the following results.Use $a=.05$ and test to determine whether the job satisfaction for computer programmersis different from the job satisfaction for IS managers.

There is sufficient evidence to reject the claim of the specific distribution.

Intro Stats / AP Statistics

Chapter 11

Comparisons Involving Proportions and a Test of Independence

Descriptive Statistics

Confidence Intervals

The Chi-Square Distribution

University of North Carolina at Chapel Hill

University of St. Thomas

Boston College

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So what we want to know is if the job satisfaction for computer programmers is different than the job satisfaction for information systems managers. So what we have to do is say that the proportion so Arnold hypothesis would be that the proportions are the same. So the proportion of information systems managers that are very satisfied with their job is equal to 0.28 Yeah, very satisfied is equal to 0.28 proportion. That would be somewhat satisfied would be 0.46 proportion, that is, neither would be 0.12 The proportion that is somewhat dissatisfied would be 0.1 and very dissatisfied. The proportion of you very dissatisfied would be equal to 0.4 So because these air the proportions for our information systems managers, we would say that these were the same four our computer scientists or computer programmers. And our alternative hypothesis would be that, um, this is not the case. This is not the case. So we're no hypothesis would be that the population proportions are not very satisfied. Equals 0.28 proportion of somewhat satisfied a 0.46 the proportion of neither is 0.12 portion of somewhat dissatisfied. Is there a 0.1 and the proportion of very dissatisfied is your appointment report. So it is not these cases Now, In order to test this, we have to come up with a chi squared ah value in order to find a p value. This will be our test statistic and we compute a chi squared value as the sum from I equals 12 k. Where Kay is the number of ah number of categories that we have and that is for each category. It is tthe e. Some of the observed frequency minus the expected frequency squared over the expected frequency. So let me just write down and create a table of what our expected frequencies observed frequencies and other things are so this will be our so there's a little small. So over here I will have our, um our category here is our proportion. Our population proportional hypothesized proportion. This will be our observed frequency. This will be our expected frequency and then this will be That's all we need. Really? So this is very satisfied, somewhat satisfied, neither somewhat dissatisfied on very good satisfied, so the proportion of people will be the same as our hypothesis. So this would be 0.28 point 46 0.12 point one and 10.0 poor. And the observed frequency is what we're given in the problem that out of a sample of 100 or 501 105 are very satisfied to 35 are very dissatisfied or 2 35 or somewhat satisfied. 55 or neither 90 are somewhat dissatisfied and 15 are very dissatisfied. The expected frequency. It's simply our total sample size times our, um hypothesized proportion. So this would be 0.28 times our end of 500. This would be 0.28 times are in a 500 which is 1 40 0.46 times a renter 500 which is 2 30 0.12 times 500 which is 60 0.1 times 500 which just 50 and 0.4 times 500 witches, uh, 20. And now we can use these values to come up with a test with acai squared value. So in a new paid, I will take the sum of the difference is squared. So it'll be 105 minus 1 40 squared over 1 40 plus 2 35 minus 2 30 squared over 2 30 plus 55 minus 60. Squared over 60 plus 90 minus 50 square over 50 plus 15 minutes. 20 squared over 20. And with this, we get a chi squared. So this is equal to work. I squared. We get akai squared value of 42 approximately 42.5. But we need to come up with a P value. But to come up with a P value, we need to find a degrees of freedom and our degrees of freedom. Degrees of freedom is equal to K minus one work A is the number of categories we have and we have How many covers? 12345 We have five categories, so five minus one is four. So our p value, given that we have a chi square to 42.5 and four degrees of freedom is equal to, um, less than 0.5 It's easier. So our P value is less than 0.5 And now we're going to test this against an Alfa of point zero five. So, um, and our Alfa is your 0.5 So our key value is less than 0.5 which is less than 0.5 So therefore we can reject the No. So what does that mean? Um, that means that computer scientists and information systems do not share the same job satisfaction readings.

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