The table shows data from 27 automotive plants on y = number of assembly defects per 100 cars and x = time (in hours) to assemble each vehicle. The data are in the Quality and Productivity file on the text CD. Number of defects in assembling 100 cars and time to assemble each vehicle

Source: Data from S. Chatterjee, M. Handcock, and J. Simonoff, A Casebook for a First Course in Statistics and Data Analysis (Wiley, 1995); based on graph in The Machine that Changed the World, by J. Womack, D. Jones, and D. Roos (Macmillan, 1990).
a. The prediction equation is = 61.3 + 0.35x. Find the predicted number of defects for a car having assembly time (i) 12 hours (the minimum) and (ii) 54 hours (the maximum).
b. The first 11 plants were Japanese facilities and the rest were not. Let x1 = time to assemble vehicle and x2 = whether facility is Japanese (1 = yes, 0 = no). The fit of the multiple regression model is = 105.0 - 0.78x1 - 36.0x2. Interpret the coefficients that estimate the effect of x1 and the effect of x2.
c. Explain why part a and part b indicate that Simpson€™s paradox has occurred.
d. Explain how Simpson€™s paradox occurred. To do this, construct a scatterplot between y and x1 in which points are identified by whether the facility is Japanese. Note that the Japanese facilities tended to have low values for both x1 and x2.

Plant Defects Time Plant Defects Time Plant Defects Time 39 27 10 89 17 19 38 23 11 42 15 12 38 26 21 50 17 13 68 21 22 55 12 14 67 26 23 56 16 15 69 30 24 57 18 16 69 32 25 100 25 56 26 17 70 31 26140 21 61 20 18 68 37 27170 28 69 54 79 29 31 84 28 87 44 98 23 48 20 20 18