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A study reported in the Journal of Small Business Management concluded that self- employed individuals do not experience higher job satisfaction than individuals who are not self-employed. In this study, job satisfaction is measured using 18 items, each of which is rated using a Likert-type scale with $1-5$ response options ranging from strong agreement to strong disagreement. A higher score on this scale indicates a higher degree of job satisfaction. The sum of the ratings for the 18 items, ranging from 18 to 90 , is used as the measure of job satisfaction. Suppose that this approach was used to measure the job satisfaction for lawyers, physical therapists, cabinetmakers, and systems analysts. The results obtained for a sample of 10 individuals from each profession follow.At the $\alpha=.05$ level of significance, test for any difference in the job satisfaction among the four professions.

There is sufficient evidence to support the claim that the means forthe four professions are not equal.

Intro Stats / AP Statistics

Chapter 10

Comparisons Involving Means, Experimental Design, and Analysis of Variance

Descriptive Statistics

The Chi-Square Distribution

Experiment

Missouri State University

University of North Carolina at Chapel Hill

Idaho State University

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So this problem the AA thing that we want to do is test for any difference in the job satisfaction among the four profession. So first thing I'm gonna do is come up with our hypotheses. So are no hypothesis is that there is no difference in job satisfaction. So what that means is that the population means for each of these jobs are equal. So on average, these jobs have the same satisfaction rating. And now my alternative hypothesis is that, um not all the jobs have, um, the same job satisfaction rating. So not ah muse O r equal. So with that in mind, we have to find a way to test these hypotheses. So we're going to use a nova, and for that we're going to create a nova table first. So on another table is going to be consisting of the source of our variation or variance. And then it's also gonna have a column for the sum of squares a column for the degrees of Freedom column for the mean squares mean of squares and then a column for the F statistic. And it would have a column for the P Value. But I'm just not going. Include that for now, Because we're gonna calculate that separately. That's the last thing we do. So we don't really need it on here. Okay, so we have three sources of various variants, one from our treatment, another from our error and one more from our total. The first thing we're going to do is calculate sum of squares for a treatment before we do that, we have to come up with the population means for each of these, so that our the sample mean sorry. Not the population means the sample means for each of these. So the sample mean for being a lawyer is equal to 50. The sample mean for being a therapist is equal to 63.7. The sample mean for being cabinetmaker is equal to 69.1. And the sample mean for being a systems analyst is 61.2. Now we're going to find the grand mean the grand mean is just the mean is the average of, um, our sample means excuse the voice cracks. Sorry. So we're gonna take 50 plus 63.7 for 69.1 61.2 and divided by four, and we get a grand mean of 61. Now, using this grand mean we're going to compute the sum of squares for our treatment. And that is just the difference between each of thes individual means. And our sample means square times the number of elements in each sample. So, for example, we have nine elements in each sample. So and for the lawyers equals nine so and that's the same for all of them. So with that in mind, we're going to take nine times the difference between our first sample mean of 50 minus our grand mean of 61 we're going to square this and then add up, Um, at this with the next sample mean minus thicker and mean squared. So 63.7 minus 61 squared plus nine times 69.1 minus. Oh, sorry. We have 10 10 elements in each sample. Um, that's my bad. I should be able to count by now. 10 done elements and then 69.1 minus 61 squared plus 10 times 61.2 9 61 squared. So we get a sum of squares for the treatment to be 1000 939.4. So we go where? No, the table. Wherever that is, we can update that value 1939.4. And now we're going to find the sum of squares for a total. The sum of squares for our total is simply the difference between each individual element and our grand means squared. So it would be 44 our first value here, 44 minus 61 squared. Plus the next item 42 minus 61 squared all the way until we get to the end of our data set 76 minus 61 squared. And lastly, 60 to 61 squared. So we take the sum and we get a value of where is this value? 6722. So we get a value of 6722 and let's update our table. And now we're going to find the air sum of squares for air, which is simply the sum of squares of the total, minus the sum of squares. For our treatment that those values in mind it would be 6722 minus 1939.4. So 67 change the color. 6722 minus 1939.4 is equal to 4780 the 82 0.6 So it's update our table or 782.6. Now we're going to find the means squares for our treatment and our air. And in doing so, we'll find the jury's of freedom for each so mean squares for treatment is equal to the sum of squares of our treatment, divided by the degrees of freedom for our treatment and the dreams of freedom for our treatment is the number of treatments that we have minus one. K represents the number of treatments and in here we have four treatments because each treatment represents a different sample. We have ah, one treatment for the lawyer. One treatment for physical therapist when treatment for the Cabinet maker maker. One treatment called systems analyst. So we have four treatments minus one equals three and we have a sum of squares of our treatment equaling 1939.4 over three, which is equal to 646 are about 646 I should say. Yeah. Um so let's update her. Table 600 46 goes here, and we found out her degrees of freedom for her treatment is three. Now, let's find the means square of our error, which is the sum of squares of our air, divided by the degrees of freedom for our error degrees of freedom for our error is the number of total elements that we have minus the number of treatments. So we have 40 total elements. Let me change this value here to be 10. So we have 10 elements in each column and we have four columns. So we have 40 elements in total and we have four treatments, so we gotta degrees of freedom of 36. So are some of squares of our error is 4782.6, divided by 36. What is equal to 133. Wrong place. We had a mean square for air of 133 and we found out that our degrees of freedom is 36. Now we have to find a degrees of freedom for our total. And this is just going to be the number of total elements that we have minus one. So we found out that we have 40 total elements. 40 minus one equals 39. So our degrees of freedom for our total is 39. Now we're gonna find an F statistic. The F statistic is simply the mean square of our treatment over the mean square of our So that is equal to 6 46 divided by 1 33 And we get an F statistic of 4.87 4.87 And now we're going to discover our p value. So, um, you can use excel for this, or you can use the F table, But ah, used excel, and I got that. Our P value is equal to 0.61 with degrees of freedom of three and 36 and with a p value of and we're comparing this, remember, against an Alfa of 0.5 to see if we should reject or accept are null so with a P value of 0.61 it is less than our Alpha of 0.5 Therefore, we reject they know, which means that job satisfaction for each of these jobs are not the same

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