Chapter 4
Understanding the regression line

• 420 districts
• Y: average district TestScore, points
• X: STR, students/teacher

step by step

1. Y = TestScore: the outcome we predict
2. X = STR: the clue used to predict Y
3. Horizontal axis: X; vertical axis: Y
4. A negative relationship is plausible; inspect the data first.

why it works

Start with the sentence, predict test score using the ratio. The thing after predict is the outcome, Y. The clue after using is X. So Test Score is the dependent variable and S T R is the independent variable or regressor. Put S T R on the horizontal axis and Test Score on the vertical axis.

The word independent here is a label for a regressor. It does not claim that X and Y are unrelated. Also, every observation is a district, not an individual student. The ratio is district enrollment divided by full-time-equivalent teachers. It is not necessarily the exact headcount in a particular classroom.

For part c, a negative relationship is a reasonable prediction: more students per teacher might be associated with lower scores. But that is a reasoned expectation, not something established before looking at the data. Other influences may matter. An answer that carefully explains uncertainty is different from claiming the sign has already been proved.

• Scatterplot of420districts

step by step

1. Overall association: negative, weak, and far from exact
2. Similar STR values can have very different scores
3. A trend does not force every observation onto a line
4. OLS supplies a common rule for fitting the line

why it works

Look at the cloud rather than choosing two convenient dots. The general tendency slopes downward, so the association appears negative, but it is weak and far from exact. At a ratio near twenty, there are districts with noticeably different scores. Therefore similar X values do not force similar Y values.

A sensible sketch slopes downward through the middle of the cloud. Two people can draw different reasonable-looking lines because eyeballing is subjective. That motivates an objective rule for choosing coefficients.

The plotted dots are observations; the line will be a summary of their average tendency. Do not confuse a trend with a law that every dot must obey. Later, a low R squared will put a number on this substantial scatter. But we already see the qualitative fact here: one predictor leaves much of the outcome's variation unexplained.

• PopulationYi=β0+β1Xi+ui
• YTestScore;XSTR

step by step

1. TestScoreᵢ = β₀ + β₁STRᵢ + uᵢ
2. Estimate the unknown β₀ and β₁
3. Other influences may include family resources and language background

why it works

Write Test Score for district i equals beta zero plus beta one times S T R for district i, plus u for district i. The i just labels a district. Beta zero is the population intercept, and beta one is the population slope. Those two unknown coefficients are the quantities we estimate.

The error term is the part of the outcome not accounted for by the population line. It does not necessarily mean a typo or a mistake in data entry. For instance, family resources and language background may be relevant to test scores beyond the staffing ratio. Those are examples of other influences, not a claim that we have measured them here.

Think of the equation as a general trend plus everything left over. Once we fit a line to our sample, we use hats for estimated coefficients. A fitted value has no added error term. The actual observed score can instead be written as the fitted value plus its residual. The population error and the observable sample residual play related but different roles.

• (18,660),(20,650),(22,652)
• Line1:695−2X
• Line2:687−1.5X

step by step

1. Line 1: A: 695 − 2(18) = 659; B: 695 − 2(20) = 655; C: 695 − 2(22) = 651
2. Residuals: 660 − 659 = 1; 650 − 655 = −5; 652 − 651 = 1. SSR = 1² + (−5)² + 1² = 27
3. Line 2: A: 687 − 1.5(18) = 660; B: 687 − 1.5(20) = 657; C: 687 − 1.5(22) = 654
4. Residuals: 660 − 660 = 0; 650 − 657 = −7; 652 − 654 = −2. SSR = 0² + (−7)² + (−2)² = 53
5. 27 < 53: line 1 fits these three observations better

why it works

We need a common score for prediction mistakes. A fitted value comes from substituting X into the line. A residual is actual minus fitted, not the other way around.

For line one, A is predicted as six hundred ninety-five minus thirty-six, or six hundred fifty-nine. Its residual is one. B's prediction is six hundred fifty-five, giving a residual of negative five. C's prediction is six hundred fifty-one, giving a residual of one. Square these residuals and add: one plus twenty-five plus one equals twenty-seven.

For line two, the predictions are six hundred sixty, six hundred fifty-seven, and six hundred fifty-four. The residuals are zero, negative seven, and negative two. The squared total is zero plus forty-nine plus four, or fifty-three.

Line one wins this comparison because twenty-seven is smaller. Squaring prevents positive and negative errors from canceling and penalizes large misses. But comparing two candidates does not establish the best line among every possible intercept and slope. The question's last sentence is testing that distinction. We have a winner among these two, not a proof of the global least-squares solution.

• Xbar19.6404
• Ybar654.1565
• sXY−8.1593
• sX²3.5790

step by step

1. β̂₁ = −8.1593 / 3.5790 = −2.2797709
2. β̂₀ = 654.1565 − (−2.2797709)(19.6404) ≈ 698.9321
3. Rounded: β̂₀ = 698.93; β̂₁ = −2.28

why it works

The slope is sample covariance divided by the sample variance of X. In plain terms, joint movement gets scaled by how much the predictor varies. Use the variance of S T R in the denominator, not its standard deviation and not the variance of Test Score.

Negative eight point one five nine three divided by three point five seven nine zero gives negative two point two seven nine seven seven zero nine after the specified seven-decimal rounding. Now compute the intercept as mean Y minus slope times mean X. That is six hundred fifty-four point one five six five minus the product of negative two point two seven nine seven seven zero nine and nineteen point six four zero four. It gives approximately six hundred ninety-eight point nine three two one.

Report the intercept as six hundred ninety-eight point nine three and the slope as negative two point two eight. The variance in the denominator is positive, so dividing by it cannot flip the covariance's sign. Negative covariance means negative slope here.

These supplied summary statistics are rounded. Slight differences from the full-precision Excel coefficients are expected; they do not mean one calculation is wrong.

• Estimatedintercept698.93,slope−2.28

step by step

1. Predicted TestScore = 698.93 − 2.28 STR
2. Actual Yᵢ = Ŷᵢ + ûᵢ
3. Population error uᵢ and sample residual ûᵢ are different

why it works

The fitted equation is predicted Test Score equals six hundred ninety-eight point nine three minus two point two eight times S T R. A hat over Test Score communicates predicted, not actual. Do not add an error term to this predicted value.

If you want to describe an actual observation, write actual Test Score equals the fitted value plus that observation's residual. Equivalently, use the estimated intercept and slope, then add the residual with a hat.

The bottom table distinguishes three things. The population model uses unknown beta coefficients and the population error u. The estimated equation uses estimated coefficients and gives Y hat. The actual outcome is Y hat plus the sample residual.

You can calculate residuals because you have the observations and fitted line. You do not directly observe the true population line, so you should not casually substitute the population error for the sample residual. In everyday terms: prediction is our rule's answer, actual is what happened, and residual is the gap.

• STR20both
• ActualA640,B670
• ŷ698.93−2.28STR

step by step

1. Ŷ = 698.93 − 2.28(20) = 653.33 for both districts
2. A: û = 640 − 653.33 = −13.33 points; overpredicted
3. B: û = 670 − 653.33 = +16.67 points; underpredicted

why it works

Substitute twenty into the line. Two point two eight times twenty is forty-five point six. Subtracting from six hundred ninety-eight point nine three gives six hundred fifty-three point three three. Both districts receive that prediction because the only predictor has the same value.

For A, subtract prediction from actual: six hundred forty minus six hundred fifty-three point three three equals negative thirteen point three three. The prediction was too high, so A is overpredicted and lies below the line.

For B, six hundred seventy minus six hundred fifty-three point three three equals positive sixteen point six seven. The prediction was too low, so B is underpredicted and lies above the line.

Do not assume that equal X values force equal actual outcomes. The one-predictor rule gives equal fitted values, but other influences can make the actual scores differ. Both residuals are measured in test-score points, not percentages or students per teacher.

• YcolumnK
• XcolumnN
• Labelsrow1
• Data2:421

step by step

1. Input Y Range: K1:K421
2. Input X Range: N1:N421
3. Check Labels; include the intercept; request residuals
4. Use all 420 observations, not the first few rows
5. SER ≈ 18.58 points; slope SE ≈ 0.48: different jobs

why it works

Input Y Range is K one through K four hundred twenty-one, because test score is the outcome. Input X Range is N one through N four hundred twenty-one, because the ratio is the predictor. Check Labels when including row one. Leave Constant is Zero unchecked because our model estimates an intercept. The worksheet also asks for residual output.

Selecting only the first few observations would estimate a different regression using a different sample. It would not reproduce the four hundred twenty-district analysis.

Now look at the actual Excel output on the next PDF page. The coefficient rows show intercept about six hundred ninety-eight point nine three and slope about negative two point two eight. The regression-statistics standard error, about eighteen point five eight, is S E R. The slope row's standard error, about zero point four eight, is uncertainty in the slope estimate. These are not interchangeable even though Excel uses the words standard error for both.

This is a reading walkthrough, not a live Excel installation demonstration. On the paper exam, the same skill is identifying the correct row and quantity in supplied output.

• ŷ698.93−2.28STR
• ObservedXabout14–25.8

step by step

1. +1 student per teacher → predicted TestScore 2.28 points lower
2. Intercept: predicted 698.93 points at STR = 0
3. STR = 0 lies far outside the observed range 14.0–25.8

why it works

A one-unit increase in the district student-to-teacher ratio, meaning one additional student per teacher, is associated with a predicted average test score two point two eight points lower. Include associated with, points, and students per teacher. Do not turn the slope into a percentage or guarantee an individual student's outcome.

The intercept is the predicted score when the ratio equals zero. Mathematically that prediction is six hundred ninety-eight point nine three. But zero lies far outside the observed range and is not a useful real-world setting for this district prediction.

An intercept can be necessary to position the fitted line even when its zero-X interpretation is not meaningful in practice. Likewise, being able to plug a number into an equation does not establish that the resulting prediction is supported by the observed data. Distinguish the equation's mathematical meaning from its empirical usefulness.

• ŷ698.93−2.28STR
• X18,22
• Change22→20

step by step

1. At STR = 18: Ŷ = 698.93 − 2.28(18) = 657.89
2. At STR = 22: Ŷ = 698.93 − 2.28(22) = 648.77
3. 648.77 − 657.89 = −9.12 points
4. 22 → 20: ΔX = −2; ΔŶ = (−2.28)(−2) = +4.56 points
5. A predicted association alone does not establish causation

why it works

At eighteen, multiply two point two eight by eighteen to get forty-one point zero four. Subtract from six hundred ninety-eight point nine three: the prediction is six hundred fifty-seven point eight nine. At twenty-two, the product is fifty point one six and the prediction is six hundred forty-eight point seven seven. The higher-ratio district has a prediction nine point one two points lower.

For the change from twenty-two to twenty, write new minus old. Twenty minus twenty-two equals negative two. Multiply the slope, negative two point two eight, by negative two. The predicted change is positive four point five six points. Two negatives make a positive increase.

Part e asks about a different issue, causation. The fitted relationship alone does not prove that hiring teachers causes that exact improvement. Other factors related to both staffing and scores may matter. Our calculation describes the model's predicted association; a causal interpretation needs additional justification.

• n420
• TSS152109.5936
• SSR144315.4836

step by step

1. ESS = TSS − SSR = 7,794.1100
2. R² = ESS / TSS ≈ 0.05124 = 5.124%
3. SER = √(144,315.4836 / 418) ≈ 18.58 score points

why it works

First recall the labels on page eleven. T S S measures squared distances from actual outcomes to the sample mean. E S S measures the part accounted for by the fitted values. S S R is the sum of squared residuals. In this course, S S R means residual, not regression sum of squares.

Explained sum of squares is total minus residual, giving seven thousand seven hundred ninety-four point one one. Divide explained by total, or compute one minus residual divided by total. R squared is approximately zero point zero five one two four, which is about five point one two four percent.

For S E R, divide the residual sum of squares by n minus two, or four hundred eighteen, then take the square root. The result is about eighteen point five eight score points. It measures a typical scale of scatter around the line, not an exact error for every observation.

Only about five percent of the sample score variation is accounted for by this line, so S T R alone does not account for most variation. That matches the broad scatter in the picture. A low R squared does not, by itself, establish statistical insignificance.

• R².05124
• SER18.58scorepoints

step by step

1. A: false — a negative slope does not make R² negative
2. B: false — R² measures variation, not exact predictions
3. C: true — about 5.124% of sample variation is accounted for
4. D: true — “explained” means statistical accounting here
5. E: false — low R² does not imply an insignificant slope
6. F: false — individual prediction errors are not all equal
7. G: false — SER has outcome units: test-score points
8. H: true — SER describes a typical scale of residual scatter

why it works

A is false: with an intercept, R squared is nonnegative even when the slope is negative. B is false because R squared is a fraction of variation, not a percentage of observations predicted exactly. C is correct. D is also correct when explained means statistical accounting, not proof of causation.

E is false. Fit and statistical significance are different questions. A relationship can account for little variation and still be estimated precisely enough to reject a zero slope.

F is false because S E R summarizes a scale of residual variation; individual misses differ. G is false because S E R uses the units of the outcome, test-score points. H is the intended correct plain-language interpretation of that typical scale. It does not mean S E R equals the average absolute residual.

The correct choices are C, D, and H. Instead of memorizing letters, connect each option to the quantity it describes: variation fraction, prediction error scale, or coefficient uncertainty. Mixing those jobs creates most of the distractors.

• Cost$1000
• Orders100s
• ŷ2.4+.85Orders
• R².36,SER1.8,n240

step by step

1. Y = Cost (thousands of dollars); X = Orders (hundreds)
2. Costᵢ = β₀ + β₁Ordersᵢ + uᵢ
3. 100 additional orders → predicted weekly cost +$850

why it works

Cost is Y, the outcome, and Orders is X, the predictor. The population model is Cost for center i equals beta zero plus beta one times Orders for center i, plus u for center i. Do not replace the unknown population coefficients with sample numbers when the question specifically asks for the population model.

One unit of Orders means one hundred actual orders. The slope is zero point eight five in thousand-dollar units, which is eight hundred fifty dollars. So one hundred additional weekly orders are associated with predicted weekly cost eight hundred fifty dollars higher.

This is not eighty-five cents per order and not an eighty-five-percent change. The units are essential inputs. Before doing any substitution, write a small conversion reminder: one X unit is one hundred orders, and one Y unit is one thousand dollars.

The assumptions paragraph above the question explains why a fitted association is not automatically causal. Conditional mean zero means that other influences left in the error do not systematically push outcomes up or down at particular X values, on average. Independently and identically distributed pairs means the observations are sampled independently from the same underlying distribution. It does not mean X and Y within a pair must be unrelated. The third assumption says very large outliers are unlikely, so a few extreme observations should not dominate. Your notes do not require fourth-moment calculations or proofs here. Simply running software does not verify these conditions.

• 1000orders→X10
• Actual$11500→Y11.5
• ŷ2.4+.85X
• R².36,SER1.8

step by step

1. 1,000 orders / 100 = 10 model units
2. Predicted Cost = 2.4 + 0.85(10) = 10.9 thousand = $10,900
3. Residual = 11.5 − 10.9 = +0.6 thousand = +$600
4. R² = 0.36: 36% of sample Cost variation accounted for
5. SER = 1.8 thousand dollars = $1,800

why it works

One thousand orders equals ten hundreds of orders, so X is ten, not one thousand. Substitute ten: two point four plus zero point eight five times ten equals ten point nine. That means ten thousand nine hundred dollars.

Actual cost is eleven point five thousand dollars. Subtract predicted cost ten point nine thousand to get a positive zero point six thousand dollars, or positive six hundred dollars. Actual cost is above the fitted value, so the model underpredicts this center.

For part d, R squared point three six means the fitted relationship with Orders accounts for thirty-six percent of the sample variation in weekly Cost. It does not mean thirty-six percent of centers are predicted exactly. S E R one point eight means the typical scale of residual variation is one point eight thousand dollars, or eighteen hundred dollars. It is not one point eight percent, and it does not say every center has exactly that error.

The calculations are simple once the units are handled. The main skill is noticing that the numbers fed into the model are scaled versions of the everyday quantities.

• Predictmonthlyrentusingfloorarea

step by step

1. Y = monthly rent
2. X = floor area

why it works

The target is monthly rent. That is the dependent variable, Y. The clue used to predict it is floor area, the independent variable, X. Choose B.

The word using is a useful signal. Predict Y using X. Reversing the variables would answer a different prediction question, even though the two measurements might be related. An intercept is a coefficient, not the predictor itself. Sample size tells us how many observations we have, not what outcome is being predicted.

Say the sentence before memorizing the letters: monthly rent is predicted from floor area.

• FiveequationchoicesA–E

step by step

1. Ŷᵢ = β̂₀ + β̂₁Xᵢ
2. The fitted prediction has no added uᵢ or ûᵢ

why it works

Y hat equals estimated beta zero plus estimated beta one times X. The hats matter. This is the sample-based line's prediction.

The population equation with unknown beta values and u describes actual outcomes in the population model, not the fitted value. Adding a population error or sample residual to the right side changes the meaning. Actual Y equals Y hat plus the residual is a valid identity, but it is not the requested equation for the prediction itself.

A practical check is to ask whether the equation gives a prediction using X and estimated coefficients alone. If it requires knowing the observation's unexplained error, it is not the fitted-value rule.

Therefore, choose D.

• sXY−9
• sX²3
• Xbar8
• Ybar42

step by step

1. β̂₁ = −9 / 3 = −3
2. β̂₀ = 42 − (−3)(8) = 66; Ŷ = 66 − 3X
3. Check: 66 − 3(8) = 42 = Ȳ

why it works

First calculate the slope: negative nine divided by three equals negative three. This immediately eliminates choices with positive three, negative nine, or other slopes.

Then compute the intercept using mean Y minus the estimated slope times mean X. Forty-two minus negative three times eight is forty-two minus negative twenty-four, which equals sixty-six. The line is Y hat equals sixty-six minus three X, choice A.

Check it at the sample mean X of eight. Sixty-six minus twenty-four equals forty-two, which matches mean Y. That is a useful verification because an O L S line with an intercept passes through the two sample means.

Do not stop after finding the slope, and do not lose the double negative while calculating the intercept.

• Earnings$1000
• Educationyears
• Slope3.4

step by step

1. +1 year of Education → fitted Earnings +3.4 thousand dollars
2. 3.4 thousand = $3,400
3. Interpret as a predicted annual earnings association

why it works

The slope says that increasing Education by one year is associated with an increase of three point four in fitted Earnings. Because Earnings is in thousands, three point four means three thousand four hundred dollars. Choose E.

The equation is in levels, not logarithms or percentage units, so the slope is not three point four percent. It also does not establish that every person gains exactly that amount by adding one year of education. We are describing a fitted average association, and causal interpretation would need additional support.

The option that reverses earnings and education describes a different regression. Keep the direction of prediction straight: education is the input, earnings is the outcome. A good answer names the increase in X, the change in predicted Y, and both units.

• ŷ5+.8X
• Xbar12
• OLSwithintercept

step by step

1. Ȳ = 5 + 0.8(12) = 14.60

why it works

Because the model includes an intercept, its fitted line passes through mean X and mean Y. Substitute mean X, twelve, into the equation. Five plus zero point eight times twelve is five plus nine point six, giving fourteen point six. The answer is C, fourteen point six zero.

We do not need an individual observation's residual. The property applies to the sample means, and the question tells us to treat the coefficients as exact, avoiding rounding ambiguity. The average fitted value equals the average actual Y in this setting.

Be careful not to return only nine point six, which omits the intercept, or five, which ignores the slope term. The words with an intercept are important because they support this means property.

• r−.60
• OneXwithintercept

step by step

1. R² = (−0.60)² = 0.36
2. 36% of sample variation is accounted for by the fitted line

why it works

Square the correlation. Negative zero point six times negative zero point six equals positive zero point three six. Choose D: the fitted line accounts for thirty-six percent of the sample variation in Y.

Squaring removes the sign, so R squared cannot tell us whether the relationship slopes upward or downward. The sign belongs to the correlation or slope. Also, zero point three six is a fraction of variation, not the fraction of data points predicted perfectly.

This shortcut relies on the stated setting, one regressor and an intercept. Do not generalize it to every regression situation without checking the conditions. Here, those conditions are explicitly given, so the calculation is immediate.

• n122
• SSR360
• Yminutes

step by step

1. n − 2 = 122 − 2 = 120
2. SSR / (n − 2) = 360 / 120 = 3
3. SER = √3 ≈ 1.73 minutes

why it works

We estimated two coefficients, an intercept and a slope, so the denominator is one hundred twenty-two minus two, which is one hundred twenty. Divide three hundred sixty by one hundred twenty and get three. Then take the square root: approximately one point seven three.

The answer is A, one point seven three minutes. The value three is the quantity inside the square root, not the S E R itself. Dividing by n rather than n minus two would use the wrong formula for this question. Reporting a percentage would also be wrong: S E R uses the dependent variable's units, and delivery time is measured in minutes.

It describes a typical scale of residual scatter. It does not say every delivery is exactly one point seven three minutes from its prediction.

• Estimationages25–50
• Newage80

step by step

1. Age 80 is outside the sample range 25–50
2. A numerical prediction can be calculated
3. Reliability outside the observed range is not guaranteed

why it works

You can substitute eighty into a linear equation and obtain a numerical prediction. But eighty lies outside the ages used to estimate the relationship, so this is extrapolation. The same relationship may not continue into that range.

Neither a straight-line formula nor an R squared above zero guarantees reliability at every age. The equation does not refuse the number eighty; the issue is evidence, not arithmetic. There is also no reason its prediction must equal the sample mean.

This final Chapter Four practice question reinforces a broader habit: ask whether your calculation answers a meaningful, supported question. A correct numerical operation can still produce an unjustified conclusion. Next we will examine a different kind of uncertainty: how coefficient estimates change when the sample changes.

The correct choice is B.