1. What is Pearson’s coefficient of correlation?

2. What proportion of the variation in SAT scores is explained by variation in class sizes?

3. What is the regression equation?

4. If class size is reduced by an average of one student, what will be the impact on SAT scores?

5. Is there a significant relationship between class size and SAT scores?

6. What test did you perform to answer 5 and what is the p-value of the test?

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.113548752R Square 0.012893319Adjusted R Square -0.007251715Standard Error 66.61227977Observations 51

ANOVAdf SS MS F Significance F

Regression 1 2839.914839 2839.9 0.640025 0.427563485Residual 49 217422.595 4437.2Total 50 220262.5098

Coefficients Standard Error t Stat P-value Lower 95%Intercept 1142.856716 95.15479298 12.011 3.27E-16 951.6360034ClassSize -3.311074206 4.138762953 -0.8 0.427563 -11.62822959

The dependent (y) variable is average SAT score; the independent (x) variable is average class size. Data are for the 50 states and DC.

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.32173199R Square 0.103511473Adjusted R Square 0.085215789Standard Error 63.48112574Observations 51

ANOVAdf SS MS F Significance F

Regression 1 22799.69688 22799.7 5.657699 0.021320976Residual 49 197462.8129 4029.853Total 50 220262.5098

Coefficients Standard Error t Stat P-value Lower 95%Intercept 1176.114279 46.68632443 25.19184 1.06E-29 1082.2946Bachelor -4.13585379 1.738782501 -2.37859 0.021321 -7.630067983

The independent (x) variable is the percentage of each state’s population who have at least a bachelor’s degree.

1. What proportion of the variation in SAT scores is explained by variation in bachelors’ degrees?

2. What is the standard error of the estimate sy.x?

3. Predict the average SAT score in a state in which 20% of the population hold bachelors’ degrees.

4. What is the t statistic for the test H0: 1 = 0? What is the p-value of the test?

5. At 5% significance should you reject H0? Is there a statistically significant relationship between SAT and Bachelor?

The independent (x) variable is Law Enforcement Officers per 1000 population (LO); the dependent variable is violent crimes per 1000 population (VC). Data are for 97 NC counties.

1. What is the correlation coefficient between LO and VC?

2. What proportion of the variation in crime is explained by variation in the number of law enforcement officers?

3. What is the regression equation?

4. Predict the crime rate for a county that has 5 law enforcement officers per 1000 population.

5. Give a 95% confidence interval for the crime rate in all counties with

LO = 5. Assume x = 2.7 and that t95 = 2.

6. Give a 95% prediction interval for the crime rate in a county in which LO = 3 using the same assumed values as in q. 5.

7. Give a 95% confidence interval for the value of 1.

8. Is there a significant relationship between LO and VC?

9. What hypothesis test did you perform to answer question 7? What was the p-value of the test?

10. Would this regression analysis support the theory that law enforcement officers tend to cause crime?

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.384827291R Square 0.148092044Adjusted R Square 0.139124592Standard Error 2.02466356Observations 97

ANOVAdf SS MS F Significance F

Regression 1 67.69683898 67.69684 16.51439 9.94331E-05Residual 95 389.4299404 4.099263Total 96 457.1267794

Coefficients Standard Error t Stat P-value Lower 95%Intercept 0.703063653 0.700137666 1.004179 0.317843 -0.686885317Per Cap LO 1.002546286 0.246702246 4.063791 9.94E-05 0.512780416