In many fast food restaurants, there is a strong correlation between a menu item

Question

In many fast food restaurants, there is a strong correlation between a menu item's fat content 12. (measured in grams) and its calorie content. We want to investigate this relationship. Using all of the food menu items at a well-known fast food restaurant, the fat content and calorie contents were measured. We decide to fit the least-squares regression line to the data, with fat content (x) as the explanatory variable and (y) as the response variable. A scatterplot of the data (with regression included) and a summary of the data are provided. One of the menu items is a hamburger with 1 grams of fat and 1410 calories. line 07 100120 0 20 40 60 Fal(grama) r = 0.979 (correlation between x and y) = 40.35 grams (mean of the values of x) 662.88 calories (mean of the values of y) 27.99 grams (standard deviation of the values of x) = 324.90 calories (standard deviation of the values of y) Refer to the example data point (107 grams, 1410 calories). What is the residual corresponding to this observation? O 10 grams O-10 grams -10 calories O 10 calories

Explanation / Answer

Let the estimated line of regression is

Y’ = mx + b

m = r sy / sx = 0.979 * 324.90/27.99 = 11.36

b = ybar - m* xbar = 662.88 - 11.36* 40.35 = 204.34

thus y’ = 11.36 x + 204.34

given x = 107

y’ = 11.36* 107 + 204.34 = 1419 .86 = 1420 calories

whereas y = 1410 calories

residual = y -y’ = 1410-1420 = -10

so answer is -10 calories

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