Suppose we add the Ford Taurus to the data in Problem 25 . A...

Suppose we add the Ford Taurus to the data in Problem $25 .$ A Ford Taurus weighs 3305 pounds and gets 19 miles per gallon. (a) Redraw the scatter diagram with the Taurus included. (b) Recompute the linear correlation coefficient with the Taurus included. (c) Compare the results of parts (a) and (b) with the results of Problem $25 .$ Why are the results here reasonable? (d) Now suppose we add the Toyota Prius to the data in Problem 25 (remove the Taurus). A Toyota Prius weighs 2890 pounds and gets 60 miles per gallon. Redraw the scatter diagram with the Prius included. What do you notice? (c) Recompute the linear correlation coefficient with the Prius included. How did this new value affect your result? (f) Why does this observation not follow the pattern of the data?

Suppose we add the Ford Taurus to the data in Problem $25 .$ A Ford Taurus weighs 3305 pounds and gets 19 miles per gallon.
(a) Redraw the scatter diagram with the Taurus included.
(b) Recompute the linear correlation coefficient with the Taurus included.
(c) Compare the results of parts (a) and (b) with the results of Problem $25 .$ Why are the results here reasonable?
(d) Now suppose we add the Toyota Prius to the data in Problem 25 (remove the Taurus). A Toyota Prius weighs 2890 pounds and gets 60 miles per gallon. Redraw the scatter diagram with the Prius included. What do you notice?
(c) Recompute the linear correlation coefficient with the Prius included. How did this new value affect your result?
(f) Why does this observation not follow the pattern of the data?

Key Concepts

Leverage and Influential Points

Leverage refers to data points with predictor values far from the mean of the predictor variable, which means they have the potential to exert a strong influence on the results of a regression analysis. When these high-leverage points also have large residuals, they become influential points that can noticeably alter the slope and correlation of the regression line. Their impact needs to be carefully considered when interpreting the overall relationship between variables.

Outlier

An outlier is an observation that significantly deviates from the overall pattern of the data. Outliers can disproportionately influence statistical estimates such as the mean, standard deviation, and correlation coefficient. In the context of regression and correlation analysis, the inclusion of an outlier can distort the perceived strength and direction of a relationship.

Linear Correlation Coefficient

The linear correlation coefficient quantifies the strength and direction of the linear relationship between two variables. Values close to 1 or -1 indicate a strong positive or negative linear relationship, respectively, while values near 0 suggest little to no linear correlation. This measure is particularly sensitive to extreme values and anomalies within the dataset.

Scatter Diagram

A scatter diagram is a graphical representation that displays the relationship between two quantitative variables. It is useful for identifying patterns, trends, clusters, or potential outliers in the data. This visual tool helps to quickly assess whether a linear relationship, a curvilinear relationship, or no relationship exists between the variables.

Transcript

00:01 And we're making a scatterplot and playing with it so this would let me delete this this is the problem the scatter plot from exercise 25 right here right there that was it and we were supposed to add in the torus it said readj out of the scatter game with the torus which has just undo there we go it has a weight of 30305 and a mile sporegall of 19 and that gets tossed in here but let me show if you blink, you'll miss it.

00:35 So here, let me just show this here.

00:40 So you look right here.

00:42 I'm going to delete there, it's right there.

00:43 It's like overlaying this other point.

00:45 So i'm going to delete that.

00:49 So there, watch right here, right there.

00:53 There's a point gone and it's back.

00:55 So there is, it's right there.

00:59 And then we're going to compute the linear correlation coefficient.

01:03 So i use this spreadsheet function, correl, to do the r value without the taurus.

01:12 And then one with the torus you can see it includes the value so without the taurus it's slightly more linear data is slightly more linear but if you add the taurus in it's it's slightly less very slightly less and then suppose we add and compare the results it was basically looking at the the previous problem was looking at same thing is it linear now suppose we had the toyota prius to the previous problem which has a different weight it has 2890 and 60 miles per gallon...

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