You can visit the official Web site of any large restaurant chain to examine the nutritional data for menu items. For fast-food restaurants, many menu items are high in fat, so most of their calorie content comes from fat (rather than carbohydrate or protein). Here we investigate the relationship between amount of fat in a menu item ("Fat," measured in grams) and the number of calories ("Calories"). To predict the number of calories in a menu item given its fat content, we use the simple linear regression model Calories = + (Fat) where the deviations are assumed to be independent and Normally distributed, with mean 0 and standard deviation .

At one major fast-food restaurant chain, there were 26 items listed under the heading of "Sandwiches" (which includes hamburgers, chicken sandwiches, and other sandwich selections) on the menu. We fit the model described above to the data using the method of least squares. We treat these 26 menu items (which came from one restaurant) as a sample from the population of all sandwich items at all fast-food restaurants. This assumption is probably dubious. The following results were obtained from software.

r₂= 0.846

s= 43.5747

Parameter

Parameter estimate

Std. err. of parameter est.

151.092

30.082

12.546

1.213

The quantitys= 43.5747 is an estimate of the standard deviation of the deviations in the simple linear regression model. The degrees of freedom forsare:

26.

12.

24.

3.