Question

the figure displays the relationship between golfers scores on the first and second rounds of the 2019 masters tournament.
the least-squares line for predicting second-round scores (y) from first-round scores (x) has equation \\(\hat{y} = 62.91 + 0.164x\\). the mean second-round score for all players was 75.02. so, a player who does well in the first round is predicted to do less well, but still better than average, in the second round. in addition, a player who does poorly in the first is predicted to do better, but still worse than average, in the second.
find the predicted second-round score for a player who shot 80 in the first round. give your answer to two decimal places.
predicted score:
find the predicted second-round score for a player who shot 70 in the first round. give your answer to two decimal places.
predicted score:
(comment: this is regression to the mean. if you select individuals with extreme scores on some measure, they tend to have less extreme scores when measured again. thats because their extreme position is partly merit and partly luck, and the luck will be different next time. regression to the mean contributes to lots of \effects\. the rookie of the year often doesnt do as well the next year; the best player in an orchestral audition may play less well once hired than the runners-up; a student who feels she needs coaching after the sat often does better on the next try without coaching.)

Explanation:

Step1: Identify the regression equation

The least - squares line for predicting second - round scores \(y\) from first - round scores \(x\) is given by \(\hat{y}=62.91 + 0.164x\).

Step2: Predict for \(x = 80\)

Substitute \(x = 80\) into the regression equation.
\(\hat{y}=62.91+0.164\times80\)
First, calculate \(0.164\times80 = 13.12\)
Then, add this to \(62.91\): \(\hat{y}=62.91 + 13.12=76.03\)

Step3: Predict for \(x = 70\)

Substitute \(x = 70\) into the regression equation.
\(\hat{y}=62.91+0.164\times70\)
First, calculate \(0.164\times70=11.48\)
Then, add this to \(62.91\): \(\hat{y}=62.91 + 11.48 = 74.39\)

Answer:

For a player who shot 80 in the first round, the predicted score is \(76.03\).
For a player who shot 70 in the first round, the predicted score is \(74.39\).