1 Fall 2016 Review Problems Question 1. In this question, we consider a large food processing center that needs to be able to switch from one type of package to another quickly to react to changes in order patterns. Consultants have developed a new method for changing the production line and used it to produce a sample of 48 change-over times (in minutes). Also available is an independent sample of 72 change-over times (in minutes) for the existing method. These two sets of times can be found on book web site in the file called changeover_times.txt. > t.test(Changeover~New) Welch Two Sample t-test data: Changeover by New t = -----, df = 93.714, p-value = 0.02964 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: 0.3203917 6.0268306 sample estimates: mean in group 0 mean in group 1 17.86111 14.68750 Method Changeover.mean Changeover.max Changeover.min Changeover.sd 1 Existing 17.86111 40 7 7.238990 2 New 14.68750 40 5 8.0110462 a) Based on your confidence interval, do you support the claim that there is a significant difference between the two means at a significance level of 0.05 (degrees of freedom is approximately 94)? (5 Points) b) If you were to conduct an equivalent hypothesis testing using SLR, what would be your: ( use the existing method "group 1" as your reference point). (2 points each) i. Hypotheses: ii. The slope: iii. The y-intercept: iv. The t-statistic: (assume unequal variances) v. The p-value of the test: c) Circle one answer: I recommend conducting simple linear regression (5 points) (i) Yes (ii) No Write the reason for your answer in simple short sentences. ________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________3 Question 2: :(20 Points, 5 each). Given the following R output for the inverse response plot for a linear regression model between X and Y variables: # library(alr3) par(mfrow=c(1,1)) # inverse.response.plot(m1,key=TRUE) inverseResponsePlot(m1,key=TRUE) ## lambda RSS ## 1 0.2621130 265.8749 ## 2 -1.0000000 46673.8798 ## 3 0.0000000 3583.8067 ## 4 1.0000000 7136.8828 a) Which of these suggested lambda values are you going to consider for your transformation? why so? b) Write down the new model after you have performed the suggested transformation in part (a). c) What do you expect happens to your R2 in your new model after using the transformation selected in part (a) compared to the one without any transformation. d) What would be the transformation if the best Lambda is zero? (Write down the mathematical expression for the regression model)4 Question 3:(20 Points, 5 each): Consider the following contingency table: The two categorical variables are: acupuncture type and pain reduction. a) State the null and the alternative hypotheses. What is the degree of freedom for this test? b) Create the expected cell count: c) Calculate the score based on your observed and expected tables. d) Perform the test based on the observed result in part (b) and report your P-value. State your conclusion based on the context.5 Question 4: Consider the following table: lm(formula = Bid.Price ~ Coupon.rate) Coefficients: (Intercept) Coupon.rate 74.787 3.066 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 74.7866 2.8267 26.458 < 2e-16 *** Coupon.rate 3.0661 0.3068 9.994 1.64e-11 *** Residual standard error: 4.175 on 33 degrees of freedom Multiple R-squared: 0.7516, Adjusted R-squared: 0.7441 F-statistic: 99.87 on 1 and 33 DF, p-value: 1.645e-11 mean(Coupon.rate)= 8.9214 sd(Coupon.rate)= 2.3340 mean(Bid.Price)= 102.1406 sd(Bid.Price)= 8.25442 a) Identify the independent and the dependent variables in the above SLR. (5 points) b) Consider the point (3.500, 94.53) in the data set then calculate it's: (2 points each) i) The leverage ii) The residual iii) and the standardized residual iv) The sample size (of the data) v) The linear correlation coefficient r between Coupon.rate and Bid.Price c) Do you consider the point (3.500, 94.53) to be: (5 points) i) A leverage point but not an outlier ii) Not a leverage point but an outlier iii) A leverage point and an outlier iv) Not a leverage point nor an outlier Why?_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________ i)_______ ii)____ iii)____6 Question 5: Consider the following R-code output: (if needed the t-distribution table is attached). (20 points) Call: lm(formula = Fare ~ Distance) Residuals: Min 1Q Median 3Q Max -18.265 -4.475 1.024 2.745 26.440 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 48.971770 4.405493 (a)_____ (b)_____ Distance 0.219687 0.004421 (c)_____ (d)_____ Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 10.41 on 15 degrees of freedom > anova(m1) Analysis of Variance Table Response: Fare Df Sum Sq Mean Sq F value Pr(>F) Distance 1 (e)_____ (f)_____ (g)_____ < 2.2e-16 Residuals (h)_____ (i)_____ (j)_____ Multiple R-squared:(k)_____ , Adjusted R-squared: (l)_____ Write down your answer here: (a) ________________________________________ (b) ________________________________________ (c) ________________________________________ (d) ________________________________________ (e) ________________________________________ (f) ________________________________________ (g) ________________________________________ (h) ________________________________________ (i) ________________________________________ (j)_________________________________________ (k)_________________________________________