Ap statistics Unit 3: Examining Relationships 2b residual Plots
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We are trying to model a real world situation with an equation, and the equation of that regression line is what models the behavior of the data.
We only really know the behavior of the data for the range that we gathered it for. Extrapolation means that you go outside the bounds of your data to predict. The trend that we see may not continue before or after the data we have collected.
A least squares regression line is a line that minimizes the sum of the squared distances from each point to the line.
Y=a+bx, where the line goes through the point  and slope b = 
The difference between what the model predicted would happen and what actually happened. Observed - Expected
Do a linear regression. When you do the linear regression, it stores the residuals automatically in your LIST menu. Then highlight on of your lists, go to the LIST menu, select residuals and it will put it into the list. Then go to statplot and create a scatterplot using your explanatory variable list, and your residuals list.
It should have a scattered random look to it with no apparent pattern.
Let y = number of manatees killed and x = number of powerboat registrations. The least-square regression equation is yˆ = − 41.43 + 0.1249x.
When 716,000 powerboats are registered, the predicted number of manatees killed will be −41.43 + 0.1249 × 716 = 47.99, or about 48 manatees.
1991 716 53 1992 716 38 1993 716 35 1994 535 49 Yes, the measures seem to be succeeding, three of the four new points are below the regression line, indicating that fewer manatees than predicted were killed. Additional evidence of success is provided by the two points for 1992 and 1993; they fall well below the overall pattern. e) In part (c) you predicted the number of manatee deaths in a year with 716,000 powerboat registrations. In fact, powerboat registrations were 716,000 for three years. Compare the mean manatee deaths in these three years with your prediction from part (c). How accurate was your prediction? The mean number of manatee deaths for the years with 716,000 powerboat registrations is 42. The prediction of 48 was too high.
The least squares regression line is yˆ = 31.9 − 0.304x . The calculator output (and Minitab output) is shown below: 
The means, standard deviations, and correlation are: x = 58.23% , sx = 13.03% , y=14.23 newbirds, sy =5.29 newbirds, r=−0.748.
The slope is b=−0.748  and the intercept is a= 14.23 − b×58.2331.9.
The slope tells us that as the percent of returning birds increases by one the number of new birds will decrease by −0.304 on average. The y intercept provides a prediction that we will see 31.9 new adults in a new colony when the percent of returning birds is zero. This value is clearly outside the range of values studied for the 13 colonies of sparrowhawks and has no practical meaning in this situation.
The predicted value for the number of new adults is 31.9 − 0.304×60 = 13.69 or about 14.
Let y = Blood Alcohol Content (BAC) and x = Number of Beers. The least-squares regression line is yˆ = −0.0127 + 0.017964x .
The slope indicates that on average, the BAC will increase by 0.017964 for each additional beer consumed. The intercept suggests that the average BAC will be −0.01270 if no beers are consumed; this is clearly ridiculous.
The predicted BAC for a student who consumed 6 beers is −0.0127 + 0.017964×6 = 0.0951.
The prediction error is 0.10 − 0.0951 = 0.0049.
12. The Trans-Alaska Oil pipeline is a tube formed from ½ inch thick steel that carries oil across 800 miles of sensitive arctic and sub-arctic terrain. The pipe and the welds that join the segments were carefully examined before installation. How accurate are field measurements of the depth of small defects? Scatterplot below compares the results of measurements on 100 defects made in the field with measurements of the same defects made in the laboratory. The line y = x is drawn on the scatterplot. The second plot is a residual plot for these data.  
There is a positive, linear association between the two variables. There is more variation in the field measurements for larger laboratory measurements. The values are scattered above and below the line y = x for small and moderate depths, indicating strong agreement, but the field measurements tend to be smaller than the laboratory measurements for large depths.
The points for the larger depths fall systematically below the line y = x showing that the field measurements are too small compared to the laboratory measurements.
In order to minimize the sum of the squared distances from the points to the regression line, the top right part of the blue line in the figure above would need to be pulled down to go through the “middle” of the group of points that are currently below the blue line. Thus, the slope would decrease and the intercept would increase. d) Discuss what the residual plot tells you about how well the least-squares regression line fits the data. The residual plot clearly shows that the prediction errors increase for larger laboratory measurements. In other words, the variability in the field measurements increases as the laboratory measurements increase. The least squares line does not provide a great fit, especially for larger depths.
The residuals in the same order as the observations are:
A scatterplot with the least squares regression line is shown below. 
We would certainly not use the regression line to predict fuel consumption. The scatterplot shows a nonlinear relationship.
The sum of the residuals provided is −0.01, which illustrates a slight roundoff error. d) Make a plot of residuals against the values of x. Draw a horizontal line at height zero on your plot. Notice that the residuals show the same pattern about this line as the data points show about the regression line in the scatterplot in (a). What do you conclude about the residual plot? The residual plot indicates that the regression line underestimates fuel consumption for slow and fast speeds and overestimates fuel consumption for moderate speeds. The quadratic pattern in the residual plot indicates that the regression model is not appropriate for these data. 
15. Below are four sets of data prepared by the statistician Frank Ascombe to illustrate the dangers of calculating without first plotting the data. 
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