AP Statistics · Free-response skill

Read regression output in context

The move. From regression output, write the line, interpret the slope, a residual and r-squared in context, and explain one limit of using the line.

How to do it

  1. Write the equation with a hat and the variable names: predicted drinks = a + b(temperature), not just y = a + bx.
  2. Slope: for each 1-unit increase in x, the predicted y changes by b. Use both variable names, the units and the word predicted.
  3. Residual: compute the prediction first, then observed minus predicted. Say whether the actual value was above or below the prediction, and by how much, in context.
  4. r and r-squared: r is the square root of r-squared with the sign of the slope. r-squared is the share of the variation in y accounted for by the linear relationship with x.
  5. Limits: compare any x you plug in with the range of the data. Outside that range is extrapolation, and the pattern may not hold there.

Practice prompt Try it

The owner of a fictional farm stand records the daily high temperature, in degrees Fahrenheit, and the number of iced drinks sold on 30 summer days. The highs ranged from 62 to 96 degrees. A scatterplot shows a fairly strong, positive, linear pattern, and a residual plot shows random scatter around 0. The owner fits a least-squares line; part of the output is below.

Regression output: iced drinks sold (y) on daily high temperature (x)

TermValue
Constant-92.4
High temperature (degrees F)7.25
S31.6
R-sq78.0%

Use the information above to answer each part. A graphing calculator is allowed.

  1. Write the equation of the least-squares line, using words that name the variables.
  2. Say what the slope tells the owner about iced drink sales.
  3. On one day the high was 70 degrees and the stand sold 438 iced drinks. Work out the residual for that day and say what it means.
  4. Work out the correlation between high temperature and drinks sold, and say what the R-sq value tells the owner.
  5. The owner wants to use the line to plan for an early spring day with a high of 45 degrees. Give one reason the prediction for that day may not be reliable.

Write each part, then check yourself against the checklist. Nothing you type here is saved or graded.

Check yourself and see a model answer

Checklist

Our own checklist of what a full answer does, one line per point. Tick each one your response earns.

0 of 7 points

Model answer, annotated

(a) Predicted iced drinks sold = -92.4 + 7.25(daily high temperature in degrees Fahrenheit).

  • Point 1 Uses both coefficients from the output, names the variables and says the output is a prediction.

(b) For each additional degree Fahrenheit in the daily high temperature, the predicted number of iced drinks sold increases by 7.25.

  • Point 2 Slope stated as a predicted change per 1-unit increase in x, in context.

(c) The predicted sales at 70 degrees are −92.4+7.25(70)=415.1-92.4 + 7.25(70) = 415.1 drinks. The residual is 438−415.1=22.9438 - 415.1 = 22.9. On that day the stand sold about 22.9 more iced drinks than the line predicts for a day with a high of 70 degrees.

  • Point 3 Predicts first, then subtracts in the right order: observed minus predicted.
  • Point 4 Explains the sign: a positive residual means actual sales were above the prediction.

(d) r = 0.78≈0.883\sqrt{0.78} \approx 0.883, positive because the slope is positive. R-sq = 78.0% means about 78% of the variation in the number of iced drinks sold is accounted for by the linear relationship with the daily high temperature.

  • Point 5 Takes the square root of r-squared and chooses the sign from the slope.
  • Point 6 Interprets r-squared as explained variation in y, naming both variables.

(e) The data only include days with highs from 62 to 96 degrees, and 45 degrees is well below that range. Using the line at 45 degrees is extrapolation: the straight-line pattern may not hold on cold days, for example if far fewer customers visit, so the prediction may be far off.

  • Point 7 Names the observed range, shows that 45 lies outside it, and calls the prediction extrapolation.