write an essay about statistics and follow the instructions

Linearregressionassignment51-1.docx

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THIS ASSIGNMENT IS DUE BY 11:59 PM WEDNESDAY, DECEMBER 20, SUBMITTED VIA BLACKBOARD (MORE DETAILS)

Submit via the assignment link on blackboard, not by email or in person. You may submit more than once (say, if you make a mistake), I will take your final submission before 11:59 PM Wednesday.

Linear regression assignment

The following assignment will give you a taste of using linear regression. The dataset supplied is called the Boston Housing Dataset. The data are from the 1970s so they are a bit dated.

The dataset is posted to blackboard in this same folder.

Each row in the dataset represents one (unnamed) suburb of Boston. The following 13 variables are provided:

1. MEDV: median value of owner-occupied homes in $1000 (so 24 means $24,000). This is the variable Y we are trying to predict by one of the X variables below. Note that because MEDV is in thousands of dollars, when you interpret Beta_1, the change in Y should be in thousands. For instance, if Beta_1=2.5, this means a change of one unit in X increases Y by 2.5, which in this case means $2,500 (2.5*1000).

2. CRIM: per-capita (per person) CRIMe rate by town. For instance, 0.006 means there were 0.006 crimes per person living in the town, or 6 crimes per 1,000 people (multiply each by 1,000)

3. ZN: proportion (between 0 and 100%) of residential land ZoNed for lots over 25,000 square feet. If more lots—land allowed for a house—are big (over 25,000 square feet) this means houses have more land so are more expensive.

4. INDUS: proportion (between 0 and 100%) of non-retail business acres per town. If 12%, this means 12% of acres in the town are zoned for businesses that are not retail (i.e., brick-and-mortar stores), for instance factories or office buildings. A higher value means the area is more INDUStrial rather than residential/commercial.

5. NOX: measure of air pollution: Nitric OXides concentration (parts per 10 million) in the air

6. RM: Average number of RooMs per residential dwelling in the town

7. AGE: proportion of (between 0 and 100%) of owner-occupied units (as opposed to rentals) that were built prior to 1940. If AGE is higher, this means more of the houses were built prior to 1940, which means they overall tend to be older. Older houses are generally less expensive, though not necessarily.

8. DIS: weighted DIStances to five Boston employment centers (higher values mean the home is generally further from major employment centers in the area). This means, for instance, that a given suburb is located 1, 5, 10, 10, and 20 miles from the 5 major employment centers. DIS is some kind of average of these distances, for instance, the mean =(1/5)*(1+5+10+10+20)=9.2 miles, in which case each weighted equally, or perhaps, the employment centers have higher weights by size. For instance, (1/3)*(1+5)+(1/6)*(10+10)+(1/2)*20, if the 5th (20 miles) is the biggest.

9. RAD: Index of accessibility of RADial highways, meaning highways that lead to urban centers (as opposed to, say, interstate highways that don’t primarily serve major cities). Higher numbers mean more accessible, generally a good thing, but you also don’t want highways right next to your house, either.

10. TAX: full-value property-TAX rate per $10,000 (higher property taxes may mean, for instance, that the schools in an area are better because public schools are often funded by property taxes). For instance, if TAX is $296, the owner pays $296 in property taxes for each $10,000 their house is worth. So a $100,000 house would cost the person $2,960 in property tax.

11. PRATIO: teacher to Pupil (student) RATIO, meaning the number of students per teacher by town (higher means more students per teacher, generally a bad thing in classrooms because students get less attention).

12. B: =1000*(Bk – 0.63)^2, where ‘Bk’ represents the proportion (between 0 and 1) of Blacks living in each town. The further Bk is from 0.63, either up or down, the higher this is (since it is squared). If Bk is low (a mostly non-black neighborhood, probably mostly white), then increasing Bk, the black proportion, probably decreases housing values if people are racist and view black neighbors negatively. However, If Bk is near 1 (mostly black neighborhood) then the relationship is different. The idea is to make a variable B from Bk that has a closer to linear relationship to MEDV.

13. LSTAT: percentage (0 to 100%) of the population that is Lower socio-economic STATus (i.e., “poorer”)

Goal: you need to perform a single variable linear regression to predict MEDV, using ONLY ONE of the remaining 12 variables. It is possible that with only one variable you will not get a good answer in terms of a model that fits MEDV well, but that is okay.

Remember we are looking for MEDV to have a (roughly) linear relationship with another variable. You should start off by plotting scatterplots of each other variable paired with MEDV. There is not necessarily a “right” answer here so I expect people to work individually and to have a variety of variables chosen. You do not have to find “the best” linear model, just pick one and talk about it.

Note: in the following examples, I will be showing you regression using a different dataset than the one you will actually be using. This dataset involves predicting car MPG ratings based on car attributes like model year, weight, horsepower, etc.

Scatterplot

To do a scatterplot, highlight the two columns of data you want to use. The left column will be on the X axis. Seeing a clearer positive or negative relationship between two variables means that they may be good candidates for performing a linear regression of one on the other.

To select a column, you may click on the tab above the worksheet with the letter names of the columns (A, B, C), etc. To select the second column, hold down the CTRL button and then click on that column.

Insert -> Scatterplot

You want to have MEDV on the Y axis. The easiest way to make sure this happens is to just cut and paste the MEDV column so it is to the right of all the other columns, so that it will automatically be the Y axis. I have already modified the spreadsheet for this.

Alternatively, right click on the points in your scatterplot and click “Select data”, then click “Edit” and manually change the inputs for the X and Y (be careful not to use the arrow keys because this will mess it up). The first solution is easier.

You should manually edit the title and axis labels so that it is clear what is being displayed.

You can add labels to the X and Y axis by left-clicking in the chart area. Axis and chart labels can be manually edited by clicking on them.

· Windows: A green “+” will appear in the right corner. Click on it to add chart elements, check “axis titles” and “chart title” (should be there already).

· Mac: Click on “Add chart element” at the left of the “Chart design” ribbon, once the chart is selected. Click on “Axis titles” and (repeating twice) click on primary horizontal/vertical axis. You can do the same for the regression output graphs.

Here I have done a scatterplot between MPG (Y) and WEIGHT (X) of the car. You can see a fairly clear relationship, where lower WEIGHT is associated with LOWER MPG, although we will see later there may be a slight heteroskedasticity (error/residual variance not equal).

Notice I have edited the graph so that the axis and chart labels are descriptive.

Regression

You will need to use the “Data Analysis” tool, which should appear on the far right under the Data menu.

If your excel does not have it loaded, you will have to add it in first:

· For Windows: File-> Options-> Add-ins -> “Analysis toolpak”. You may have to click on “Go…” next to “Manage” menu in the Add-in window to load it.

In this menu, check “Analysis Toolpak” then click OK to load it

· For Mac: Tools -> Add-ins -> “Data analysis toolpak”. If you have a Mac with Excel 2016 or above, you will be fine with this. If you have Excel 2011 or earlier you will not be able to load the Toolpak onto your computer, and will likely need to use a school computer to do so. You can do the initial regression on a school computer, then save the Excel workbook with the results and do the rest of the work on your own computer. On a Mac, you may have to click “save” or reload to use the toolpak for the first time.

To perform regression, click on Data-> Data Analysis -> select Regression

Input Y range: select the MEDV column (excluding the variable name in the top row), click ENTER

Input X range: select the variable you want (excluding the variable name in the top row), click ENTER

If you include the variable name you will get a warning that your data includes non-numeric (text) values

Confidence level: check the box and type 95 in the box (this tells you it will do 95% confidence intervals for parameters)

Under output options, select “New worksheet ply”. This will output to another sheet in the same workbook

Under “Residuals”, check the boxes “Residuals”, “Residual Plots”, and “Line fit plots”.

The icon may flash for a few seconds as it calculates. Then go to the new worksheet.

You will see a bunch of numbers and plots.

Regression statistics:

1. R square and Adjusted R square: give you a measure of how well the X variable explains the variation in the Y variable (MPG or MEDV). This number will be between 0 and 1, with higher meaning better. In our case the R square is relatively high, so weight and MPG can be well-explained by a linear relationship. Use the “R square”, not “Adjusted R square”. The correlation (R, not shown) is the square root of R^2.

2. Standard error: The standard deviation of the error term in the regression (the bigger this is, the more error in predicting Y from your model).

The interpretation of standard error (s) is that 95% of the time, you expect your prediction to be within 2*s of the actual value of Y.

Regression Statistics
Multiple R	0.831740933
R Square	0.69179298
Adjusted R Square	0.691014679
Standard Error	4.344628058
Observations	398

Skip the ANOVA box

You will now see a box with the regression model (you may have to widen or shorten the column widths in Word to get – signs on the same line):

	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	46.31736442	0.79524523	58.24286987	2.7534E-196	44.75393408	47.88079476
X Variable 1	-0.00767661	0.000257487	-29.81359895	2.9728E-103	-0.008182822	-0.007170398

You can use the following tool (under Home -> Number -> Decrease decimal) to round so the numbers are more readable. Alternatively, format the numbers to have fewer decimal places by highlighting the numbers, right click -> Format Cells 0-> Number.

Note, the last two columns are duplicates of the ones next to them.

	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	46.3174	0.7952	58.2429	0.0000	44.7539	47.8808
X Variable 1	-0.0077	0.0003	-29.8136	0.0000	-0.0082	-0.0072

Note: the table above has two rows, the first for the intercept (Beta_0), the second for the slope (Beta_1). In #5-6, there is also a hypothesis test being conducted on if Beta_0=0, while we are only concerned with the same test about the value of Beta_1. You can ignore the test and confidence interval done in the first row.

1. The first column tells you if the statistics are for the intercept (Beta_0) or X variable (Beta_1) term.

2. Coefficients: the value of the estimate of Beta_0 or Beta_1. For instance, this means that an increase in weight of 1 pound will decrease MPG by 0.0077 (Beta_1).

Beta_0, your intercept, is what your model says Y (MPG) would be if X (WEIGHT) is 0. It is not meaningful if 0 is out of range for your X variable. For instance, it is impossible to have a car that weighs 0 pounds (WEIGHT=0), so in our case it is not meaningful due to logic. However, in some cases it may be logically meaningful but you may not have 0 in the range just because your data doesn’t include any of these examples. For instance for the Boston variable LSTAT (socio-economic status), 0 would mean there are no people of low socio-economic status in that suburb. In theory Beta_0 there would be what the model predicts for housing value if there are no such people. But because LSTAT is always positive (between about 5 and 45 in the data), Beta_0 would not have a meaningful interpretation there.

3. Standard error: the standard deviation of the estimate

4. T Stat: the T-statistic calculated for the coefficient, for testing if it =0.

5. P-value: the p-value. This is the p-value for the hypothesis test that the coefficient=0. Note, if you see something like 2.7534E-196, this means 0. Reformat the cell as a number to show more decimal places (right click -> Format Cells ->Number -> “Number” under category). For Beta_1, the p-value being low (below 0.05) means that we statistically say there IS a relationship between the variables because we are saying the slope IS NOT ZERO.

6. Lower 95% and Upper 95% are the values of the 95% confidence interval for the term. The interpretation here is related to the p-value. For instance, the p-value for Beta_1 is zero, so we say there is a relationship (slope is NOT 0). Therefore the confidence interval for Beta_1 does not include 0, since it indicates the relationship between WEIGHT and MPG is firmly positive.

The tables below show the residuals for the model and the predicted value of MPG (predicted Y) for each observation.

Plots: You should change the default plot titles so they are actually descriptive.

· X variable 1 residual plot: shows the residuals (errors) plotted on the y-axis. Remember these should basically be distributed equally above and below the x-axis (0) along the range of the data (homoscedasticity).

In this case what you see is that there tends to be more error in the prediction when WEIGHT is lower than higher. This is some evidence of heteroscedasticity (the error is not evenly distributed at all values of WEIGHT), but not a whole lot.

The fact that the error is not evenly distributed means a linear model may not be very appropriate (see the line fit model). This also affects our interpretation of the standard error of the error term (4.344 above). The interpretation is that about 95% of the time, our prediction for MPG is within about 2*4.344 (=8.688) of the actual MPG (remember this is like the empirical rule for a normal distribution). However, if there is heteroscedasticity, this may be off, but not necessarily. Even if it is still accurate, it means that the value of the error depends on what WEIGHT is.

For a similar but more extreme example, see a regression where X is HORSEPOWER instead: The scatterplot and line fit plot suggest that the relationship between horsepower and MPG is not a line, but rather a curve (more horsepower decreases MPG, but at higher horsepower the amount of decrease is less).

The residuals (errors) confirm this. You see at low and high values of horsepower, the errors tend to be positive not negative. This means a linear model may not be the best in our situation.

· X variable 1 line plot: shows a scatterplot of the variables with the best regression line plotted.

Note that in this plot, the regression line is plotted as the predicted points, rather than as a solid line. If you want to change this, follow these steps:

· Left click on the orange points so they are highlighted, then right-click on the orange points, and select “Add trendline”. The “linear” option should be selected by default.

· This will overlay the line onto the points, as shown below:

· To get rid of the orange points, left click on them again, so they are selected, then right click, and click on “Format data series”

· In the “Format data series” on the right, click on the left icon (that looks like a graduation hat), then on “Marker”, then “Marker Options”. Click on “None.”

· You can remove the entry in the legend for “Predicted Y”, since the orange dots don’t appear, by right clicking on it and then “Delete” (NOT “Delete series”).

· Your graph should now look like this:

Assignment: The following should take you 2-3 pages to do, and no more than 3 pages.

When you write up your assignment, please write it in essay form like it is a report or paper. Do not answer the questions below in a numbered way as if they were individual problems.

Conduct scatterplot analysis to try to determine which variables may be good to try in regression. Pick one only, and explain your observations about this variable ONLY. Make sure to address/complete all the following items in your analysis.

1. Choose one variable to perform regression with. Include the initial scatterplot without the regression line in your document. Explain your observations from the scatterplot.

2. Explain logically (before doing any regression) why you think that variable would have a relationship with MEDV, and what it should be. Address the following aspects:

a. Why should the relationship be positive or negative?

b. Should it be weak or strong?

c. Do you think that there is some causal relationship (e.g. X directly causes Y) or it is just an association? If an association, is there another variable that might explain the relationship? If you think there is another variable (it doesn’t have to be one of the other 11 variables in the dataset) that may be behind the relationship, you don’t have to do additional analysis (like more scatterplots), just explain your reasoning. For instance, “I think that there is a relationship between MEDV and property tax that is negative, where higher property taxes reduce median house value on average. Since TAX is per $10,000 of value, not total amount paid, it is possible that higher property values pay higher total tax but lower per-dollar tax. Maybe property tax goes mostly to public schools, and maybe people in richer areas (with higher MEDV) send their kids to private schools, so the per-house-dollar-value property tax assessment is lower.”

3. Perform regression as indicated above. Include the linear regression plot (the scatterplot plus the line)

4. In your document, copy ONLY the tables for regression statistics and the coefficient estimates.

5. In the text body, mention what the R-squared value (coefficient of determination) is and say what that means about the quality of your regression. If it is big or small, what does this mean?

6. In the text body, report the coefficient estimates for Beta_0 and Beta_1 are. Explain what the value of Beta_0 and Beta_1 mean in this and say if this means the relationship is positive or negative. Also explain if Beta_0 is meaningful in this situation (it is meaningful if the variable X value 0 has a real meaning or is within the range of X values in your dataset). For instance, in the example I gave you would say

a. Beta_0=4.317 is not meaningful because WEIGHT can never be 0

b. Beta_1=-0.0077 means an increase in WEIGHT of 1 pound will decrease MPG by 0.0077.

7. State what the p-value for Beta_1 in your regression is and what it means. Does it mean there definitely is or isn’t a relationship between the two variables? If Beta_1=0, the regression is a horizontal line so there is no relationship.

8. In the text body, report the 95% confidence interval for the value of Beta_1. Does this support your answer to #7 about the hypothesis test on Beta_1? Explain.

9. Include the residual (error) plot. Do the residuals show it is a well-fit model (remember we need homoscedasticity)? If there is an even distribution of the residuals above and below the horizontal 0 line, that is good. Do they show that a linear model is appropriate in this case? What about the line plot? Describe what you see.

MPG of car and weight

MPG 3504 3693 3436 3433 3449 4341 4354 4312 4425 3850 3563 3609 3761 3086 2372 2833 2774 2587 2130 1835 2672 2430 2375 2234 2648 4615 4376 4382 4732 2130 2264 2228 2046 2634 3439 3329 3302 3288 4209 4464 4154 4096 4955 4746 5140 2962 2408 3282 3139 2220 2123 2074 2065 1773 1613 1834 1955 2278 2126 2254 2408 2226 4274 4385 4135 4129 3672 4633 4502 4456 4422 2330 3892 4098 4294 4077 2933 2511 2979 2189 2395 2288 2506 2164 2100 4100 3672 3988 4042 3777 4952 4464 4363 4237 4735 4951 3821 3121 3278 2945 3021 2904 1950 4997 4906 4654 4499 2789 2279 2401 2379 2124 2310 2472 2265 4082 4278 1867 2158 2582 2868 3399 2660 2807 3664 3102 2875 2901 3336 1950 2451 1836 2542 3781 3632 3613 4141 4699 4457 4638 4257 2219 1963 2300 1649 2003 2125 2108 2246 2489 2391 2000 3264 3459 3432 3158 4668 4440 4498 4657 3907 3897 3730 3785 3039 3221 3169 2171 2639 2914 2592 2702 2223 2545 2984 1937 3211 2694 2957 2945 2671 1795 2464 2220 2572 2255 2202 4215 4190 3962 4215 3233 3353 3012 3085 2035 2164 1937 1795 3651 3574 3645 3193 1825 1990 2155 2565 3150 3940 3270 2930 3820 4380 4055 3870 3755 2045 2155 1825 2300 1945 3880 4060 4140 4295 3520 3425 3630 3525 4220 4165 4325 4335 1940 2740 2265 2755 2051 2075 1985 2190 2815 2600 2720 1985 1800 1985 2070 1800 3365 3735 3570 3535 3155 2965 2720 3430 3210 3380 3070 3620 3410 3425 3445 3205 4080 2155 2560 2300 2230 2515 2745 2855 2405 2830 3140 2795 3410 1990 2135 3245 2990 2890 3265 3360 3840 3725 3955 3830 4360 4054 3605 3940 1925 1975 1915 2670 3530 3900 3190 3420 2200 2150 2020 2130 2670 2595 2700 2556 2144 1968 2120 2019 2678 2870 3003 3381 2188 2711 2542 2434 2265 2110 2800 2110 2085 2335 2950 3250 1850 1835 2145 1845 2910 2420 2500 2905 2290 2490 2635 2620 2725 2385 1755 1875 1760 2065 1975 2050 1985 2215 2045 2380 2190 2320 2210 2350 2615 2635 3230 3160 2900 2930 3415 3725 3060 3465 2605 2640 2395 2575 2525 2735 2865 3035 1980 2025 1970 2125 2125 2160 2205 2245 1965 1965 1995 2945 3015 2585 2835 2665 2370 2950 2790 2130 2295 2625 2720 18 15 18 16 17 15 14 14 14 15 15 14 15 14 24 22 18 21 27 26 25 24 25 26 21 10 10 11 9 27 28 25 25 19 16 17 19 18 14 14 14 14 12 13 13 18 22 19 18 23 28 30 30 31 35 27 26 24 25 23 20 21 13 14 15 14 17 11 13 12 13 19 15 13 13 14 18 22 21 26 22 28 23 28 27 13 14 13 14 15 12 13 13 14 13 12 13 18 16 18 18 23 26 11 12 13 12 18 20 21 22 18 19 21 26 15 16 29 24 20 19 15 24 20 11 20 21 19 15 31 26 32 25 16 16 18 16 13 14 14 14 29 26 26 31 32 28 24 26 24 26 31 19 18 15 15 16 15 16 14 17 16 15 18 21 20 13 29 23 20 23 24 25 24 18 29 19 23 23 22 25 33 28 25 25 26 27 17.5 16 15.5 14.5 22 22 24 22.5 29 24.5 29 33 20 18 18.5 17.5 29.5 32 28 26.5 20 13 19 19 16.5 16.5 13 13 13 31.5 30 36 25.5 33.5 17.5 17 15.5 15 17.5 20.5 19 18.5 16 15.5 15.5 16 29 24.5 26 25.5 30.5 33.5 30 30.5 22 21.5 21.5 43.1 36.1 32.79999999999 9997 39.4 36.1 19.899999999999999 19.399999999999999 20.2 19.2 20.5 20.2 25.1 20.5 19.399999999999999 20.6 20.8 18.600000000000001 18.100000000000001 19.2 17.7 18.100000000000001 17.5 30 27.5 27.2 30.9 21.1 23.2 23.8 23.9 20.3 17 21.6 16.2 31.5 29.5 21.5 19.8 22.3 20.2 20.6 17 17.600000000000001 16.5 18.2 16.899999999999999 15.5 19.2 18.5 31.9 34.1 35.700000000000003 27.4 25.4 23 27.2 23.9 34.200000000000003 34.5 31.8 37.299999999999997 28.4 28.8 26.8 33.5 41.5 38.1 32.1 37.200000000000003 28 26.4 24.3 19.100000000000001 34.299999999999997 29.8 31.3 37 32.200000000000003 46.6 27.9 40.799999999999997 44.3 43.4 36.4 30 44.6 40.9 33.799999999999997 29.8 32.700000000000003 23.7 35 23.6 32.4 27.2 26.6 25.8 23.5 30 39.1 39 35.1 32.299999999999997 37 37.700000000000003 34.1 34.700000000000003 34.4 29.9 33 34.5 33.700000000000003 32.4 32.9 31.6 28.1 30.7 25.4 24.2 22.4 26.6 20.2 17.600000000000001 28 27 34 31 29 27 24 23 36 37 31 38 36 36 36 34 38 32 38 25 38 26 22 32 36 27 27 44 32 28 31

Weight of vehicle (pounds)

miles per gallon (MPG) rating

Weight Residual Plot

3504 3693 3436 3433 3449 4341 4354 4312 4425 3850 3563 3609 3761 3086 2372 2833 2774 2587 2130 1835 2672 2430 2375 2234 2648 4615 4376 4382 4732 2130 2264 2228 2046 2634 3439 3329 3302 3288 4209 4464 4154 4096 4955 4746 5140 2962 2408 3282 3139 2220 2123 2074 2065 1773 1613 1834 1955 2278 2126 2254 2408 2226 4274 4385 4135 4129 3672 4633 4502 4456 4422 2330 3892 4098 4294 4077 2933 2511 2979 2189 2395 2288 2506 2164 2100 4100 3672 3988 4042 3777 4952 4464 4363 4237 4735 4951 3821 3121 3278 2945 3021 2904 1950 4997 4906 4654 4499 2789 2279 2401 2379 2124 2310 2472 2265 4082 4278 1867 2158 2582 2868 3399 2660 2807 3664 3102 2875 2901 3336 1950 2451 1836 2542 3781 3632 3613 4141 4699 4457 4638 4257 2219 1963 2300 1649 2003 2125 2108 2246 2489 2391 2000 3264 3459 3432 3158 4668 4440 4498 4657 3907 3897 3730 3785 3039 3221 3169 2171 2639 2914 2592 2702 2223 2545 2984 1937 3211 2694 2957 2945 2671 1795 2464 2220 2572 2255 2202 4215 4190 3962 4215 3233 3353 3012 3085 2035 2164 1937 1795 3651 3574 3645 3193 1825 1990 2155 2565 3150 3940 3270 2930 3820 4380 4055 3870 3755 2045 2155 1825 2300 1945 3880 4060 4140 4295 3520 3425 3630 3525 4220 4165 4325 4335 1940 2740 2265 2755 2051 2075 1985 2190 2815 2600 2720 1985 1800 1985 2070 1800 3365 3735 3570 3535 3155 2965 2720 3430 3210 3380 3070 3620 3410 3425 3445 3205 4080 2155 2560 2300 2230 2515 2745 2855 2405 2830 3140 2795 3410 1990 2135 3245 2990 2890 3265 3360 3840 3725 3955 3830 4360 4054 3605 3940 1925 1975 1915 2670 3530 3900 3190 3420 2200 2150 2020 2130 2670 2595 2700 2556 2144 1968 2120 2019 2678 2870 3003 3381 2188 2711 2542 2434 2265 2110 2800 2110 2085 2335 2950 3250 1850 1835 2145 1845 2910 2420 2500 2905 2290 2490 2635 2620 2725 2385 1755 1875 1760 2065 1975 20 50 1985 2215 2045 2380 2190 2320 2210 2350 2615 2635 3230 3160 2900 2930 3415 3725 3060 3465 2605 2640 2395 2575 2525 2735 2865 3035 1980 2025 1970 2125 2125 2160 2205 2245 1965 1965 1995 2945 3015 2585 2835 2665 2370 2950 2790 2130 2295 2625 2720 -1.4185227562673042 -2.9676434541852004 -1.9405322406143029 -3.9635620708060841 -2.8407363097832601 2.0067998672391525 1.1065957980701953 0.78417817538527856 1.6516351126089717 -1.7624156741487482 -3.9656027624956423 -4.6124786995550267 -2.4456339698382017 -8.6273457629885684 -4.1084453486320669 -2.5695281091619648 -7.0224481029336268 -5.457974184887874 -2.9661849841022701 -6.2307849529605761 -0.80546232945412655 -3.6632019649243297 -3.0854155184402856 -3.1678175374539173 -4.9897009709883626 -0.88980897524499625 -2.7245187805234252 -1.6784591201398626 -0.99164559776560424 -2.9661849841022701 -0.93751923553612571 -4.2138771978374763 -5.6110202294720963 -7.0971735118833301 -3.9175024104225251 -3.7619295174544369 -1.9691979891804507 -3.0766705300754218 -6.5126611991459527E-3 1.9510229051021071 -0.42872621471510186 -0.8739695984228355 3.7202384464900007 3.1158269431293704 6.1404113083163949 -5.5792454109154512 -5.8320873863307128 -2.1227301904589808 -4.2204854296004655 -6.2752900783488883 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Weight

Residuals

Horsepower Line Fit Plot

Horsepower

MPG

Horsepower Residual Plot

46 46 48 48 48 49 52 52 52 52 53 53 54 58 58 60 60 60 60 60 61 62 62 63 63 63 64 65 65 65 65 65 65 65 65 65 65 66 67 67 67 67 67 67 67 67 67 67 67 67 68 68 68 68 68 68 69 69 69 70 70 70 70 70 70 70 70 70 70 70 70 71 71 71 71 71 72 72 72 72 72 72 74 74 74 75 75 75 75 75 75 75 75 75 75 75 75 75 75 76 76 76 76 77 78 78 78 78 78 78 79 79 80 80 80 80 80 80 80 81 81 82 83 83 83 83 84 84 84 84 84 84 85 85 85 85 85 85 85 85 85 86 86 86 86 86 87 87 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 89 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 91 92 92 92 92 92 92 93 94 95 95 95 95 95 95 95 95 95 95 95 95 95 95 96 96 96 97 97 97 97 97 97 97 97 97 98 98 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 102 103 105 105 105 105 105 105 105 105 105 105 105 105 107 108 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 112 112 112 113 115 115 115 115 115 116 120 120 120 120 122 125 125 125 129 129 130 130 130 130 130 132 133 135 137 138 139 139 140 140 140 140 140 140 140 142 145 145 145 145 145 145 145 148 149 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 152 153 153 155 155 158 160 160 165 165 165 165 167 170 170 170 170 170 175 175 175 175 175 180 180 180 180 180 190 190 190 193 198 198 200 208 210 215 215 215 220 225 225 225 230 -6.6750032869024309 -6.6750032869024309 10.740686179804882 11.940686179804878 11.040686179804879 -3.2014690868414633 -0.72793488678050267 -2.7279348867805027 1.0720651132194945 12.272065113219497 1.4299098465731497 1.4299098465731497 -8.412245420073198 5.2191335133414185 8.3191335133414199 -3.4651770199512733 -5.9651770199512733 5.6348229800487282 7.6348229800487282 4.6348229800487282 1.6926677134023791 -0.3494875532439643 7.5505124467560378 0.50835718010969089 4.7083571801096937 8.0083571801096909 9.1662019134633432 1.3240466468169956 2.3240466468169956 4.424046646816997 2.1240466468169963 7.5240466468169984 16.924046646816997 11.124046646816993 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-2.151387685805112 -4.151387685805112 -5.151387685805112 -3.651387685805112 -0.45138768580511268 8.7486123141948866 -3.8356982190978073 -3.3778534857441507 -7.3621640190368431 -5.3621640190368431 -5.3621640190368431 -5.3621640190368431 -7.3621640190368431 -1.3621640190368431 -2.8621640190368431 -4.1621640190368439 -2.7621640190368417 -0.16216401903684385 4.5378359809631554 3.2378359809631583 -2.0464745523295385 -3.8886298189758826 -4.5729403522685743 1.4270596477314257 -6.5729403522685743 -5.5729403522685743 -7.5729403522685743 -1.5729403522685743 -2.5729403522685743 -4.0729403522685743 -5.5729403522685743 -5.0729403522685743 -1.0729403522685743 -1.0729403522685743 -2.6729403522685757 -3.9729403522685729 -1.9729403522685729 0.92705964773142568 -0.17294035226857574 2.4270596477314257 -4.2572508855612661 -3.2572508855612661 -0.25725088556126607 3.9005938477923863 3.2162833144996945 -0.18371668550030407 -0.28371668550030549 7.0162833144996952 5.0162833144996952 3.774128047853349 -5.4944930187320367 -4.4944930187320367 -2.8944930187320352 3.2055069812679626 -0.67880355202472842 -3.2052693519637643 -1.005269351963765 2.7947306480362357 -6.5738904185491513 -1.9738904185491499 -1.4160456851954955 -6.4160456851954955 -6.4160456851954955 -4.4160456851954955 -2.4160456851954955 13.599643781511816 -2.7425114851345356 -0.42682201842722733 -4.3111325517199184 -1.6532878183662625 2.2045569149873891 0.10455691498739128 -0.8375983516589578 -4.8375983516589578 -1.8375983516589578 -3.8375983516589578 -0.3375983516589578 1.5624016483410408 -0.3375983516589578 -2.0219088849516496 -4.0483746848906854 -2.0483746848906854 -2.0483746848906854 -4.0483746848906854 0.45162531510931458 -1.5483746848906854 2.1516253151093139 -2.5748404848297248 -0.41699575147606893 1.7408489818775834 -0.25915101812241659 -1.259151018 1224166 -2.2591510181224166 -1.2591510181224166 0.74084898187758341 -1.2591510181224166 -2.2591510181224166 -2.2591510181224166 -1.2591510181224166 -3.2591510181224166 -2.2591510181224166 -5.2591510181224166 -1.2591510181224166 -3.2591510181224166 -2.2591510181224166 -2.2591510181224166 -0.25915101812241659 -0.25915101812241659 -3.2591510181224166 -3.2591510181224166 2.2408489818775834 -1.4434615514151083 -1.785616818061456 -1.785616818061456 -2.4699273513541478 1.4300726486458508 -1.9963931512931836 -0.68070368458587893 -2.6807036845858789 1.1085199821823934 0.10851998218239345 -0.89148001781760655 3.8085199821823927 -1.5757905511102983 1.8977436489506623 -0.10225635104933772 -0.10225635104933772 2.8977436489506623 2.3977436489506623 1.6869673157189311 0.6869673157189311 1.6869673157189311 0.6869673157189311 0.6869673157189311 0.47619098248720348 0.47619098248720348 -0.52380901751279652 4.9761909824872035 4.4761909824872035 5.0546383160237411 3.0546383160237411 5.5546383160237411 -0.47182748391529827 6.3173961828529741 3.3173961828529741 1.6330856495602823 3.8958435163895118 4.2115329830968236 8.0007566498650888 4.0007566498650888 7.0007566498650888 8.7899803166333612 9.5792039834016265 9.5792039834016265 7.5792039834016265 12.368427650169899

Horsepower

Residuals

Weight Line Fit Plot

Y 3504 3693 3436 3433 3449 4341 4354 4312 4425 3850 3563 3609 3761 3086 2372 2833 2774 2587 2130 1835 2672 2430 2375 2234 2648 4615 4376 4382 4732 2130 2264 2228 2046 2634 3439 3329 3302 3288 4209 4464 4154 4096 4955 4746 5140 2962 2408 3282 3139 2220 2123 2074 2065 1773 1613 1834 1955 2278 2126 2254 2408 2226 4274 4385 4135 4129 3672 4633 4502 4456 4422 2330 3892 4098 4294 4077 2933 2511 2979 2189 2395 2288 2506 2164 2100 4100 3672 3988 4042 3777 4952 4464 4363 4237 4735 4951 3821 3121 3278 2945 3021 2904 1950 4997 4906 4654 4499 2789 2279 2401 2379 2124 2310 2472 2265 4082 4278 1867 2158 2582 2868 3399 2660 2807 3664 3102 2875 2901 3336 1950 2451 1836 2542 3781 3632 3613 4141 4699 4457 4638 4257 2219 1963 2300 1649 2003 2125 2108 2246 2489 2391 2000 3264 3459 3432 3158 4668 4440 4498 4657 3907 3897 3730 3785 3039 3221 3169 2171 2639 2914 2592 2702 2223 2545 2984 1937 3211 2694 2957 2945 2671 1795 2464 2220 2572 2255 2202 4215 4190 3962 4215 3233 3353 3012 3085 2035 2164 1937 1795 3651 3574 3645 3193 1825 1990 2155 2565 3150 3940 3270 2930 3820 4380 4055 3870 3755 2045 2155 1825 2300 1945 3880 4060 4140 4295 3520 3425 3630 3525 4220 4165 4325 4335 1940 2740 2265 2755 2051 2075 1985 2190 2815 2600 2720 1985 1800 1985 2070 1800 3365 3735 3570 3535 3155 2965 2720 3430 3210 3380 3070 3620 3410 3425 3445 3205 4080 2155 2560 2300 2230 2515 2745 2855 2405 2830 3140 2795 3410 1990 2135 3245 2990 2890 3265 3360 3840 3725 3955 3830 4360 4054 3605 3940 1925 1975 1915 2670 3530 3900 3190 3420 2200 2150 2020 2130 2670 2595 2700 2556 2144 1968 2120 2019 2678 2870 3003 3381 2188 2711 2542 2434 2265 2110 2800 2110 2085 2335 2950 3250 1850 1835 2145 1845 2910 2420 2500 2905 2290 2490 2635 2620 2725 2385 1755 1875 1760 2065 1975 2050 1985 2215 2045 2380 2190 2320 2210 2350 2615 2635 3230 3160 2900 2930 3415 3725 3060 3465 2605 2640 2395 2575 2525 2735 2865 3035 1980 2025 1970 2125 2125 2160 2205 2245 1965 1965 1995 2945 3015 2585 2835 2665 2370 2950 2790 2130 2295 2625 2720 18 15 18 16 17 15 14 14 14 15 15 14 15 14 24 22 18 21 27 26 25 24 25 26 21 10 10 11 9 27 28 25 25 19 16 17 19 18 14 14 14 14 12 13 13 18 22 19 18 23 28 30 30 31 35 27 26 24 25 23 20 21 13 14 15 14 17 11 13 12 13 19 15 13 13 14 18 22 21 26 22 28 23 28 27 13 14 13 14 15 12 13 13 14 13 12 13 18 16 18 18 23 26 11 12 13 12 18 20 21 22 18 19 21 26 15 16 29 24 20 19 15 24 20 11 20 21 19 15 31 26 32 25 16 16 18 16 13 14 14 14 29 26 26 31 32 28 24 26 24 26 31 19 18 15 15 16 15 16 14 17 16 15 18 21 20 13 29 23 20 23 24 25 24 18 29 19 23 23 22 25 33 28 25 25 26 27 17.5 16 15.5 14.5 22 22 24 22.5 29 24.5 29 33 20 18 18.5 17.5 29.5 32 28 26.5 20 13 19 19 16.5 16.5 13 13 13 31.5 30 36 25.5 33.5 17.5 17 15.5 15 17.5 20.5 19 18.5 16 15.5 15.5 16 29 24.5 26 25.5 30.5 33.5 30 30.5 22 21.5 21.5 43.1 36.1 32.799999999999997 39.4 36.1 19.899999999999999 19.399999999999999 20.2 19.2 20.5 20.2 25.1 20.5 19.399999999999999 20.6 20.8 18.600000000000001 18.100000000000001 19.2 17.7 18.100000000000001 17.5 30 27.5 27.2 30.9 21.1 23.2 23.8 23.9 20.3 17 21.6 16.2 31.5 29.5 21.5 19.8 22.3 20.2 20.6 17 17.600000000000001 16.5 18.2 16.899999999999999 15.5 19.2 18.5 31.9 34.1 35.700000000000003 27.4 25.4 23 27.2 23.9 34.200000000000003 34.5 31.8 37.299999999999997 28.4 28.8 26.8 33.5 41.5 38.1 32.1 37.200000000000003 28 26.4 24.3 19.100000000000001 34.299999999999997 29.8 31.3 37 32.200000000000003 46.6 27.9 40.799999999999997 44.3 43.4 36.4 30 44.6 40.9 33.799999999999997 29.8 32.700000000000003 23.7 35 23.6 32.4 27.2 26.6 25.8 23.5 30 39.1 39 35.1 32.299999999999997 37 37.700000000000003 34.1 34.700000000000003 34.4 29.9 33 34.5 33.700000000000003 32.4 32.9 31.6 28.1 30.7 25.4 24.2 22.4 26.6 20.2 17.600000000000001 28 27 34 31 29 27 24 23 36 37 31 38 36 36 36 34 38 32 38 25 38 26 22 32 36 27 27 44 32 28 31 Predicted Y 3504 3693 3436 3433 3449 4341 4354 4312 4425 3850 3563 3609 3761 3086 2372 2833 2774 2587 2130 1835 2672 2430 2375 2234 2648 4615 4376 4382 4732 2130 2264 2228 2046 2634 3439 3329 3302 3288 4209 4464 4154 4096 4955 4746 5140 2962 2408 3282 3139 2220 2123 2074 2065 1773 1613 1834 1955 2278 2126 2254 2408 2226 4274 4385 4135 4129 3672 4633 4502 4456 4422 2330 3892 4098 4294 4077 2933 2511 2979 2189 2395 2288 2506 2164 2100 4100 3672 3988 4042 3777 4952 4464 4363 4237 4735 4951 3821 3121 3278 2945 3021 2904 1950 4997 4906 4654 4499 2789 2279 2401 2379 2124 2310 2472 2265 4082 4278 1867 2158 2582 2868 3399 2660 2807 3664 3102 2875 2901 3336 1950 2451 1836 2542 3781 3632 3613 4141 4699 4457 4638 4257 2219 1963 2300 1649 2003 2125 2108 2246 2489 2391 2000 3264 3459 3432 3158 4668 4440 4498 4657 3907 3897 3730 3785 3039 3221 3169 2171 2639 2914 2592 2702 2223 2545 2984 1937 3211 2694 2957 2945 2671 1795 2464 2220 2572 2255 2202 4215 4190 3962 4215 32 33 3353 3012 3085 2035 2164 1937 1795 3651 3574 3645 3193 1825 1990 2155 2565 3150 3940 3270 2930 3820 4380 4055 3870 3755 2045 2155 1825 2300 1945 3880 4060 4140 4295 3520 3425 3630 3525 4220 4165 4325 4335 1940 2740 2265 2755 2051 2075 1985 2190 2815 2600 2720 1985 1800 1985 2070 1800 3365 3735 3570 3535 3155 2965 2720 3430 3210 3380 3070 3620 3410 3425 3445 3205 4080 2155 2560 2300 2230 2515 2745 2855 2405 2830 3140 2795 3410 1990 2135 3245 2990 2890 3265 3360 3840 3725 3955 3830 4360 4054 3605 3940 1925 1975 1915 2670 3530 3900 3190 3420 2200 2150 2020 2130 2670 2595 2700 2556 2144 1968 2120 2019 2678 2870 3003 3381 2188 2711 2542 2434 2265 2110 2800 2110 2085 2335 2950 3250 1850 1835 2145 1845 2910 2420 2500 2905 2290 2490 2635 2620 2725 2385 1755 1875 1760 2065 1975 2050 1985 2215 2045 2380 2190 2320 2210 2350 2615 2635 3230 3160 2900 2930 3415 3725 3060 3465 2605 2640 2395 2575 2525 2735 2865 3035 1980 2025 1970 2125 2125 2160 2205 2245 1965 1965 1995 2945 3015 2585 2835 2665 2370 2950 2790 2130 2295 2625 2720 19.418522756267304 17.9676434541852 19.940532240614303 19.963562070806084 19.84073630978326 12.993200132760848 12.893404201929805 13.215821824614721 12.348364887391028 16.762415674148748 18.965602762495642 18.612478699555027 17.445633969838202 22.627345762988568 28.108445348632067 24.569528109161965 25.022448102933627 26.457974184887874 29.96618498410227 32.230784952960576 25.805462329454127 27.66320196492433 28.085415518440286 29.167817537453917 25.989700970988363 10.889808975244996 12.724518780523425 12.678459120139863 9.9916455977656042 29.96618498410227 28.937519235536126 29.213877197837476 30.611020229472096 26.09717351188333 19.917502410422525 20.761929517454437 20.969197989180451 21.076670530075422 14.006512661199146 12.048977094897893 14.428726214715102 14.873969598422836 8.2797615535099993 9.8841730568706296 6.8595886916836051 23.579245410915451 27.832087386330713 21.122730190458981 22.220485429600465 29.275290078348888 30.019921254549757 30.396075147682154 30.465164638257491 32.706734776924023 33.934992387152256 32.238461563024508 31.309591745289403 28.830046694641155 29.996891424357976 29.014285336175391 27.832087386330713 29.229230417965329 13.507533007043925 12.655429289948088 14.574581805929704 14.620641466313263 18.128852265527659 10.751629994094323 11.757265912468689 12.110389975409305 12.37139471758281 28.430862971316976 16.439998051463835 14.858616378294982 13.354000805765395 15.019825189637437 23.801867102769318 27.041396549746288 23.448743039828699 29.513264990330608 27.931883317161756 28.75328059400189 27.079779600065919 29.705180241928772 30.1 96483286020065 14.843263158167129 18.128852265527659 15.703043485326894 15.288506541874863 17.322808208815378 8.3027913837017806 12.048977094897893 12.824314711354468 13.791567579409204 9.9686157675738229 8.3104679937657053 16.985037366002615 22.358664410751143 21.153436630714687 23.7097477820022 23.126325417143789 24.024488794623185 31.347974795609034 7.9573439308250897 8.6559154466423962 10.590421182751868 11.780295742660471 24.907298951974727 28.822370084577226 27.8858236567782 28.05470907818458 30.012244644485829 28.584395172595507 27.34078434223942 28.929842625472197 14.981442139317807 13.476826566788219 31.985133430914932 29.751239902312332 26.496357235207508 24.300846756924539 20.224566812979585 25.897581650221245 24.769119970824054 18.190265146039071 22.504520001965744 24.247110486477052 24.047518624814966 20.70819324700695 31.347974795609034 27.501993153581875 32.223108342896651 26.803421637764565 17.292101768559672 18.435916668084715 18.581772259299321 14.528522145546145 10.244973729875177 12.10271336534538 10.713246943774692 13.638035378130674 29.282966688412817 31.248178864777991 28.661161273234772 33.658634424850902 30.941114462220931 30.004568034421904 30.135070405508653 29.075698216686799 27.210281971152668 27.962589757417462 30.964144292412712 21.260909171609658 19.763970209143995 19.971238680870009 22.074629838385864 10.482948641856893 12.233215736432129 11.787972352724395 10.567391352560087 16.324848900504939 16.401615001144204 17.683608881819922 17.261395328303 966 22.988146435993112 21.591003404358496 21.99018712768267 29.651443971481285 26.058790461563699 23.94772269398392 26.419591134568243 25.575164027536331 29.252260248157111 26.780391807572787 23.410359989509068 31.44777072644008 21.667769504997761 25.636576908047743 23.617628461235082 23.7097477820022 25.813138939518051 32.537849355517636 27.402197222750832 29.275290078348888 26.573123335846773 29.006608726111462 29.413469059499565 13.960453000815583 14.152368252413744 15.902635346988983 13.960453000815583 21.498884083591378 20.577690875920201 23.195414907719126 22.635022373052493 30.695462940175286 29.705180241928772 31.44777072644008 32.537849355517636 18.290061076870114 18.881160051792452 18.336120737253673 21.805948486148434 32.307551053599845 31.040910393051977 29.774269732504109 26.626859606294257 22.136042718897276 16.071520768395363 21.214849511226099 23.824896932961096 16.99271397606654 12.693812340267719 15.18871061104382 16.608883472870218 17.491693630221761 30.618696839536021 29.774269732504109 32.307551053599845 28.661161273234772 31.386357845928668 16.532117372230953 15.15032756072419 14.536198755610069 13.34632419570147 19.29569699524448 20.024974951317496 18.451269888212568 19.257313944924849 13.922069950495953 14.344283504011909 13.116025893783672 13.03925979314441 31.424740896248302 25.283452845107124 28.929842625472197 25.168303694148229 30.572637179152462 30.38839853761823 31.079293443371611 29.505588380266683 24.707707090312642 26.3581782540568 31 25.436985046385654 31.079293443371611 32.499466305198006 31.079293443371611 30.42678158793786 32.499466305198006 20.485571555153083 17.645225831500291 18.911866492048159 19.180547844285584 22.097659668577641 23.55621558072367 25.436985046385654 19.986591900997862 21.675446115061686 20.370422404194187 22.750171524011392 18.528035988851833 20.140124102276392 20.024974951317496 19.871442750038966 21.71382916538132 14.99679535944566 29.774269732504109 26.665242656613891 28.661161273234772 29.198523977709623 27.010690109490582 25.245069794787494 24.400642687755582 27.855117216522494 24.592557939353743 22.212808819536537 24.861239291591168 20.140124102276392 31.040910393051977 29.927801933782639 21.40676476282426 23.364300329125509 24.131961335518156 21.25323256154573 20.523954605472717 16.83918177478801 17.721991932139556 15.956371617436467 16.915947875427275 12.847344541546249 15.196387221107749 18.643185139810733 16.071520768395363 31.539890047207198 31.156059544010873 31.616656147846463 25.82081554958198 19.218930894605215 16.378585170952423 21.828978316340216 20.063358001637127 29.428822279627418 29.812652782823744 30.810612091134182 29.96618498410227 25.82081554958198 26.396561304376462 25.590517247664184 26.695949096869597 29.858712443207299 31.20979581445836 30.042951084741535 30.818288701198107 25.759402669070568 24.285493536796686 23.264504398294466 20.362745794130259 29.520941600394536 25.506074536960995 26.803421637764565 27.632495524668624 28.929842625472 197 30.1197171853808 24.822856241271538 30.1197171853808 30.311632436978961 28.392479920997346 23.671364731682569 21.368381712504629 32.115635802001684 32.230784952960576 29.851035833143374 32.154018852321315 23.978429134239626 27.739968065563595 27.125839260449478 24.01681218455926 28.737927373874037 27.202605361088743 26.089496901819405 26.204646052778301 25.398601996066024 28.008649417801021 32.844913758074696 31.923720550403523 32.806530707755066 30.465164638257491 31.156059544010873 30.58031378921639 31.079293443371611 29.313673128668523 30.618696839536021 28.047032468120655 29.505588380266683 28.507629071956242 29.352056178988153 28.27733077003845 26.243029103097935 26.089496901819405 21.521913913783155 22.059276618258011 24.055195234878891 23.824896932961096 20.101741051956761 17.721991932139556 22.826937624650657 19.717910548760436 26.3197952037372 26.051113851499771 27.931883317161756 26.550093505654992 26.933924008851317 25.321835895426759 24.323876587116317 23.018852876248818 31.117676493691242 30.772229040814551 31.194442594330503 30.004568034421904 30.004568034421904 29.735886682184479 29.390439229307788 29.083374826750727 31.232825644650138 31.232825644650138 31.002527342732343 23.7097477820022 23.172385077527348 26.473327405015727 24.554174889034112 25.85919859990161 28.12379856875992 23.671364731682569 24.899622341910803 29.96618498410227 28.699544323554402 26.16626300245867 25.436985046385654

Weight

MPG

X Variable 1 Line Fit Plot

Y 3504 3693 3436 3433 3449 4341 4354 4312 4425 3850 3563 3609 3761 3086 2372 2833 2774 2587 2130 1835 2672 2430 23 75 2234 2648 4615 4376 4382 4732 2130 2264 2228 2046 2634 3439 3329 3302 3288 4209 4464 4154 4096 4955 4746 5140 2962 2408 3282 3139 2220 2123 2074 2065 1773 1613 1834 1955 2278 2126 2254 2408 2226 4274 4385 4135 4129 3672 4633 4502 4456 4422 2330 3892 4098 4294 4077 2933 2511 2979 2189 2395 2288 2506 2164 2100 4100 3672 3988 4042 3777 4952 4464 4363 4237 4735 4951 3821 3121 3278 2945 3021 2904 1950 4997 4906 4654 4499 2789 2279 2401 2379 2124 2310 2472 2265 4082 4278 1867 2158 2582 2868 3399 2660 2807 3664 3102 2875 2901 3336 1950 2451 1836 2542 3781 3632 3613 4141 4699 4457 4638 4257 2219 1963 2300 1649 2003 2125 2108 2246 2489 2391 2000 3264 3459 3432 3158 4668 4440 4498 4657 3907 3897 3730 3785 3039 3221 3169 2171 2639 2914 2592 2702 2223 2545 2984 1937 3211 2694 2957 2945 2671 1795 2464 2220 2572 2255 2202 4215 4190 3962 4215 3233 3353 3012 3085 2035 2164 1937 1795 3651 3574 3645 3193 1825 1990 2155 2565 3150 3940 3270 2930 3820 4380 4055 3870 3755 2045 2155 1825 2300 1945 3880 4060 4140 4295 3520 3425 3630 3525 4220 4165 4325 4335 1940 2740 2265 2755 2051 2075 1985 2190 2815 2600 2720 1985 1800 1985 2070 1800 3365 3735 3570 3535 3155 2965 2720 3430 3210 3380 3070 3620 3410 3425 3445 3205 4080 2155 2560 2300 2230 2515 2745 2855 2405 2830 3140 2795 3410 1990 2135 3245 2990 2890 3265 3360 3840 3725 3955 3830 4360 4054 3605 3940 1925 1975 1915 2670 3530 3900 3190 3420 2200 2150 2020 2130 2670 2595 2700 2556 2144 1968 2120 2019 2678 2870 3003 3381 2188 2711 2542 2434 2265 2110 2800 2110 2085 2335 2950 3250 1850 1835 2145 1845 2910 2420 2500 2905 2290 2490 2635 2620 2725 2385 1755 1875 1760 2065 1975 2050 1985 2215 2045 2380 2190 2320 2210 2350 2615 2635 3230 3160 2900 2930 3415 3725 3060 3465 2605 2640 2395 2575 2525 2735 2865 3035 1980 2025 1970 2125 2125 2160 2205 2245 1965 1965 1995 2945 3015 2585 2835 2665 2370 2950 2790 2130 2295 2625 2720 18 15 18 16 17 15 14 14 14 15 15 14 15 14 24 22 18 21 27 26 25 24 25 26 21 10 10 11 9 27 28 25 25 19 16 17 19 18 14 14 14 14 12 13 13 18 22 19 18 23 28 30 30 31 35 27 26 24 25 23 20 21 13 14 15 14 17 11 13 12 13 19 15 13 13 14 18 22 21 26 22 28 23 28 27 13 14 13 14 15 12 13 13 14 13 12 13 18 16 18 18 23 26 11 12 13 12 18 20 21 22 18 19 21 26 15 16 29 24 20 19 15 24 20 11 20 21 19 15 31 26 32 25 16 16 18 16 13 14 14 14 29 26 26 31 32 28 24 26 24 26 31 19 18 15 15 16 15 16 14 17 16 15 18 21 20 13 29 23 20 23 24 25 24 18 29 19 23 23 22 25 33 28 25 25 26 27 17.5 16 15.5 14.5 22 22 24 22.5 29 24.5 29 33 20 18 18.5 17.5 29.5 32 28 26.5 20 13 19 19 16.5 16.5 13 13 13 31.5 30 36 25.5 33.5 17.5 17 15.5 15 17.5 20.5 19 18.5 16 15.5 15.5 16 29 24.5 26 25.5 30.5 33.5 30 30.5 22 21.5 21.5 43.1 36.1 32.799999999999997 39.4 36.1 19.899999999999999 19.399999999999999 20.2 19.2 20.5 20.2 25.1 20.5 19.399999999999999 20.6 20.8 18.600000000000001 18.100000000000001 19.2 17.7 18.100000000000001 17.5 30 27.5 27.2 30.9 21.1 23.2 23.8 23.9 20.3 17 21.6 16.2 31.5 29.5 21 .5 19.8 22.3 20.2 20.6 17 17.600000000000001 16.5 18.2 16.899999999999999 15.5 19.2 18.5 31.9 34.1 35.700000000000003 27.4 25.4 23 27.2 23.9 34.200000000000003 34.5 31.8 37.299999999999997 28.4 28.8 26.8 33.5 41.5 38.1 32.1 37.200000000000003 28 26.4 24.3 19.100000000000001 34.299999999999997 29.8 31.3 37 32.200000000000003 46.6 27.9 40.799999999999997 44.3 43.4 36.4 30 44.6 40.9 33.799999999999997 29.8 32.700000000000003 23.7 35 23.6 32.4 27.2 26.6 25.8 23.5 30 39.1 39 35.1 32.299999999999997 37 37.700000000000003 34.1 34.700000000000003 34.4 29.9 33 34.5 33.700000000000003 32.4 32.9 31.6 28.1 30.7 25.4 24.2 22.4 26.6 20.2 17.600000000000001 28 27 34 31 29 27 24 23 36 37 31 38 36 36 36 34 38 32 38 25 38 26 22 32 36 27 27 44 32 28 31 Predicted Y 3504 3693 3436 3433 3449 4341 4354 4312 4425 3850 3563 3609 3761 3086 2372 2833 2774 2587 2130 1835 2672 2430 2375 2234 2648 4615 4376 4382 4732 2130 2264 2228 2046 2634 3439 3329 3302 3288 4209 4464 4154 4096 4955 4746 5140 2962 2408 3282 3139 2220 2123 2074 2065 1773 1613 1834 1955 2278 2126 2254 2408 2226 4274 4385 4135 4129 3672 4633 4502 4456 4422 2330 3892 4098 4294 4077 2933 2511 2979 2189 2395 2288 2506 2164 2100 4100 3672 3988 4042 3777 4952 4464 4363 4237 4735 4951 3821 3121 3278 2945 3021 2904 1950 4997 4906 4654 4499 2789 2279 2401 2379 2124 2310 2472 2265 4082 4278 1867 2158 2582 2868 3399 2660 2807 3664 3102 2875 2901 3336 1950 2451 1836 2542 37 81 3632 3613 4141 4699 4457 4638 4257 2219 1963 2300 1649 2003 2125 2108 2246 2489 2391 2000 3264 3459 3432 3158 4668 4440 4498 4657 3907 3897 3730 3785 3039 3221 3169 2171 2639 2914 2592 2702 2223 2545 2984 1937 3211 2694 2957 2945 2671 1795 2464 2220 2572 2255 2202 4215 4190 3962 4215 3233 3353 3012 3085 2035 2164 1937 1795 3651 3574 3645 3193 1825 1990 2155 2565 3150 3940 3270 2930 3820 4380 4055 3870 3755 2045 2155 1825 2300 1945 3880 4060 4140 4295 3520 3425 3630 3525 4220 4165 4325 4335 1940 2740 2265 2755 2051 2075 1985 2190 2815 2600 2720 1985 1800 1985 2070 1800 3365 3735 3570 3535 3155 2965 2720 3430 3210 3380 3070 3620 3410 3425 3445 3205 4080 2155 2560 2300 2230 2515 2745 2855 2405 2830 3140 2795 3410 1990 2135 3245 2990 2890 3265 3360 3840 3725 3955 3830 4360 4054 3605 3940 1925 1975 1915 2670 3530 3900 3190 3420 2200 2150 2020 2130 2670 2595 2700 2556 2144 1968 2120 2019 2678 2870 3003 3381 2188 2711 2542 2434 2265 2110 2800 2110 2085 2335 2950 3250 1850 1835 2145 1845 2910 2420 2500 2905 2290 2490 2635 2620 2725 2385 1755 1875 1760 2065 1975 2050 1985 2215 2045 2380 2190 2320 2210 2350 2615 2635 3230 3160 2900 2930 3415 3725 3060 3465 2605 2640 2395 2575 2525 2735 2865 3035 1980 2025 1970 2125 2125 2160 2205 2245 1965 1965 1995 2945 3015 2585 2835 2665 2370 2950 2790 2130 2295 2625 2720 19.41852275626730 4 17.9676434541852 19.940532240614303 19.963562070806084 19.84073630978326 12.993200132760848 12.893404201929805 13.215821824614721 12.348364887391028 16.762415674148748 18.965602762495642 18.612478699555027 17.445633969838202 22.627345762988568 28.108445348632067 24.569528109161965 25.022448102933627 26.457974184887874 29.96618498410227 32.230784952960576 25.805462329454127 27.66320196492433 28.085415518440286 29.167817537453917 25.989700970988363 10.889808975244996 12.724518780523425 12.678459120139863 9.9916455977656042 29.96618498410227 28.937519235536126 29.213877197837476 30.611020229472096 26.09717351188333 19.917502410422525 20.761929517454437 20.969197989180451 21.076670530075422 14.006512661199146 12.048977094897893 14.428726214715102 14.873969598422836 8.2797615535099993 9.8841730568706296 6.8595886916836051 23.579245410915451 27.832087386330713 21.122730190458981 22.220485429600465 29.275290078348888 30.019921254549757 30.396075147682154 30.465164638257491 32.706734776924023 33.934992387152256 32.238461563024508 31.309591745289403 28.830046694641155 29.996891424357976 29.014285336175391 27.832087386330713 29.229230417965329 13.507533007043925 12.655429289948088 14.574581805929704 14.620641466313263 18.128852265527659 10.751629994094323 11.757265912468689 12.110389975409305 12.37139471758281 28.430862971316976 16.439998051463835 14.858616378294982 13.354000805765395 15.019825189637437 23.801867102769318 27.041396549746288 23.448743039828699 29.513264990330608 27.931883317161756 28.75328059400189 27.079779600065919 29.705180241928772 30.196483286020065 14.843263158167129 18.128852265527659 15.703043485326894 15.288506541874863 17.322808208815378 8.3027913837017806 12.048977094897893 12.824314711354468 13.791567579409204 9.9686157675738229 8.3104679937657053 16.985037366002615 22.358664410751143 21.153436630714687 23.7097477820022 23.126325417143789 24.024488794623185 31.347974795609034 7.9573439308250897 8.6559154466423962 10.590421182751868 11.780295742660471 24.907298951974727 28.822370084577226 27.8858236567782 28.05470907818458 30.012244644485829 28.584395172595507 27.34078434223942 28.929842625472197 14.981442139317807 13.476826566788219 31.985133430914932 29.751239902312332 26.496357235207508 24.300846756924539 20.224566812979585 25.897581650221245 24.769119970824054 18.190265146039071 22.504520001965744 24.247110486477052 24.047518624814966 20.70819324700695 31.347974795609034 27.501993153581875 32.223108342896651 26.803421637764565 17.292101768559672 18.435916668084715 18.581772259299321 14.528522145546145 10.244973729875177 12.10271336534538 10.713246943774692 13.638035378130674 29.282966688412817 31.248178864777991 28.661161273234772 33.658634424850902 30.941114462220931 30.004568034421904 30.135070405508653 29.075698216686799 27.210281971152668 27.962589757417462 30.964144292412712 21.260909171609658 19.763970209143995 19.971238680870009 22.074629838385864 10.482948641856893 12.233215736432129 11.787972352724395 10.567391352560087 16.32484890 0504939 16.401615001144204 17.683608881819922 17.261395328303966 22.988146435993112 21.591003404358496 21.99018712768267 29.651443971481285 26.058790461563699 23.94772269398392 26.419591134568243 25.575164027536331 29.252260248157111 26.780391807572787 23.410359989509068 31.44777072644008 21.667769504997761 25.636576908047743 23.617628461235082 23.7097477820022 25.813138939518051 32.537849355517636 27.402197222750832 29.275290078348888 26.573123335846773 29.006608726111462 29.413469059499565 13.960453000815583 14.152368252413744 15.902635346988983 13.960453000815583 21.498884083591378 20.577690875920201 23.195414907719126 22.635022373052493 30.695462940175286 29.705180241928772 31.44777072644008 32.537849355517636 18.290061076870114 18.881160051792452 18.336120737253673 21.805948486148434 32.307551053599845 31.040910393051977 29.774269732504109 26.626859606294257 22.136042718897276 16.071520768395363 21.214849511226099 23.824896932961096 16.99271397606654 12.693812340267719 15.18871061104382 16.608883472870218 17.491693630221761 30.618696839536021 29.774269732504109 32.307551053599845 28.661161273234772 31.386357845928668 16.532117372230953 15.15032756072419 14.536198755610069 13.34632419570147 19.29569699524448 20.024974951317496 18.451269888212568 19.257313944924849 13.922069950495953 14.344283504011909 13.116025893783672 13.03925979314441 31.424740896248302 25.283452845107124 28.929842625472197 25.168303694148229 30.572637179152462 30.38839853761823 31.079293443 371611 29.505588380266683 24.707707090312642 26.358178254056831 25.436985046385654 31.079293443371611 32.499466305198006 31.079293443371611 30.42678158793786 32.499466305198006 20.485571555153083 17.645225831500291 18.911866492048159 19.180547844285584 22.097659668577641 23.55621558072367 25.436985046385654 19.986591900997862 21.675446115061686 20.370422404194187 22.750171524011392 18.528035988851833 20.140124102276392 20.024974951317496 19.871442750038966 21.71382916538132 14.99679535944566 29.774269732504109 26.665242656613891 28.661161273234772 29.198523977709623 27.010690109490582 25.245069794787494 24.400642687755582 27.855117216522494 24.592557939353743 22.212808819536537 24.861239291591168 20.140124102276392 31.040910393051977 29.927801933782639 21.40676476282426 23.364300329125509 24.131961335518156 21.25323256154573 20.523954605472717 16.83918177478801 17.721991932139556 15.956371617436467 16.915947875427275 12.847344541546249 15.196387221107749 18.643185139810733 16.071520768395363 31.539890047207198 31.156059544010873 31.616656147846463 25.82081554958198 19.218930894605215 16.378585170952423 21.828978316340216 20.063358001637127 29.428822279627418 29.812652782823744 30.810612091134182 29.96618498410227 25.82081554958198 26.396561304376462 25.590517247664184 26.695949096869597 29.858712443207299 31.20979581445836 30.042951084741535 30.818288701198107 25.759402669070568 24.285493536796686 23.264504398294466 20.362745794130259 29.520941600394536 25.5060 74536960995 26.803421637764565 27.632495524668624 28.929842625472197 30.1197171853808 24.822856241271538 30.1197171853808 30.311632436978961 28.392479920997346 23.671364731682569 21.368381712504629 32.115635802001684 32.230784952960576 29.851035833143374 32.154018852321315 23.978429134239626 27.739968065563595 27.125839260449478 24.01681218455926 28.737927373874037 27.202605361088743 26.089496901819405 26.204646052778301 25.398601996066024 28.008649417801021 32.844913758074696 31.923720550403523 32.806530707755066 30.465164638257491 31.156059544010873 30.58031378921639 31.079293443371611 29.313673128668523 30.618696839536021 28.047032468120655 29.505588380266683 28.507629071956242 29.352056178988153 28.27733077003845 26.243029103097935 26.089496901819405 21.521913913783155 22.059276618258011 24.055195234878891 23.824896932961096 20.101741051956761 17.721991932139556 22.826937624650657 19.717910548760436 26.3197952037372 26.051113851499771 27.931883317161756 26.550093505654992 26.933924008851317 25.321835895426759 24.323876587116317 23.018852876248818 31.117676493691242 30.772229040814551 31.194442594330503 30.004568034421904 30.004568034421904 29.735886682184479 29.390439229307788 29.083374826750727 31.232825644650138 31.232825644650138 31.002527342732343 23.7097477820022 23.172385077527348 26.473327405015727 24.554174889034112 25.85919859990161 28.12379856875992 23.671364731682569 24.899622341910803 29.96618498410227 28.699544323554402 26.16626300245867 25.43698 5046385654

X Variable 1

MPG 3504 3693 3436 3433 3449 4341 4354 4312 4425 3850 3563 3609 3761 3086 2372 2833 2774 2587 2130 1835 2672 2430 2375 2234 2648 4615 4376 4382 4732 2130 2264 2228 2046 2634 3439 3329 3302 3288 4209 4464 4154 4096 4955 4746 5140 2962 2408 3282 3139 2220 2123 2074 2065 1773 1613 1834 1955 2278 2126 2254 2408 2226 4274 4385 4135 4129 3672 4633 4502 4456 4422 2330 3892 4098 4294 4077 2933 2511 2979 2189 2395 2288 2506 2164 2100 4100 3672 3988 4042 3777 4952 4464 4363 4237 4735 4951 3821 3121 3278 2945 3021 2904 1950 4997 4906 4654 4499 2789 2279 2401 2379 2124 2310 2472 2265 4082 4278 1867 2158 2582 2868 3399 2660 2807 3664 3102 2875 2901 3336 1950 2451 1836 2542 3781 3632 3613 4141 4699 4457 4638 4257 2219 1963 2300 1649 2003 2125 2108 2246 2489 2391 2000 3264 3459 3432 3158 4668 4440 4498 4657 3907 3897 3730 3785 3039 3221 3169 2171 2639 2914 2592 2702 2223 2545 2984 1937 3211 2694 2957 2945 2671 1795 2464 2220 2572 2255 2202 4215 4190 3962 4215 3233 3353 3012 3085 2035 2164 1937 1795 3651 3574 3645 3193 1825 1990 2155 2565 3150 3940 3270 2930 3820 4380 4055 3870 3755 2045 2155 1825 2300 1945 3880 4060 4140 4295 3520 3425 3630 3525 4220 4165 4325 4335 1940 2740 2265 2755 2051 2075 1985 2190 2815 2600 2720 1985 1800 1985 2070 1800 3365 3735 3570 3535 3155 2965 2720 3430 3210 3380 3070 3620 3410 3425 3445 3205 4080 2155 2560 2300 2230 2515 2745 2855 2405 2830 3140 2795 3410 1990 2135 3245 2990 2890 3265 3360 3840 3725 3955 3830 4360 4054 3605 3940 1925 1975 1915 2670 3530 3900 3190 3420 2200 2150 2020 2130 2670 2595 2700 2556 2144 1968 2120 2019 2678 2870 3003 3381 2188 2711 2542 2434 2265 2110 2800 2110 2085 2335 29 50 3250 1850 1835 2145 1845 2910 2420 2500 2905 2290 2490 2635 2620 2725 2385 1755 1875 1760 2065 1975 2050 1985 2215 2045 2380 2190 2320 2210 2350 2615 2635 3230 3160 2900 2930 3415 3725 3060 3465 2605 2640 2395 2575 2525 2735 2865 3035 1980 2025 1970 2125 2125 2160 2205 2245 1965 1965 1995 2945 3015 2585 2835 2665 2370 2950 2790 2130 2295 2625 2720 18 15 18 16 17 15 14 14 14 15 15 14 15 14 24 22 18 21 27 26 25 24 25 26 21 10 10 11 9 27 28 25 25 19 16 17 19 18 14 14 14 14 12 13 13 18 22 19 18 23 28 30 30 31 35 27 26 24 25 23 20 21 13 14 15 14 17 11 13 12 13 19 15 13 13 14 18 22 21 26 22 28 23 28 27 13 14 13 14 15 12 13 13 14 13 12 13 18 16 18 18 23 26 11 12 13 12 18 20 21 22 18 19 21 26 15 16 29 24 20 19 15 24 20 11 20 21 19 15 31 26 32 25 16 16 18 16 13 14 14 14 29 26 26 31 32 28 24 26 24 26 31 19 18 15 15 16 15 16 14 17 16 15 18 21 20 13 29 23 20 23 24 25 24 18 29 19 23 23 22 25 33 28 25 25 26 27 17.5 16 15.5 14.5 22 22 24 22.5 29 24.5 29 33 20 18 18.5 17.5 29.5 32 28 26.5 20 13 19 19 16.5 16.5 13 13 13 31.5 30 36 25.5 33.5 17.5 17 15.5 15 17.5 20.5 19 18.5 16 15.5 15.5 16 29 24.5 26 25.5 30.5 33.5 30 30.5 22 21.5 21.5 43.1 36.1 32.799999999999997 39.4 36.1 19.899999999999999 19.399999999999999 20.2 19.2 20.5 20.2 25.1 20.5 19.399999999999999 20.6 20.8 18.600000000000001 18.100000000000001 19.2 17.7 18.100000000000001 17.5 30 27.5 27.2 30.9 21.1 23.2 23.8 23.9 20.3 17 21.6 16.2 31.5 29.5 21.5 19.8 22.3 20.2 20.6 17 17.600000000000001 16.5 18.2 16.899999999999999 15.5 19.2 18.5 31.9 34.1 35.700000000000003 27.4 25.4 23 27.2 23.9 34.200000000000003 34.5 31.8 37.299999999999997 28.4 28.8 26.8 33.5 41.5 38.1 32.1 37.200000000 000003 28 26.4 24.3 19.100000000000001 34.299999999999997 29.8 31.3 37 32.200000000000003 46.6 27.9 40.799999999999997 44.3 43.4 36.4 30 44.6 40.9 33.799999999999997 29.8 32.700000000000003 23.7 35 23.6 32.4 27.2 26.6 25.8 23.5 30 39.1 39 35.1 32.299999999999997 37 37.700000000000003 34.1 34.700000000000003 34.4 29.9 33 34.5 33.700000000000003 32.4 32.9 31.6 28.1 30.7 25.4 24.2 22.4 26.6 20.2 17.600000000000001 28 27 34 31 29 27 24 23 36 37 31 38 36 36 36 34 38 32 38 25 38 26 22 32 36 27 27 44 32 28 31