A study reported in the Journal of Small Business Management...

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.

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.

Key Concepts

Likert Scale Measurement

A Likert scale is a psychometric scale commonly used to measure attitudes or opinions. In research, participants are asked to rate their agreement with a series of statements on a fixed scale (often from 1 to 5), and these ratings can be summed or averaged to create a composite measure of an underlying construct.

Significance Level (?)

The significance level, denoted by alpha (?), is the threshold for making a decision in hypothesis testing. It defines the probability of committing a Type I error—rejecting a true null hypothesis. A common choice is 0.05, meaning there is a 5% risk of incorrectly concluding that a significant effect exists.

F-Test

The F-test is a statistical test that compares the variances of different groups and is a central component of ANOVA. It assesses whether the ratio of the mean square between groups to the mean square within groups is large enough to suggest that not all group means are equal, thereby providing evidence against the null hypothesis.

One-Way Analysis of Variance (ANOVA)

One-way ANOVA is a statistical technique used to test whether there are any significant differences between the means of three or more independent groups. It partitions the total variability into components attributed to between-group and within-group variances, and uses the F-statistic to determine if the observed differences in group means are greater than would be expected by chance.

Hypothesis Testing

Hypothesis testing is a systematic method used in statistics to decide between competing hypotheses about a given data sample. It involves stating a null hypothesis (usually representing no difference or effect) and an alternative hypothesis, then using statistical tests to determine whether there is sufficient evidence to reject the null hypothesis at a predefined significance level.

Transcript

00:01 So in this problem, the thing that we want to do is test for any difference in the job satisfaction among the four professions.

00:07 So the first thing i'm going to do is come up with our hypotheses.

00:10 So our null hypothesis is that there is no difference in job satisfaction.

00:15 So what that means is that the population means for each of these jobs are equal.

00:23 So on average, these jobs have the same satisfaction rating.

00:27 And now my alternative hypothesis is that not all the jobs have.

00:32 The same job satisfaction rating.

00:36 So not all muse are equal.

00:46 So with that in mind, we have to find a way to test these hypotheses.

00:50 So we're going to use an an anova.

00:54 And for that, we're going to create an an annova table first.

00:57 So our annova table is going to be consisting of the source of our variation or of variance.

01:06 And then it's all also going to have a column for the sum of squares, a column for the degrees of freedom, a column for the mean of squares, and then a column for the f statistic.

01:18 And it would have a column for the p value, but i'm just not going to include that for now because we're going to calculate that separately.

01:25 That'll be the last thing we do, so we don't really need it on here.

01:33 Okay, so we have three sources of variance, one from our treatment, another from our error, and one more from our total.

01:43 So the first thing we're going to do is calculate a sum of squares for a treatment.

01:48 Before we do that, we have to come up with the population means for each of these.

01:53 So that, or the sample means, sorry, not the population means, the sample means for each of these.

01:58 So the sample mean for being a lawyer is equal to 50.

02:04 The sample mean for being a therapist is equal to 63 .7.

02:10 The sample mean for being a cabinet maker is equal to 69 .1.

02:18 And the sample mean for being a systems analyst is 61 .2.

02:24 Now we're going to find the grand mean.

02:26 The grand mean is just the mean, this is the average of our sample means.

02:32 Excuse the voice crack, sorry.

02:34 So we're going to take 50 plus 63 .7 plus 69 .1 plus 61 .2 and divided by 4.

02:39 And we get a grand mean of 61.

02:43 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 these individual means and our sample means squared times the number of elements in each sample.

03:03 So, for example, we have nine elements in each sample, so n for the lawyer is equal to nine.

03:12 And that's the same for all of them.

03:15 So with that in mind, we're going to take nine times the difference between our first sample of 50 minus our grand mean of 61.

03:30 We're going to square this and then add up, add this with the next sample mean minus the grand mean squared.

03:38 So 63 .7 minus 61 squared.

03:43 Plus 9 times 69 .1 minus, oh, sorry, we have 10, 10 elements in each sample.

03:56 That's my bad.

03:57 I should be able to count by now.

03:59 10 elements, and then 69 .1 minus 61 squared plus 10 times 61 .2 minus 61 squared.

04:15 Squared so we get a sum of squares for the treatment to be 1 ,939 .4 so if we go to our nova table wherever that is we can update that value 1939 .4 and now we're going to find the sum of squares for our total the sum of squares for our total is simply the difference between each individual element and and our grand mean squared.

04:51 So it would be 44, our first value here, 44 minus 61 squared plus the next item.

05:07 42 minus 61 squared all the way until we get to the end of our data set.

05:15 76 minus 61 squared.

05:21 And lastly, 62 minus 61 squared.

05:24 So we take the sum and we get a value...

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