Daily Regents: Scatter Plot Residuals (August 2014)

I'll be reviewing a New York State Regents Exam Question every day from now until the Regents exams begin next month. At least, that is the plan.

August 2014, Questions 31

31. The table below represents the residuals for a line of best fit.
Plot these residuals on the set of axes below.
Using the plot, assess the fit of the line for these residuals and justify your answer.

Plot the points on the table. It should look like this:

If a regression line is a "good fit", the residuals should be scattered about, above and below the x-axis, randomly.
If a pattern emerges, then the regression is not a good fit for the data. (For example, suppose you did a linear regression, but the data suggested that you should have done a quadratic regression instead. Then a pattern will emerge in the residuals.)

The graph below shows that there is a pattern among the residuals, and it looks like it forms a curve.
DO NOT draw the curve! Just explain that it is not a good fit because of the pattern.

Any questions?

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Regents: Residuals and the Line of Best Fit (August 2014 Common Core Algebra 1)

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New York State Common Core Algebra 1 Regents Exam, August 2014.

31. The table below represents the residuals for a line of best fit. Plot these residuals on the set of axes below.

Using the plot, assess the fit of the line for these residuals and justify your answer.

This is another new topic in Algebra 1. The residuals for a line of best fit are distance a point is away from the line. It is the actual value – the expected value. In other words, subtract the y-value of the data point from the y-value of the trend line for that x-value. This could give you a negative value; that’s okay.

A trend line should go through the middle of the data on a scatter plot. That means that some values are above and some values are below the line. Therefore, there should be many positive and negative residuals. If the residual points are plotted, they should be scattered out in such a way that there is no pattern (almost like a scatter plot with no correlation). If there is a pattern, then there was a mistake.

If the pattern for the residuals is somewhat linear, there was a mistake calculating the line of best fit.

If the pattern for the residuals is a curve (such as a parabola), then it was incorrect to try to use a linear regression in the first place, instead of, say, a quadratic or some other regression.

When the residuals are plotted, you will get the graph below. As you can see, it is a poor fit because there is a pattern.

To get both points for this question, you needed a correct plot, stated that it is a poor fit, and gave a correct justification for the poor fit (a pattern was formed). One mistake or one thing missing would cost you one point point. More than one mistake would result in zero points for this question.