AP Statistics Chapter 3 Review

AP Statistics Chapter 3 Review ŷ = a + bx sy b=r a = y − bx sx r= ∑Z Z x y n −1 Dependent variable is: % with no HS Diploma No Selector R squared = 6.8% R squared (adjusted) = 4.9% s = 5.455 with 51 - 2 = 49 degrees of freedom % w i t h 30.0 n o H S D i p l o m a 22.5 Source Sum of Squares df Regression Residual 105.702 1457.87 1 49 Mean Square 105.702 29.7525 Variable Coefficient s.e. of Coeff t-ratio Constant Ave. Teac… 32.3874 -0.229313 4.632 0.1217 6.99 -1.88 F-ratio 3.55 prob ≤ 0.0001 0.0654 15.0 30.0 37.5 45.0 Ave. Teacher's Pay (1000s) Data was collected state-by-state comparing average teacher’s salary’s and the "percent of the population lacking a high school diploma". 1. Give the equation for the line of best fit. ŷ = 32.3874 − 0.229313x where x = Ave. teacher’s pay(in 1000s); y = % with no HS diploma 2. Interpret r2. 6.8% of the variation in the percent of people without hs diplomas is accounted for (explained by) average teacher’s pay. r e s i d u a l s 3. What is the correlation coefficient and what does it mean (in context). r = -0.26 There is a weak negative linear relationship between the ave. teacher’s pay(in 1000s and the % with no HS diploma 4. The following is the residual plot for the data above. Give a reasonable interpretation. The residual plot is patternless, confirming the linear model. 5. If a state has an annual teachers salary of $38,000, what is the predicted % of the population lacking a high school diploma? ŷ = 32.3874 − 0.229313(38) ≈ 23.674 About 24% of the population can be expected to have no HS diploma if average teacher’s pay is $38,000. 10 5 0 -5 30.0 37.5 45.0 Ave. Teacher's Pay (1000s) 6. What is the meaning of the slope in the context of this problem? As average teacher’s pay increases by $1000, the percent of the population without a HS diploma decreases by about 0.2%. 7. What is the meaning of the y-intercept in the context of this problem? This is the projected percent of the population without HS diplomas if average teacher were zero. 8. If a state has an annual teachers salary of $98,000, what is the predicted % of the population lacking a high school diploma? Is this a reasonable value? Discuss. If the average teacher’s pay increased to $98,000, the percent of the population without a HS diploma decreases would be around 10%. This calculation involves extrapolation and so caution should be used, the teacher salary is outside the range of values used in the linear regression. 9. Fill in the missing cells: x y sx sy 10 -9 2 2.5 500 1.2 6 r -.7666 100 -0.6 equation yˆ ŷ = 13.998 − 2.2998x x 6 y 1 0.199 residual 0.801 ŷ = 625 − 50x 3.2 510 465 45 10. Weights and ages of children in the Egyptian village of Nahya were recorded. Age (months) 1 2 3 4 5 6 1 7 8 9 10 11 12 Weight (kg) 4.3 5.1 5.7 6.3 6.8 7.1 7.2 7.2 7.2 Propose a reasonable model for this data using sound statistical methods. Reminder: show your work. 7.2 7.5 7.8 This graph is shows a strong positive nonlinear relationship between age and weight. The regression equation does not fit the curve very well in spite of the .91 correlation. This is shown clearly in the residual plot with a striking curve, meaning that a better model is needed. The regression equation is ŷ = 4.880303 + 0.2671329x r = .91 In terms of answering the question on a test, you can stop here. As an analysis of the data, you can go further. Examining this data, it can be seen that the graph shows a strong positive linear relationship for children between the ages of 1 and 6 months. After that the weight stabilizes with less additional growth, and a nonlinear increase. Using the data from the first six months: ŷ = 3.9133333 + 0.5628571x The graph looks good and the correlation r = .99 is very strong. The residual plot, however, shows a striking pattern, with the interpretation that the linear model is not the best fit. Examining the residuals, however, shows that the values are very small, so the linear model is not the best, but truly is not so bad. 2