Experience and Sales. Refer to exercise 7, where an...

Experience and Sales. Refer to exercise $7,$ where an estimated regression equation relating years of experience and annual sales was developed. a. Compute the residuals and construct a residual plot for this problem. b. Do the assumptions about the error terms seem reasonable in light of the residual plot?

Experience and Sales. Refer to exercise $7,$ where an estimated regression equation relating years of experience and annual sales was developed.
a. Compute the residuals and construct a residual plot for this problem.
b. Do the assumptions about the error terms seem reasonable in light of the residual plot?

Key Concepts

Residuals

Residuals are the differences between the observed values and the values predicted by a regression model. They are used to assess how well the model fits the data, and each residual corresponds to one observation's discrepancy, providing insights into the model's performance in capturing the underlying relationship between the predictors and the response variable.

Residual Plot

A residual plot is a graphical representation where residuals are plotted against predicted values or an independent variable. This plot is used as a diagnostic tool to identify non-random patterns, which can indicate problems in the model such as non-linearity, heteroscedasticity, or the presence of outliers. An ideal residual plot should show no systematic patterns, implying that the model has appropriately captured the underlying process.

Assumptions About Error Terms

The assumptions about error terms in regression analysis include normality, independence, homoscedasticity (constant variance), and a mean of zero. Checking these assumptions is crucial because violations can affect the validity of inference, such as confidence intervals and hypothesis tests. A proper residual plot can help assess these assumptions; for instance, a random dispersion of residuals supports the assumption of homoscedasticity and independence, while patterns or trends might suggest otherwise.

Transcript

00:01 The following is a solution to number 48, and we're given a price and a score.

00:07 I can't remember what it's about, but we're supposed to compute the residuals and construct a residual plot.

00:15 So i'm going to do this shorter way, just because it's, you know, we know how to do it now.

00:20 Once i can get this thing to work, there we go.

00:24 So here's my data.

00:25 And if you go to data and then data analysis and go to regression, so your y input, at least for me is from b2 to b11, and then the x input is from a2 to a11.

00:35 And i go ahead and check the residuals and the residual plots.

00:38 You can do all this stuff if you want, but really, for this problem, you just need those two.

00:43 And go ahead and click ok, and it brings you to this sheet here.

00:46 So you can see, this looks to be a pretty significant regression line here.

00:55 And here are the residuals.

00:57 So the residuals are the actual y value minus the predicted.

01:00 So you would just take the y -value, minus 70 .65.

01:03 So you get that predicted by plugging in the x value into your regression line.

01:07 And you should get these residuals here.

01:09 And we plotted them against the x variable...