This problem uses data from bulbs.com, which sells dif t types of light bulbs on their website only. Since it does not have physical stores, all purchases can be recorded and linked to the reviews that were shown at the time of ordering. We restrict our attention in this study to the six largest brands (Bulbrite, Eiko, Philips, Plusrite, TCP and Ushio)) and the f e largest categories of bulbs (CFL, Halogen, Incandescent, LED, and linear), which gives 919 unique stock keeping units (SKUs). These 919 SKUs were displayed 275,289 times during our study period resulting in 4546 purchases. For each exposure we know the following about the reviews to which the customer was exposed: (1) the log number of reviews for the SKU (logreview), (2) the average number of stars across the reviews for the SKU (avgstar), and (3) the average length of the reviews, measured in the number of characters. Our interest is in an interaction between length and the number of reviews, and we divide length at the mean so that there are long or short reviews; the dummy variable long equals 1 for long reviews and 0 for short ones. The variable buy equals 1 if a customer bought the SKU and 0 otherwise. The number of reviews is logged because it is right skewed with outliers. Call: glm(formula = buy ~ cat + brand + bs(avgstar) + logreview * long, family = binomial, data = bulb, weights = wt) Estimat Std. (Intercept) e 0.1351 Error 4.4219 5 catHal 0.5425 0.0849 6 5 catInc 0.7058 0.0827 9 3 catLED 0.0866 0.4011 8 catLin 0.0976 0.0827 7 8 brandEiko 0.0659 3 brandPhilips 0.1791 0.0483 0.0384 9 brandPlusri 9 0.4990 0.0929 te 2 6 z value 32.719 6.38 7 8.53 -3 4.627 1.18 -0 2.718 1.25 7 5.36 8 1 Pr(>| < z|)2e16 1.69e10 2e< 16 3.70e06 0.23806 7 0.00656 8 0.20870 4 7.96e08 *** *** *** *** ** *** 0.0851 0.0815 9 0 0.2510 9 0.1427 2 0.1053 7 0.0310 4 0.0437 4 0.0394 3 Null deviance: 47117 degrees Residual deviance: 46356 degrees 3.93 1.259 -8 4.015 3.44 -7 0.586 4.37 -8 1.343 2.091 0.20796 8.23e3 05 5.95e05 0.00056 6 0.55817 8 1.20e05 0.17923 2 0.03650 7 *** *** *** *** * on 1538 of freedom on 1523 of freedom 0.2 (a) (3 points) Test the overall signifcance of the model (H0 : 1 = = 15 = 0)? Give details. (d) (2 points) Which brand is most likely to be purchased? (e) (3 points) The marginal plot for the average number of stars is shown to the right. Briefly describe what the plot tells you. s(Avg Stars) (c) (2 points) What is the dif in the log odds of buying Plusrite over TCP? 0.1 (b) (2 points) What does the coefficient for halogen tell you (catHal = 0.54256)? 0.0 -0.10727 0.32093 -1.00807 0.49201 -0.06170 0.13590 -0.05874 -0.08246 0.30.20.1 brandTCP brandUshio bs(avgstar)1 bs(avgstar)2 bs(avgstar)3 logreview long logreview:lo ng 1 2 3 4 Average Number of Stars (f) (4 points) Is the interaction between the log number of reviews and the dummy long signifcant at the .05 level? State the null and alternative and describe how you know whether or not to reject the null. 5 (g) (6 points) The logreview variable ranges from 0 to about 3. Construct an in- teraction plot showing logreview on the horizontal axis with dif t lines for short and long reviews. Give the equations of both lines. Hint: the specifc values of the intercepts are determined by the other variables and are unimportant; the difference in intercepts is what really matters. In you equation, label the intercept for short reviews b0. (h) (4 points) Write 1-2 concise sentences summarizing how the length of the review and the log number of reviews afects the log odds of purchasing. (i) (4 points) The following line is from the output of drop1(ft, test="Chisq"). What specif does the LRT statistic (396.54) and P -value tell you? cat Df Deviance AIC LRT Pr(>Chi) 4 46752 46776 396.54 < 2.2e-16 ***