Slide 1

Least Squares RegressionFitting a Line to Bivariate Data

Correlation tells us about strength (scatter) and direction of the linear relationship between two quantitative variables.In addition, we would like to have a numerical description of how both variables vary together. For instance, is one variable increasing faster than the other one? And we would like to make predictions based on that numerical description.

But which line best describes our data?

Distances between the points and line are squared so all are positive values. This is done so that distances can be properly added (Pythagoras).The regression lineThe least-squares regression line is the unique line such that the sum of the squared vertical (y) distances between the data points and the line is the smallest possible. Properties is the predicted y value (y hat) b1 is the slopeb0 is the y-intercept

b0" is in units of y"b1" is in units of y / units of xThe least-squares regression line can be shown to have this equation:

where

First we calculate the slope of the line, b; from statistics we already know:r is the correlation.sy is the standard deviation of the response variable y.sx is the the standard deviation of the explanatory variable x.Once we know b1, the slope, we can calculate b0, the y-intercept:

where x and y are the sample means of the x and y variablesHow to:This means that we don't have to calculate a lot of squared distances to find the least-squares regression line for a data set. We can instead rely on the equation.

But typically, we use a 2-var stats calculator or stats software.BEWARE!!!Not all calculators and software use the same convention:

Some use instead:

Make sure you know what YOUR calculator gives you for a and b before you answer homework or exam questions.

Software outputinterceptslope

R2absolute value of rR2

interceptslope

The equation completely describes the regression line.

NOTE: The regression line always passes through the point with coordinates (xbar, ybar).

The distinction between explanatory and response variables is crucial in regression. If you exchange y for x in calculating the regression line, you will get the wrong line. Recall that Regression examines the distance of all points from the line in the y direction only.

Hubble telescope data about galaxies moving away from earth:

These two lines are the two regression lines calculated either correctly (x = distance, y = velocity, solid line) or incorrectly (x = velocity, y = distance, dotted line).

In regression we examine the variation in the response variable (y) given change in the explanatory variable (x).

The correlation is a measure of spread (scatter) in both the x and y directions in the linear relationship.Correlation versus regressionMaking predictions: interpolationThe equation of the least-squares regression allows to predict y for any x within the range studied. This is called interpolating.

Nobody in the study drank 6.5 beers, but by finding the value of from the regression line for x = 6.5 we would expect a blood alcohol content of 0.094 mg/ml.

There is a positive linear relationship between the number of powerboats registered and the number of manatee deaths. (in 1000s)

The least squares regression line has the equation:

Roughly 21 manatees.

Thus if we were to limit the number of powerboat registrations to 500,000, what could we expect for the number of manatee deaths? y =0.125x-41.4 y =0.125x-41.4

Extrapolation is the use of a regression line for predictions outside the range of x values used to obtain the line.

This can be a very stupid thing to do, as seen here.Height in InchesHeight in Inches!!!!!!ExtrapolationSometimes the y-intercept is not a realistic possibility. Here we have negative blood alcohol content, which makes no sense

y-intercept shows negative blood alcoholBut the negative value is appropriate for the equation of the regression line.

There is a lot of scatter in the data, and the line is just an estimate. The y interceptR-squared = r2; the proportion of y-variation explained by changes in x.r2 represents the proportion of the variation in y (vertical scatter from the regression line) that can be explained by changes in x.

r2, the coefficient of determination, is the square of the correlation coefficient.15Go over point that missing zero is OK - this is messy data, a prediction based on messy data. Residuals should be scattered randomly around the line, or there is something wrong with your data - not linear, outliers, etc.

r = -1r2 = 1Changes in x explain 100% of the variations in y. Y can be entirely predicted for any given value of x.

r = 0r2 = 0Changes in x explain 0% of the variations in y. The value(s) y takes is (are) entirely independent of what value x takes.

Here the change in x only explains 76% of the change in y. The rest of the change in y (the vertical scatter, shown as red arrows) must be explained by something other than x.r = 0.87r2 = 0.76Example: SAT scores

SAT scores: calculations

SAT scores: result

r2 = (-.868)2 = .7534About 75% of the variation in state mean SAT scores is explained by differences in the % of seniors that take the test.If 57% of NC seniors take the SAT, the predicted mean score is

There is a great deal of variation in BAC for the same number of beers drunk. A persons blood volume is a factor in the equation that was overlooked here.

In the first plot, number of beers only explains 49% of the variation in blood alcohol content.But number of beers / weight explains 81% of the variation in blood alcohol content.Additional factors contribute to variations in BAC among individuals (like maybe some genetic ability to process alcohol).

We changed number of beers to number of beers/weight of person in lb.r =0.7r2 =0.49

r =0.9r2 =0.8120So lets look at this scatterplot. Overall pattern: in general, the BAC increases with the number of beers you drink. Grade performance

If class attendance explains 16% of the variation in grades, what is the correlation between percent of classes attended and grade?

1. We need to make an assumption: attendance and grades are positively correlated. So r will be positive too.

2. r2 = 0.16, so r = +0.16 = + 0.4

A weak correlation.

Transforming relationshipsA scatterplot might show a clear relationship between two quantitative variables, but issues of influential points or non linearity prevent us from using correlation and regression tools.

Transforming the data changing the scale in which one or both of the variables are expressed can make the shape of the relationship linear in some cases.

Example: Patterns of growth are often exponential, at least in their initial phase. Changing the response variable y into log(y) or ln(y) will transform the pattern from an upward-curved exponential to a straight line.

Exponential bacterial growthlog(2n) = n*log(2) 0.3nTaking the log changes the growth pattern into a straight line.In ideal environments, bacteria multiply through binary fission. The number of bacteria can double every 20 minutes in that way.

1 - 2 - 4 - 8 - 16 - 32 - 64 - Exponential growth 2n,not suitable for regression.

r = 0.86, but this is misleading. The elephant is an influential point. Most mammals are very small in comparison. Without this point, r = 0.50 only. Body weight and brain weight in 96 mammal species

Now we plot the log of brain weight against the log of body weight.

The pattern is linear, with r = 0.96. The vertical scatter is homogenous good for predictions of brain weight from body weight (in the log scale).Inference for least squares linesInference for simple linear regressionSimple linear regression modelConditions for inference Confidence interval for regression parameters Significance test for the slope Confidence interval for E(y) for a given xPrediction interval for y for a given xThe data in a scatterplot are a random sample from a population that may exhibit a linear relationship between x and y. Different sample different plot.

Now we want to describe the population mean response E(y) as a function of the explanatory variable x: E(y)= b0 + b1x.

And to assess whether the observed relationship is statistically significant (not entirely explained by chance events due to random sampling).

Simple linear regression modelIn the population, the linear regression equation is E(y) = b0 + b1x.

Sample data then fits the model:

Data = fit + residual yi = (b0 + b1xi) + (ei)

where the ei are independent and Normally distributed N(0,s).

Linear regression assumes equal standard deviation of y (s is the same for all values of x).E(y) = b0 + b1x

The intercept b0, the slope b1, and the standard deviation s of y are the unknown parameters of the regression model. We rely on the random sample data to provide unbiased estimates of these parameters.

The value of from the least-squares regression line is really a prediction of the mean value of y (E(y)) for a given value of x. The least-squares regression line ( = b0 + b1x) obtained from sample data is the best estimate of the true population regression line (E(y) = b0 + b1x). unbiased estimate for mean response E(y)b0 unbiased estimate for intercept b0b1 unbiased estimate for slope b1The regression standard error, s, for n sample data points is calculated from the residuals (yi i):

se is an unbiased estimate of the regression standard deviation s.The population standard deviation s for y at any given value of x represents the spread of the normal distribution of the ei around the mean E(y) .

Conditions for inferenceThe observations are independent.The relationship is indeed linear.The standard deviation of y, , is the same for all values of x.The response y varies normally around its mean.Using residual plots to check for regression validityThe residuals (y ) give useful information about the contribution of individual data points to the overall pattern of scatter.

We view the residuals in a residual plot:

If residuals are scattered randomly around 0 with uniform variation, it indicates that the data fit a linear model, have normally distributed residuals for each value of x, and constant standard deviation .

Residuals are randomly scattered good!

Curved pattern the relationship is not linear.

Change in variability across plot not equal for all values of x.

What is the relationship between the average speed a car is driven and its fuel efficiency?

We plot fuel efficiency (in miles per gallon, MPG) against average speed (in miles per hour, MPH) for a random sample of 60 cars. The relationship is curved. When speed is log transformed (log of miles per hour, LOGMPH) the new scatterplot shows a positive, linear relationship.

Residual plot:The spread of the residuals is reasonably randomno clear pattern. The relationship is indeed linear. But we see one low residual (3.8, 4) and one potentially influential point (2.5, 0.5). Normal quantile plot for residuals:The plot is fairly straight, supporting the assumption of normally distributed residuals. Data okay for inference.Slide 1- 35Standard Error for the SlopeThree aspects of the scatterplot affect the standard error of the regression slope: spread around the line, sespread of x values, sxsample size, n.

The formula for the standard error (which you will probably never have to calculate by hand) is:

35Confidence interval for 1Estimating the regression parameters b0, b1 is a case of one-sample inference with unknown population standard deviation. We rely on the t distribution, with n 2 degrees of freedom.

A level C confidence interval for the slope, b1, is proportional to the standard error of the least-squares slope:b1 t* SE(b1)t* is the t critical for the t (n 2) distribution with area C between t* and +t*.We estimate the standard error of b1 with

where n is the sample size, sx is the ordinary standard deviation of the x values

Confidence interval for 0A level C confidence interval for the intercept, b0 , is proportional to the standard error of the least-squares intercept:b0 t* SEb0 The intercept usually isnt interesting. Most hypothesis tests and confidence intervals for regression are about the slope.

Hypothesis test for the slopeWe may look for evidence of a significant relationship between variables x and y in the population from which our data were drawn.

For that, we can test the hypothesis that the regression slope parameter 1 is equal to zero.

H0: 1 = 0 vs. Ha: 1 0

Testing H0: 1 = 0 also allows to test the hypothesis of no correlation between x and y in the population.

Note: A test of hypothesis for b0 is irrelevant (b0 is often not even achievable).

Hypothesis test for the slope (cont.)We usually test the hypothesis H0: 1 = 0 vs. Ha: 1 0 but we can also test H0: 1 = 0 vs. Ha: 1 < 0 or H0: 1 = 0 vs. Ha: 1 > 0 To do this we calculate the test statistic

Use the t dist. with n 2df to find the P-value of the test.

Note: Software typically providestwo-sided p-values.

SPSSUsing technologyComputer software runs all the computations for regression analysis. Here is some software output for the car speed/gas efficiency example.SlopeInterceptp-value for tests of significanceConfidence intervalsThe t-test for regression slope is highly significant (p < 0.001). There is a significant relationship between average car speed and gas efficiency.

Excel SASintercept: interceptlogmph: slopeP-value for tests of significanceconfidence intervals

Confidence Intervals and Prediction Intervals for Predicted ValuesOnce we have a useful regression, how can we indulge our natural desire to predict, without being irresponsible?Now we have standard errorswe can use those to construct a confidence interval for the predictions and to report our uncertainty honestly.

Slide 1- 43An Example: Body Fat and Waist SizeConsider an example that involves investigating the relationship in adult males between % Body Fat and Waist size (in inches). Here is a scatterplot of the data for 250 adult males of various ages:

43Confidence Intervals and Prediction Intervals for Predicted Values(cont.)For our %body fat and waist size example, there are two questions we could ask:Do we want to know the mean %body fat for all men with a waist size of, say, 38 inches?Do we want to estimate the %body fat for a particular man with a 38-inch waist?

The predicted %body fat is the same in both questions, but we can predict the mean %body fat for all men whose waist size is 38 inches with a lot more precision than we can predict the %body fat of a particular individual whose waist size happens to be 38 inches.

Confidence Intervals and Prediction Intervals for Predicted Values(cont.)We start with the same prediction in both cases.We are predicting for a new individual, one that was not in the original data set.Call his x-value x.The regression predicts %body fat as

Confidence Intervals and Prediction Intervals for Predicted Values(cont.)Both intervals take the form

The SEs will be different for the two questions we have posed.

Confidence Intervals and Prediction Intervals for Predicted Values(cont.)The standard error of the mean predicted value is:

Individuals vary more than means, so the standard error for a single predicted value is larger than the standard error for the mean:

Confidence Intervals and Prediction Intervals for Predicted Values (cont.)Confidence interval for Prediction interval for y

Confidence Intervals for Predicted ValuesHeres a look at the difference between predicting for a mean and predicting for an individual.The solid green lines near the regression line show the 95% confidence intervals for the mean predicted value, and the dashed red lines show the prediction intervals for individuals.The solid green lines and the dashed red lines curve away from the least squares line as x moves farther away from xbar.

More on confidence intervals for As seen on the preceding slides, we can calculate a confidence interval for the population mean of all responses y when x takes the value x (within the range of data tested): denote this expected value E(y) by

This interval is centered on , the unbiased estimate of .

The true value of the population mean at a particularvalue x, will indeed be within our confidenceinterval in C% of all intervals calculated from many different random samples.The level C confidence interval for the mean response at a given value x of x is centered on (unbiased estimate of ):

A separate confidence interval is calculated for along all the values that x takes. Graphically, the series of confidence intervals is shown as a continuous interval on either side of .t* is the t critical for the t (n 2) distribution with area C between t* and +t*.

95% confidence bands for m as x varies over all x values

More on prediction intervals for yOne use of regression is for predicting the value of y, , for any value of x within the range of data tested: = b0 + b1x.But the regression equation depends on the particular sample drawn. More reliable predictions require statistical inference:

To estimate an individual response y for a given value of x, we use a prediction interval. If we randomly sampled many times, there would be many different values of y obtained for a particular x following N(0, ) around the mean response .The level C prediction interval for a single observation on y when x takes the value x is:

The prediction interval represents mainly the error from the normal distribution of the residuals ei.

Graphically, the series confidence intervals is shown as a continuous interval on either side of .

95% prediction interval for as x varies over all x values

t* is the t critical for the t (n 2) distribution with area C between t* and +t*.

The confidence interval for contains with C% confidence the population mean of all responses at a particular value x.

The prediction interval contains C% of all the individual values taken by y at a particular value x. 95% prediction interval for y95% confidence interval for m Estimating m uses a smaller confidence interval than estimating an individual in the population (sampling distribution narrower than population distribution).

1918 flu epidemicsThe line graph suggests that 7 to 9% of those diagnosed with the flu died within about a week of diagnosis.

We look at the relationship between the number of deaths in a given week and the number of new diagnosed cases one week earlier. EXCEL Regression Statistics Multiple R 0.911 R Square 0.830 Adjusted R Square 0.82 Standard Error 85.07 Observations 16.00

Coefficients St. Error t Stat P-value Lower 95% Upper 95% Intercept 49.292 29.845 1.652 0.1209 (14.720) 113.304 FluCases0 0.072 0.009 8.263 0.0000 0.053 0.0911918 flu epidemic: Relationship between the number of deaths in a given week and the number of new diagnosed cases one week earlier.

s

r = 0.91P-value for H0: 1 = 0b1P-value very small reject H0 1 significantly different from 0There is a significant relationship between the number of flu cases and the number of deaths from flu a week later.SPSS

Least squares regression line95% prediction interval for y95% confidence interval for my CI for mean weekly death count one week after x= 4000 flu cases are diagnosed: within about 300380.Prediction interval for a weekly death count one week after x= 4000 flu cases are diagnosed: y within about 180500 deaths.What is this?

A 90% prediction interval for the height (above) and a 90% prediction interval for the weight (below) of male children, ages 3 to 18.

58So, any individual boy may be well within the range of height for his age, but also be within the distributions for boys year or two older or younger as well. Two possible size distributions are shown.YearPowerboatsDead Manatees

197744713

197846021

197948124

198049816

198151324

198251220

198352615

198455934

198558533

198661433

198764539

198867543

198971150

199071947

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% of Seniors Taking TestMean SAT ScoreSAT Mean per State vs % Seniors Taking Test

lecunit1ht7121120.5116129127.5118.5131134.5125.5BinFrequency132141.5132.5127.51132148.5139.5134.58133155.5146.5141.545133162.5153.5148.545134160.5155.56134162.51135More0136136136136BinFrequency1371211137124.70137128.40138132.14138135.85138139.521138143.240138146.919138150.613138154.32139More1139139BinFrequency139118.50139125.51139132.54139139.526140146.559140153.515140160.51140More0140140140140140140140141141141141141141141141141141141141142142142142142142142142143143143143143143143143143144144144144144145145145145145145145145145146146146146146147148148148148148148148148149149149150153153158

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&APage &PFrequencyAbsences from WorkFrequencyHistogram

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&APage &PFrequencyAbsences from WorkFrequencyHistogram

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&APage &PFrequencyAbsences from WorkFrequencyHistogram

lecunit3(2)WeaponsMurder Rate11.613.18.310.63.610.10.64.46.911.52.56.62.43.62.65.3

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Murder RateRegistered automatic Weapons (1000's)Murder Rate (per 100,000)Automatic Weapons and Murder Rate

lecunit3(3)ht7FUEL CONSUMPTION3.45.5SUMMARY OUTPUT3.85.94.16.5Regression Statistics2.23.3Multiple R0.97662997182.63.6R Square0.95380610192.94.6Adjusted R Square0.948031864622.9Standard Error0.29024187762.73.6Observations101.93.13.44.9ANOVAdfSSMSFSignificance FRegression113.915077220113.9150772201165.1830463030.0000012688Residual80.67392277990.0842403475Total914.589RESIDUAL OUTPUTCoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%CarObservedPredictedIntercept-0.36308880310.3810415067-0.95288517580.3685477163-1.24177266140.5155950552-1.24177266140.5155950552ObservationWeightFuel ConsumptionFuel ConsumptionResidualsCAR WEIGHT1.6389961390.127524959712.85235567140.00000126881.34492286441.93306941361.34492286441.933069413613.45.55.20949806950.290501930523.85.95.86509652510.034903474934.16.56.35679536680.143204633242.23.33.24270270270.0572972973RESIDUAL OUTPUT52.63.63.8983011583-0.298301158362.94.64.390.21ObservationPredicted FUEL CONSUMPTIONResidualsStandard Residuals722.92.9149034749-0.014903474915.20949806950.29050193051.00089598782.73.64.0622007722-0.462200772225.86509652510.03490347490.120256508891.93.12.7510038610.34899613936.35679536680.14320463320.4933975566103.44.95.2094980695-0.309498069543.24270270270.05729729730.197412233553.8983011583-0.2983011583-1.027767463564.390.210.723534459572.9149034749-0.0149034749-0.05134846584.0622007722-0.4622007722-1.592467551892.7510038610.3489961391.2024320609105.2094980695-0.3094980695-1.0663453258IncomeConsumption Expenditure1756SUMMARY OUTPUT99138Regression Statistics1710Multiple R0.8R Square0.64Adjusted R Square0.52Standard Error1.095445115Observations5ANOVAdfSSMSFSignificance FRegression16.46.45.33333333330.1040880386Residual33.61.2Total410CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.000%Upper 95.000%Intercept6.20.9205976326.73475553780.00668445483.2702447199.1297552813.2702447199.129755281Income0.20.08660254042.30940107680.1040880386-0.07560819320.4756081932-0.07560819320.4756081932RESIDUAL OUTPUTObservationPredicted Consumption ExpenditureResiduals16.40.627.2-1.238148.8-0.859.60.4

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&APage &PFUEL CONSUMPTIONCAR WEIGHTFUEL CONSUMPTIONCAR WEIGHT Line Fit Plot

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&APage &PFUEL CONSUMPTIONWEIGHT (1000 lbs)FUEL CONSUMP. (gal/100 miles)FUEL CONSUMPTION vs CAR WEIGHT

olympic0.6-1.21-0.80.4

&APage &PIncomeResidualsIncome Residual Plot

SAT76.467.29888.8109.6

&APage &PConsumption ExpenditurePredicted Consumption ExpenditureIncomeConsumption ExpenditureIncome Line Fit Plot

STUDATA0.29050193050.03490347490.14320463320.0572972973-0.29830115830.21-0.0149034749-0.46220077220.348996139-0.3094980695

&APage &PCAR WEIGHT(1000 lbs)ResidualsCAR WEIGHT Residual Plot

1

FUEL CONSUMPTIONPredicted FUEL CONSUMPTIONCAR WEIGHTFUEL CONSUMPTIONCAR WEIGHT Line Fit Plot

ht7MinFoulsPoints0.18362898290.2399662060.8397097446685633449841241.4MinFoulsPoints47847180Min142728152Fouls0.839709744134636117.8Points0.2399662060.1836289829140043116431741083213080293406216335362602017653741288406582403534819830820156.836829176.74802616456243152303291182144168.494937415213449.62322.21710.86096927357345262.562538218.453051203.76685317032636113.435044109.250219.83929.9961536.496933.81272433.6601117.642210502.141067340423.7686343526156527633821136578471345944911422728401461722.525212.632515.419252214.8202259748298.352367279.351956248.954645239.428936134.936140110.25043998.83163358.951123.245616.822111.2244778165.265954470.46503233671328334.465017191.166943184.849973125.42092472.6751740.367928.865319.6603.973289178.52412.669048302.166445193.859155188.137145133.537137138.764637129.276520.12483582.64437303677128.41702.1739604417754635046271170.1704531276842421.24545786.13323758.81812545.919917453173143.71652786310050728291.647716190.441834162.437332147.640935129.63585064.83232164.82702149.32352750.41571437.58415201001.8300100

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ht7% Below Gr. Level% Fed. Support% MinoritySchoolReadingLunchEnrollmentAversboro21.735.252.5Brentwood24.44070.50.7702866331Brooks20.838.555.1Bugg21.425.548.4ReadingLunchEnrollmentCarver35.445.342.6Reading1Cary26.83744.5Lunch0.63449361611Conn24.839.354Enrollment0.42953208640.77028663311Creech Rd33.647.258.2Douglas22.546.556.1Fox Rd18.332.554.6Fuller26.625.854.7Fuqua-Varina26.526.632.7Hodge Rd31.936.255.6Jones Dairy12.37.611.6Knightdale29.438.653.1Lincoln Heights26.728.933.2Lockhart28.937.349Millbrook19.537.966.7Olive Chapel15.99.314.8Olds14.821.345.9Pleasant Union6.57.514.8Poe36.629.947Powell12.830.259.4Rand Rd28.441.346.3Reedy Creek29.226.846.8Rolesville26.832.642.1Smith21.646.859Vance25.636.440.3Vandora Spr33.144.853.2Wendell26.94448.6West Lake10.47.913.8Wilburn20.431.756.2Wiley21.335.246.4Zebulon22.652.449.5

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12111111.210.810.810.810.610.810.310.310.310.410.510.2109.9510.1410.0610.259.999.929.969.84

YearWinning Time (secs)Olympic 100 M Winning Times vs Yeary = -0.0141x + 37.817

yrTIME189612190011190411190611.2SUMMARY OUTPUT190810.8191210.8Regression Statistics192010.8Multiple R0.8858266554192410.6R Square0.7846888634192810.8Adjusted R Square0.7749019936193210.3Standard Error0.2435110366193610.3Observations24194810.3195210.4ANOVA195610.5dfSSMSFSignificance F196010.2Regression14.7543480854.75434808580.1777152310.0000000087196410Residual221.30454774830.059297624919689.95Total236.0588958333197210.14197610.06CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%198010.25Intercept37.81707887973.055301417612.3775280112031.480764754844.153393004731.480764754844.153393004719849.99YEAR-0.01405721150.0015699013-8.95420098230.0000000087-0.017312991-0.0108014321-0.017312991-0.010801432119889.9219929.9619969.84RESIDUAL OUTPUTPROBABILITY OUTPUTResid norm plotObservationPredicted TIMEResidualsPercentileTIMESample Percentilenorm valueresids111.16460580910.83539419092.08333333339.840.961.75068635140.8353941909211.108376963-0.1083769636.259.920.347-0.3934326287-0.108376963311.0521481168-0.052148116810.41666666679.950.434-0.1661987881-0.0521481168411.02403369380.175966306214.58333333339.960.8691.1216775420.1759663062510.9959192707-0.195919270718.759.990.173-0.942375209-0.1959192707610.9396904246-0.139690424622.9166666667100.26-0.6433447197-0.1396904246710.8272327323-0.027232732327.083333333310.060.478-0.0551733592-0.0272327323810.7710038861-0.171003886131.2510.140.217-0.7823655324-0.1710038861910.714775040.0852249635.416666666710.20.7820.77896629590.085224961010.6585461939-0.358546193939.583333333310.250.04-1.7506863514-0.35854619391110.6023173477-0.302317347743.7510.30.043-1.7168849809-0.30231734771210.4336308093-0.133630809347.916666666710.30.304-0.5129300007-0.13363080931310.37740196320.022598036852.083333333310.30.5650.16365902410.02259803681410.3211731170.17882688356.2510.40.9131.3594626580.1788268831510.2649442709-0.064944270960.416666666710.50.391-0.276713763-0.06494427091610.2087154247-0.208715424764.583333333310.60.086-1.3658063835-0.20871542471710.1524865786-0.202486578668.7510.80.13-1.1263909983-0.20248657861810.09625773250.043742267572.916666666710.80.6080.27411033440.04374226751910.04002888630.019971113777.083333333310.80.5210.05266315380.0199711137209.98380004020.266199959881.2510.80.9561.7060438040.2661999598219.9275711940.06242880685.4166666667110.6950.51007418730.062428806229.87134234790.048657652189.5833333333110.6520.39072574510.0486576521239.81511350180.144886498293.7511.20.8260.93847575040.1448864982249.75888465560.081115344497.9166666667120.7390.64026608020.0811153444

&APage &P

YEARResidualsYEAR Residual Plot

TIMEPredicted TIMEYEARTIMEYEAR Line Fit Plot

Sample PercentileTIMENormal Probability Plot

N(0,1) Value That Corresponds to Sample Percentile Value of ResidualResidualNormal Probability Plot of Residuals

SAT%TAKE1084541242115.871.5258849326644572116.272.3267031067535191915.466.4222499274947892318.475.6293851041632491915.467.9205259146052012516.867.43390010401040762015.273.3259929087562302013.170.3376591036524542824.580230239064438402822.773.5351721030103943201573.4238459056352071913.6712681910151136912318.568.9245549057254712113.772.2322001013124747221669.6310469036050172316.468.63158510091230682519.356.2256199025941492216.162.4290561006154386251773.1313958975942582014.668.7249381002827182618.756.5251908956353292214.661.1342341001830932116.566234008946765642013.667.4328629961030112417.853.1249208906971512214.966.3366549922042462015.874.4244218885239192621.173.8313079891337862215.963.5259818874740922517.166.726971988425482418.454.8225798866349892315.964.731248986931382418.257.7224698774336082217.462.626513986729892215.755.5217368715537942317.866.4291699821743692317.166.5311958475934342518.556.4269209781450512014.677.9276898466761322613.367.1367879721246922519.868355308385534082317.253.7250609662944622317.878.6295578365733682517.554.8256509651726672420.673.722732na nanana nana na9522337442518.272.428499na nanana nana na9482339982417.66729166na nanana nana na9393941642520.477.629146na nanana nana na9391538582115.15621904na nanana nana na9326644572116.272.326703na nanana nana na9274947892318.475.629385na nanana nana na9262336232620.375.528836na nanana nana na923427971211782.541832na nanana nana na9146052012516.867.433900na nanana nana na9087562302013.170.337659na nanana nana na9064438402822.773.535172na nanana nana na9056352071913.67126819na nanana nana na9057254712113.772.232200na nanana nana na9036050172316.468.631585na nanana nana na9025941492216.162.429056na nanana nana na8975942582014.668.724938na nanana nana na8956353292214.661.134234na nanana nana na8946765642013.667.432862na nanana nana na8906971512214.966.336654na nanana nana na8885239192621.173.831307na nanana nana na8874740922517.166.726971na nanana nana na8866349892315.964.731248na nanana nana na8774336082217.462.626513na nanana nana na8715537942317.866.429169na nanana nana na8475934342518.556.426920na nanana nana na8466761322613.367.136787na nanana nana na8385534082317.253.725060na nanana nana na8365733682517.554.825650na nanana nana naSUMMARY OUTPUTRegression StatisticsMultiple R0.8684259238R Square0.7541635852Adjusted R Square0.7491465155Standard Error31.1031278139Observations51ANOVAdfSSMSFSignificance FRegression1145419.804020362145419.804020362150.31953544770Residual4947402.8234306189967.4045598085Total50192822.62745098CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%Intercept1023.36006728727.5632468184135.30697752701008.16116298691038.55897158741008.16116298691038.5589715874%TAKE-2.23747883780.1824951066-12.26048675410-2.6042162625-1.870741413-2.6042162625-1.870741413RESIDUAL OUTPUTObservationPredicted 88-89 SATResiduals11012.172673098471.827326901621012.172673098454.827326901631009.935194260631.064805739441000.985278909639.014721090451012.172673098423.827326901661000.985278909629.01472109047998.747800071816.25219992828996.510321234116.48967876599996.510321234112.489678765910989.797884720816.2021152792111005.4602365851-3.4602365851121005.4602365851-4.4602365851131000.9852789096-4.985278909614978.61049053213.38950946815994.2728423963-5.2728423963161014.4101519361-26.4101519361171003.2227577473-17.2227577473181007.6977154228-21.697715422919985.3229270453-3.322927045320992.0353635586-14.035363558621996.5103212341-24.510321234122958.47318099227.526819007823985.3229270453-20.322927045324971.8980540187-19.898054018725971.8980540187-23.898054018726936.09839261462.901607385427989.7978847208-50.797884720828875.686463995256.313536004829913.72360423713.27639576330971.8980540187-45.898054018731929.3859561013-6.385956101332889.111337021724.888662978333855.549154455352.450845544734924.9109984258-18.910998425835882.398900508422.601099491636862.261590968642.738409031437889.111337021713.888662978338891.348815859510.651184140539891.34881585955.651184140540882.398900508412.601099491641873.448985157420.551014842642868.974027481921.025972518143907.0111677238-19.011167723844918.1985619126-31.198561912645882.39890050843.601099491646927.1484772636-50.148477263647900.2987312105-29.298731210548891.3488158595-44.348815859549873.4489851574-27.448985157450900.2987312105-62.298731210551895.823773535-59.823773535

&APage &P

&APage &P%TAKEResiduals%TAKE Residual Plot

&APage &P88-89 SATPredicted 88-89 SAT%TAKE88-89 SAT%TAKE Line Fit Plot

% of Seniors Taking TestMean SAT ScoreSAT Mean per State vs % Seniors Taking Test

ht7HTGPA$BOOKSHRS$/HRDR/WKEX/WEEKBPULSEEPULSEHEDCIRSLEEP16513.12501319.23077209172235SUMMARY OUTPUT070322251416.0714312797423607212.772001910.5263216776622.56Regression Statistics073.512.72001612.57126054224.5Multiple R0.126546567307112.92001216.6666737757022.57.5R Square0.016014033706712.11801313.846153.509091226Adjusted R Square-0.000385732406812.61801313.84615056362226Standard Error13.813656210707411.950170367062236Observations6207113.772501913.157924706822.5706813.83001618.7574706522.57ANOVA07612.62101316.15385438778237.5dfSSMSFSignificance F16232.518717111057063216Regression1186.328964311186.3289643110.9764793950.327037874706612.8na 10.00000na5207578220Residual6011449.0258743986190.817097906607313.12301219.16667058076226Total6111635.354838709707412.9120101245757223707312.61501311.53846549678238CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%068232001811.11111316260238Intercept71.60421271622.72138135726.3117157515066.160641282977.047784149466.160641282977.047784149407423.44115157.66666763645722.57EX/WEEK-0.6127941750.6201304926-0.98816971970.3270378747-1.85323944570.6276510957-1.85323944570.627651095707212.82001513.3333341947423707032.3na 18.00000na360707022.5007212.282501319.2307742969324007112.662001612.543746622.58RESIDUAL OUTPUT16721.7100137.69230730726923706712.83501720.5882423717322.56ObservationPredicted EPULSEResiduals07032.43na 17.00000na5005960220171.60421271620.395787283807112.93101916.31579959474237270.37862436613.6213756339168742101316.1538523969522.57.5367.9274476661-1.927447666106713.142301614.375137670207464.250682616-10.25068261616312.92501714.70588437068236.5567.31465349112.685346508916013.33001421.4285735676221.55671.604212716219.395787283807231.72001315.3846122847623.58.5768.5402418411-6.540241841116411.89150101540857723.54867.9274476661-5.927447666107012.22501615.6252.50706222.56969.1530360161-1.153036016106713.22501319.2307701579812361069.1530360161-4.153036016107013.2466164.12533605923.7561169.76583019118.2341698089168.51316016108410310422.571268.5402418411-5.540241841116413.92101613.12513807621.756.51371.60421271626.395787283806712.6150101513717122.571468.54024184117.4597581589167.512.52161812126928622.581568.54024184113.459758158906912.293185.166667827860237.51669.15303601618.846963983907313.22351912.368423363652371770.9914185412-10.991418541216112.9na 19.00000na22077682101869.7658301911-12.765830191116612.92001315.384613473602261970.99141854123.008581458816413.465154.3333333374642282071.6042127162-1.604212716207213.12001315.384614462692382170.378624366122.621375633907432.12001216.666677484762372269.7658301911-3.765830191106912.482001414.2857135806522.7562371.6042127162-2.604212716216723.12001711.764717277802262469.76583019113.234169808917012.42001414.2857152776422.7572571.6042127162-11.604212716206813.141501212.50575652102668.54024184115.459758158906812.810010102764692382769.765830191125.234169808906412.6316082033687625na na2869.76583019110.2341698089074na 3.250000 na12.00000 nana na 98.00000750987522.5na na2969.7658301911-1.7658301911071121801611.253378602363068.5402418411-6.540241841107112.12601715.294123.210656423.563170.37862436615.621375633916473.1na 15.00000na230988622na n3271.60421271625.3957872838164.52na na15.00000 na00 5.00000248517na3371.6042127162-9.6042127162163232501516.666673193792273462.41230009118.58769990907232.6017053383722.583569.7658301911-10.765830191116522.4na 17.00000na220957920na n3669.153036016134.8469639839076126012522857322.173769.76583019116.234169808907223.292201712.9411835726522.76.53869.76583019111.23416980893967.927447666118.07255233394070.3786243661-10.37862436614169.7658301911-4.76583019114271.6042127162-3.60421271624369.1530360161-9.15303601614469.7658301911-5.76583019114569.1530360161-0.15303601614669.15303601616.84696398394768.5402418411-3.54024184114870.37862436619.62137563394970.3786243661-6.37862436615068.5402418411-3.54024184115167.31465349111.68534650895269.76583019116.23416980895371.60421271623.39578728385469.7658301911-9.76583019115565.4762709661-1.47627096615671.604212716214.39578728385769.1530360161-68.15303601615870.99141854128.00858145885969.7658301911-32.76583019116071.60421271627.39578728386170.37862436612.62137563396268.5402418411-3.5402418411

&APage &P

00000000000000000000000000000000000000000000000000000000000000

&APage &PEX/WEEKResidualsEX/WEEK Residual Plot

0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

&APage &PEPULSEPredicted EPULSEEX/WEEKEPULSEEX/WEEK Line Fit Plot

Chart3108410671041104010361030101510131009100610021001996992989988986986982978972966965952948939939932927926923914908906905905903902897895894890888887886877871847846838836

% of Seniors Taking TestMean SAT ScoreSAT Mean per State vs % Seniors Taking Test

lecunit1ht7121120.5116129127.5118.5131134.5125.5BinFrequency132141.5132.5127.51132148.5139.5134.58133155.5146.5141.545133162.5153.5148.545134160.5155.56134162.51135More0136136136136BinFrequency1371211137124.70137128.40138132.14138135.85138139.521138143.240138146.919138150.613138154.32139More1139139BinFrequency139118.50139125.51139132.54139139.526140146.559140153.515140160.51140More0140140140140140140140141141141141141141141141141141141141142142142142142142142142143143143143143143143143143144144144144144145145145145145145145145145146146146146146147148148148148148148148148149149149150153153158

&APage &P

lecunit10000000

&APage &PFrequencyAbsences from WorkFrequencyHistogram

wksht700000000000

&APage &PFrequencyAbsences from WorkFrequencyHistogram

lecunit300000000

&APage &PFrequencyAbsences from WorkFrequencyHistogram

lecunit3(2)WeaponsMurder Rate11.613.18.310.63.610.10.64.46.911.52.56.62.43.62.65.3

lecunit3(2)00000000

Murder RateRegistered automatic Weapons (1000's)Murder Rate (per 100,000)Automatic Weapons and Murder Rate

lecunit3(3)ht7FUEL CONSUMPTION3.45.5SUMMARY OUTPUT3.85.94.16.5Regression Statistics2.23.3Multiple R0.97662997182.63.6R Square0.95380610192.94.6Adjusted R Square0.948031864622.9Standard Error0.29024187762.73.6Observations101.93.13.44.9ANOVAdfSSMSFSignificance FRegression113.915077220113.9150772201165.1830463030.0000012688Residual80.67392277990.0842403475Total914.589RESIDUAL OUTPUTCoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%CarObservedPredictedIntercept-0.36308880310.3810415067-0.95288517580.3685477163-1.24177266140.5155950552-1.24177266140.5155950552ObservationWeightFuel ConsumptionFuel ConsumptionResidualsCAR WEIGHT1.6389961390.127524959712.85235567140.00000126881.34492286441.93306941361.34492286441.933069413613.45.55.20949806950.290501930523.85.95.86509652510.034903474934.16.56.35679536680.143204633242.23.33.24270270270.0572972973RESIDUAL OUTPUT52.63.63.8983011583-0.298301158362.94.64.390.21ObservationPredicted FUEL CONSUMPTIONResidualsStandard Residuals722.92.9149034749-0.014903474915.20949806950.29050193051.00089598782.73.64.0622007722-0.462200772225.86509652510.03490347490.120256508891.93.12.7510038610.34899613936.35679536680.14320463320.4933975566103.44.95.2094980695-0.309498069543.24270270270.05729729730.197412233553.8983011583-0.2983011583-1.027767463564.390.210.723534459572.9149034749-0.0149034749-0.05134846584.0622007722-0.4622007722-1.592467551892.7510038610.3489961391.2024320609105.2094980695-0.3094980695-1.0663453258IncomeConsumption Expenditure1756SUMMARY OUTPUT99138Regression Statistics1710Multiple R0.8R Square0.64Adjusted R Square0.52Standard Error1.095445115Observations5ANOVAdfSSMSFSignificance FRegression16.46.45.33333333330.1040880386Residual33.61.2Total410CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.000%Upper 95.000%Intercept6.20.9205976326.73475553780.00668445483.2702447199.1297552813.2702447199.129755281Income0.20.08660254042.30940107680.1040880386-0.07560819320.4756081932-0.07560819320.4756081932RESIDUAL OUTPUTObservationPredicted Consumption ExpenditureResiduals16.40.627.2-1.238148.8-0.859.60.4

&APage &P

lecunit3(3)

&APage &PFUEL CONSUMPTIONCAR WEIGHTFUEL CONSUMPTIONCAR WEIGHT Line Fit Plot

Chart1

&APage &PFUEL CONSUMPTIONWEIGHT (1000 lbs)FUEL CONSUMP. (gal/100 miles)FUEL CONSUMPTION vs CAR WEIGHT

olympic0.6-1.21-0.80.4

&APage &PIncomeResidualsIncome Residual Plot

SAT76.467.29888.8109.6

&APage &PConsumption ExpenditurePredicted Consumption ExpenditureIncomeConsumption ExpenditureIncome Line Fit Plot

STUDATA0.29050193050.03490347490.14320463320.0572972973-0.29830115830.21-0.0149034749-0.46220077220.348996139-0.3094980695

&APage &PCAR WEIGHT(1000 lbs)ResidualsCAR WEIGHT Residual Plot

1

FUEL CONSUMPTIONPredicted FUEL CONSUMPTIONCAR WEIGHTFUEL CONSUMPTIONCAR WEIGHT Line Fit Plot

ht7MinFoulsPoints0.18362898290.2399662060.8397097446685633449841241.4MinFoulsPoints47847180Min142728152Fouls0.839709744134636117.8Points0.2399662060.1836289829140043116431741083213080293406216335362602017653741288406582403534819830820156.836829176.74802616456243152303291182144168.494937415213449.62322.21710.86096927357345262.562538218.453051203.76685317032636113.435044109.250219.83929.9961536.496933.81272433.6601117.642210502.141067340423.7686343526156527633821136578471345944911422728401461722.525212.632515.419252214.8202259748298.352367279.351956248.954645239.428936134.936140110.25043998.83163358.951123.245616.822111.2244778165.265954470.46503233671328334.465017191.166943184.849973125.42092472.6751740.367928.865319.6603.973289178.52412.669048302.166445193.859155188.137145133.537137138.764637129.276520.12483582.64437303677128.41702.1739604417754635046271170.1704531276842421.24545786.13323758.81812545.919917453173143.71652786310050728291.647716190.441834162.437332147.640935129.63585064.83232164.82702149.32352750.41571437.58415201001.8300100

&APage &P

ht7% Below Gr. Level% Fed. Support% MinoritySchoolReadingLunchEnrollmentAversboro21.735.252.5Brentwood24.44070.50.7702866331Brooks20.838.555.1Bugg21.425.548.4ReadingLunchEnrollmentCarver35.445.342.6Reading1Cary26.83744.5Lunch0.63449361611Conn24.839.354Enrollment0.42953208640.77028663311Creech Rd33.647.258.2Douglas22.546.556.1Fox Rd18.332.554.6Fuller26.625.854.7Fuqua-Varina26.526.632.7Hodge Rd31.936.255.6Jones Dairy12.37.611.6Knightdale29.438.653.1Lincoln Heights26.728.933.2Lockhart28.937.349Millbrook19.537.966.7Olive Chapel15.99.314.8Olds14.821.345.9Pleasant Union6.57.514.8Poe36.629.947Powell12.830.259.4Rand Rd28.441.346.3Reedy Creek29.226.846.8Rolesville26.832.642.1Smith21.646.859Vance25.636.440.3Vandora Spr33.144.853.2Wendell26.94448.6West Lake10.47.913.8Wilburn20.431.756.2Wiley21.335.246.4Zebulon22.652.449.5

&APage &P

12111111.210.810.810.810.610.810.310.310.310.410.510.2109.9510.1410.0610.259.999.929.969.84

YearWinning Time (secs)Olympic 100 M Winning Times vs Yeary = -0.0141x + 37.817

yrTIME189612190011190411190611.2SUMMARY OUTPUT190810.8191210.8Regression Statistics192010.8Multiple R0.8858266554192410.6R Square0.7846888634192810.8Adjusted R Square0.7749019936193210.3Standard Error0.2435110366193610.3Observations24194810.3195210.4ANOVA195610.5dfSSMSFSignificance F196010.2Regression14.7543480854.75434808580.1777152310.0000000087196410Residual221.30454774830.059297624919689.95Total236.0588958333197210.14197610.06CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%198010.25Intercept37.81707887973.055301417612.3775280112031.480764754844.153393004731.480764754844.153393004719849.99YEAR-0.01405721150.0015699013-8.95420098230.0000000087-0.017312991-0.0108014321-0.017312991-0.010801432119889.9219929.9619969.84RESIDUAL OUTPUTPROBABILITY OUTPUTResid norm plotObservationPredicted TIMEResidualsPercentileTIMESample Percentilenorm valueresids111.16460580910.83539419092.08333333339.840.961.75068635140.8353941909211.108376963-0.1083769636.259.920.347-0.3934326287-0.108376963311.0521481168-0.052148116810.41666666679.950.434-0.1661987881-0.0521481168411.02403369380.175966306214.58333333339.960.8691.1216775420.1759663062510.9959192707-0.195919270718.759.990.173-0.942375209-0.1959192707610.9396904246-0.139690424622.9166666667100.26-0.6433447197-0.1396904246710.8272327323-0.027232732327.083333333310.060.478-0.0551733592-0.0272327323810.7710038861-0.171003886131.2510.140.217-0.7823655324-0.1710038861910.714775040.0852249635.416666666710.20.7820.77896629590.085224961010.6585461939-0.358546193939.583333333310.250.04-1.7506863514-0.35854619391110.6023173477-0.302317347743.7510.30.043-1.7168849809-0.30231734771210.4336308093-0.133630809347.916666666710.30.304-0.5129300007-0.13363080931310.37740196320.022598036852.083333333310.30.5650.16365902410.02259803681410.3211731170.17882688356.2510.40.9131.3594626580.1788268831510.2649442709-0.064944270960.416666666710.50.391-0.276713763-0.06494427091610.2087154247-0.208715424764.583333333310.60.086-1.3658063835-0.20871542471710.1524865786-0.202486578668.7510.80.13-1.1263909983-0.20248657861810.09625773250.043742267572.916666666710.80.6080.27411033440.04374226751910.04002888630.019971113777.083333333310.80.5210.05266315380.0199711137209.98380004020.266199959881.2510.80.9561.7060438040.2661999598219.9275711940.06242880685.4166666667110.6950.51007418730.062428806229.87134234790.048657652189.5833333333110.6520.39072574510.0486576521239.81511350180.144886498293.7511.20.8260.93847575040.1448864982249.75888465560.081115344497.9166666667120.7390.64026608020.0811153444

&APage &P

YEARResidualsYEAR Residual Plot

TIMEPredicted TIMEYEARTIMEYEAR Line Fit Plot

Sample PercentileTIMENormal Probability Plot

N(0,1) Value That Corresponds to Sample Percentile Value of ResidualResidualNormal Probability Plot of Residuals

SAT%TAKE1084541242115.871.5258849326644572116.272.3267031067535191915.466.4222499274947892318.475.6293851041632491915.467.9205259146052012516.867.43390010401040762015.273.3259929087562302013.170.3376591036524542824.580230239064438402822.773.5351721030103943201573.4238459056352071913.6712681910151136912318.568.9245549057254712113.772.2322001013124747221669.6310469036050172316.468.63158510091230682519.356.2256199025941492216.162.4290561006154386251773.1313958975942582014.668.7249381002827182618.756.5251908956353292214.661.1342341001830932116.566234008946765642013.667.4328629961030112417.853.1249208906971512214.966.3366549922042462015.874.4244218885239192621.173.8313079891337862215.963.5259818874740922517.166.726971988425482418.454.8225798866349892315.964.731248986931382418.257.7224698774336082217.462.626513986729892215.755.5217368715537942317.866.4291699821743692317.166.5311958475934342518.556.4269209781450512014.677.9276898466761322613.367.1367879721246922519.868355308385534082317.253.7250609662944622317.878.6295578365733682517.554.8256509651726672420.673.722732na nanana nana na9522337442518.272.428499na nanana nana na9482339982417.66729166na nanana nana na9393941642520.477.629146na nanana nana na9391538582115.15621904na nanana nana na9326644572116.272.326703na nanana nana na9274947892318.475.629385na nanana nana na9262336232620.375.528836na nanana nana na923427971211782.541832na nanana nana na9146052012516.867.433900na nanana nana na9087562302013.170.337659na nanana nana na9064438402822.773.535172na nanana nana na9056352071913.67126819na nanana nana na9057254712113.772.232200na nanana nana na9036050172316.468.631585na nanana nana na9025941492216.162.429056na nanana nana na8975942582014.668.724938na nanana nana na8956353292214.661.134234na nanana nana na8946765642013.667.432862na nanana nana na8906971512214.966.336654na nanana nana na8885239192621.173.831307na nanana nana na8874740922517.166.726971na nanana nana na8866349892315.964.731248na nanana nana na8774336082217.462.626513na nanana nana na8715537942317.866.429169na nanana nana na8475934342518.556.426920na nanana nana na8466761322613.367.136787na nanana nana na8385534082317.253.725060na nanana nana na8365733682517.554.825650na nanana nana naSUMMARY OUTPUTRegression StatisticsMultiple R0.8684259238R Square0.7541635852Adjusted R Square0.7491465155Standard Error31.1031278139Observations51ANOVAdfSSMSFSignificance FRegression1145419.804020362145419.804020362150.31953544770Residual4947402.8234306189967.4045598085Total50192822.62745098CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%Intercept1023.36006728727.5632468184135.30697752701008.16116298691038.55897158741008.16116298691038.5589715874%TAKE-2.23747883780.1824951066-12.26048675410-2.6042162625-1.870741413-2.6042162625-1.870741413RESIDUAL OUTPUTObservationPredicted 88-89 SATResiduals11012.172673098471.827326901621012.172673098454.827326901631009.935194260631.064805739441000.985278909639.014721090451012.172673098423.827326901661000.985278909629.01472109047998.747800071816.25219992828996.510321234116.48967876599996.510321234112.489678765910989.797884720816.2021152792111005.4602365851-3.4602365851121005.4602365851-4.4602365851131000.9852789096-4.985278909614978.61049053213.38950946815994.2728423963-5.2728423963161014.4101519361-26.4101519361171003.2227577473-17.2227577473181007.6977154228-21.697715422919985.3229270453-3.322927045320992.0353635586-14.035363558621996.5103212341-24.510321234122958.47318099227.526819007823985.3229270453-20.322927045324971.8980540187-19.898054018725971.8980540187-23.898054018726936.09839261462.901607385427989.7978847208-50.797884720828875.686463995256.313536004829913.72360423713.27639576330971.8980540187-45.898054018731929.3859561013-6.385956101332889.111337021724.888662978333855.549154455352.450845544734924.9109984258-18.910998425835882.398900508422.601099491636862.261590968642.738409031437889.111337021713.888662978338891.348815859510.651184140539891.34881585955.651184140540882.398900508412.601099491641873.448985157420.551014842642868.974027481921.025972518143907.0111677238-19.011167723844918.1985619126-31.198561912645882.39890050843.601099491646927.1484772636-50.148477263647900.2987312105-29.298731210548891.3488158595-44.348815859549873.4489851574-27.448985157450900.2987312105-62.298731210551895.823773535-59.823773535

&APage &P

&APage &P%TAKEResiduals%TAKE Residual Plot

&APage &P88-89 SATPredicted 88-89 SAT%TAKE88-89 SAT%TAKE Line Fit Plot

% of Seniors Taking TestMean SAT ScoreSAT Mean per State vs % Seniors Taking Test

ht7HTGPA$BOOKSHRS$/HRDR/WKEX/WEEKBPULSEEPULSEHEDCIRSLEEP16513.12501319.23077209172235SUMMARY OUTPUT070322251416.0714312797423607212.772001910.5263216776622.56Regression Statistics073.512.72001612.57126054224.5Multiple R0.126546567307112.92001216.6666737757022.57.5R Square0.016014033706712.11801313.846153.509091226Adjusted R Square-0.000385732406812.61801313.84615056362226Standard Error13.813656210707411.950170367062236Observations6207113.772501913.157924706822.5706813.83001618.7574706522.57ANOVA07612.62101316.15385438778237.5dfSSMSFSignificance F16232.518717111057063216Regression1186.328964311186.3289643110.9764793950.327037874706612.8na 10.00000na5207578220Residual6011449.0258743986190.817097906607313.12301219.16667058076226Total6111635.354838709707412.9120101245757223707312.61501311.53846549678238CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%068232001811.11111316260238Intercept71.60421271622.72138135726.3117157515066.160641282977.047784149466.160641282977.047784149407423.44115157.66666763645722.57EX/WEEK-0.6127941750.6201304926-0.98816971970.3270378747-1.85323944570.6276510957-1.85323944570.627651095707212.82001513.3333341947423707032.3na 18.00000na360707022.5007212.282501319.2307742969324007112.662001612.543746622.58RESIDUAL OUTPUT16721.7100137.69230730726923706712.83501720.5882423717322.56ObservationPredicted EPULSEResiduals07032.43na 17.00000na5005960220171.60421271620.395787283807112.93101916.31579959474237270.37862436613.6213756339168742101316.1538523969522.57.5367.9274476661-1.927447666106713.142301614.375137670207464.250682616-10.25068261616312.92501714.70588437068236.5567.31465349112.685346508916013.33001421.4285735676221.55671.604212716219.395787283807231.72001315.3846122847623.58.5768.5402418411-6.540241841116411.89150101540857723.54867.9274476661-5.927447666107012.22501615.6252.50706222.56969.1530360161-1.153036016106713.22501319.2307701579812361069.1530360161-4.153036016107013.2466164.12533605923.7561169.76583019118.2341698089168.51316016108410310422.571268.5402418411-5.540241841116413.92101613.12513807621.756.51371.60421271626.395787283806712.6150101513717122.571468.54024184117.4597581589167.512.52161812126928622.581568.54024184113.459758158906912.293185.166667827860237.51669.15303601618.846963983907313.22351912.368423363652371770.9914185412-10.991418541216112.9na 19.00000na22077682101869.7658301911-12.765830191116612.92001315.384613473602261970.99141854123.008581458816413.465154.3333333374642282071.6042127162-1.604212716207213.12001315.384614462692382170.378624366122.621375633907432.12001216.666677484762372269.7658301911-3.765830191106912.482001414.2857135806522.7562371.6042127162-2.604212716216723.12001711.764717277802262469.76583019113.234169808917012.42001414.2857152776422.7572571.6042127162-11.604212716206813.141501212.50575652102668.54024184115.459758158906812.810010102764692382769.765830191125.234169808906412.6316082033687625na na2869.76583019110.2341698089074na 3.250000 na12.00000 nana na 98.00000750987522.5na na2969.7658301911-1.7658301911071121801611.253378602363068.5402418411-6.540241841107112.12601715.294123.210656423.563170.37862436615.621375633916473.1na 15.00000na230988622na n3271.60421271625.3957872838164.52na na15.00000 na00 5.00000248517na3371.6042127162-9.6042127162163232501516.666673193792273462.41230009118.58769990907232.6017053383722.583569.7658301911-10.765830191116522.4na 17.00000na220957920na n3669.153036016134.8469639839076126012522857322.173769.76583019116.234169808907223.292201712.9411835726522.76.53869.76583019111.23416980893967.927447666118.07255233394070.3786243661-10.37862436614169.7658301911-4.76583019114271.6042127162-3.60421271624369.1530360161-9.15303601614469.7658301911-5.76583019114569.1530360161-0.15303601614669.15303601616.84696398394768.5402418411-3.54024184114870.37862436619.62137563394970.3786243661-6.37862436615068.5402418411-3.54024184115167.31465349111.68534650895269.76583019116.23416980895371.60421271623.39578728385469.7658301911-9.76583019115565.4762709661-1.47627096615671.604212716214.39578728385769.1530360161-68.15303601615870.99141854128.00858145885969.7658301911-32.76583019116071.60421271627.39578728386170.37862436612.62137563396268.5402418411-3.5402418411

&APage &P

00000000000000000000000000000000000000000000000000000000000000

&APage &PEX/WEEKResidualsEX/WEEK Residual Plot

0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

&APage &PEPULSEPredicted EPULSEEX/WEEKEPULSEEX/WEEK Line Fit Plot

Chart4248163264128256512102420484096

bacterial countTime (min)Bacterial count

Sheet1timebacterial countlog of bacterial count12020.301029995724040.602059991336080.903089987480161.20411998275100321.50514997836120641.80617997471401282.107209969681602562.408239965391805122.7092699611020010243.01029995661122020483.31132995231224040963.6123599481326081923.913389943614280163844.214419939315300327684.51544993516320655364.8164799306173401310725.1175099263183602621445.418539922193805242885.71956991762040010485766.02059991332142020971526.32162990892244041943046.62265990462346083886086.923689900324480167772167.224719895925500335544327.525749891626520671088647.8267798873275401342177288.1278098829285602684354568.4288398786

Sheet1000000000000

bacterial counttime (min)bacterial count

Sheet2000000000000

log of bacterial counttime (min)log of bacterial count

Sheet3

Chart50.30102999570.60205999130.9030899871.20411998271.50514997831.8061799742.10720996962.40823996532.7092699613.01029995663.31132995233.612359948

log of bacterial countTime (min)Log of bacterial count

Sheet1timebacterial countlog of bacterial count12020.301029995724040.602059991336080.903089987480161.20411998275100321.50514997836120641.80617997471401282.107209969681602562.408239965391805122.7092699611020010243.01029995661122020483.31132995231224040963.6123599481326081923.913389943614280163844.214419939315300327684.51544993516320655364.8164799306173401310725.1175099263183602621445.418539922193805242885.71956991762040010485766.02059991332142020971526.32162990892244041943046.62265990462346083886086.923689900324480167772167.224719895925500335544327.525749891626520671088647.8267798873275401342177288.1278098829285602684354568.4288398786

Sheet1000000000000

bacterial counttime (min)bacterial count

Sheet2000000000000

log of bacterial counttime (min)log of bacterial count

Sheet3