Regression Analysis unit test
A test on this unit alone, marked as a percentage and a letter grade — for the test your class is actually sitting, rather than for May. Answer everything, then submit once: seeing the answer to question 3 before attempting question 4 makes the final percentage meaningless.
Extrapolation
Why r² is not r squared conceptually
Why slope inference uses df = n − 2
Reading regression output
Coefficient of determination r²
Checking conditions with a residual plot
Effect of removing an influential point
Interpreting the y-intercept
Sign of a residual
Population regression model
Hypotheses for slope inference
Interpreting a significant slope
Short answer 1. Define or explain: Testing whether a point is influential
3 ptsShort answer 2. Define or explain: Influential point vs outlier
3 ptsShort answer 3. Define or explain: r is unitless and bounded
3 ptsShort answer 4. Define or explain: t-interval for the slope
3 ptsFree response
4 ptsINVESTIGATIVE TASK. To win a game, a baseball team needs to score runs, and to score runs, players need to get on base by hitting the ball. FIGURE 1 is a scatterplot of the number of runs against the number of hits for 30 randomly selected professional teams: the points show a moderately strong, positive, roughly linear association, with hits ranging from about 1,150 to 1,550 and runs from about 580 to 950. The least-squares regression line for predicting runs from hits is: predicted runs = −372.2 + 0.823 × (number of hits) FIGURE 2 repeats the scatterplot with each team classified by total salary: dots are teams above the median total salary of $160 million, squares are teams below it. The circled square labeled A sits at about (1,430 hits, 855 runs) — among the highest values on both axes for below-median teams, and well above the other below-median teams. Both salary classes show positive linear patterns of visually similar spread about their trends. Interval formulas. All three intervals below use Point Estimate ± t*·(standard error), where t* has n − 2 degrees of freedom: • a confidence interval for the slope; • a confidence interval for the MEAN number of runs for all teams with a given number of hits (standard error 17.48 at 1,250 hits); • a prediction interval for the number of runs of a SINGLE team with a given number of hits (standard error 56.78 at 1,250 hits). The standard error for the mean-runs confidence interval is s·√(1/n + (x − x̄)²/Σ(xi − x̄)²); the prediction-interval standard error is s·√(1 + 1/n + (x − x̄)²/Σ(xi − x̄)²).
A. (i) Describe the relationship between the number of hits and the number of runs for these baseball teams, in context. (ii) Using the given regression equation, calculate the point estimate for the predicted number of runs for a team achieving 1,250 hits. Show your work.
B. Consider Figure 2. (i) Compare the team represented by point A with the other teams that have the same total salary classification. (ii) For teams with total salary greater than the median, is the strength of the linear relationship between hits and runs stronger than, weaker than, or similar to that for teams below the median? Explain.
C. (i) What is the critical value t* for a 95% confidence level used for both the mean-runs confidence interval and the prediction interval? Give two decimal places. (ii) The standard error for the mean-runs confidence interval is 17.48; calculate the 95% confidence interval for the mean number of runs for all teams with 1,250 hits. Show your work. (iii) The standard error for the prediction interval is 56.78; calculate the 95% prediction interval for a single team with 1,250 hits. Show your work.
D. (i) Would a distribution of sample means be expected to have more or less variability than a distribution of individual observations? Explain. (ii) Using your answer and the two standard-error formulas, explain why the prediction interval in part C(iii) is wider than the confidence interval in part C(ii).