Chicken sandwiches are often advertised as a healthier alternative to beef because many are lower in fat. Tests on 11 brands of fast-food chicken sandwiches produced the following summary statistics and scatterplot from a graphing calculator:

Fat: mean 20.6, standard deviation 9.8

Calories: mean 472.7, standard deviation 144.2

Correlation: 0.947

Write the equation of the regression line. Explain the meaning of the slope and the y-intercept. What does it mean if a certain sandwich has a negative residual? If a chicken sandwich and a burger each advertised 35 grams of fat, which would you expect to have more calories?

 

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We are given the following information about chicken sandwiches.

Let x represent the grams of fat. Then the mean of 11 sandwiches is `bar(x)=20.6` with a standard deviation of `s_x=9.8`

Let y (the dependent variable) represent the total calories. Then the mean is `bar(y)=472.7` with a standard deviation of `s_y=144.2`

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We are given the following information about chicken sandwiches.

Let x represent the grams of fat. Then the mean of 11 sandwiches is `bar(x)=20.6` with a standard deviation of `s_x=9.8`

Let y (the dependent variable) represent the total calories. Then the mean is `bar(y)=472.7` with a standard deviation of `s_y=144.2`

The correlation coefficient `r=0.947` indicates a strong, positive linear relationship between these variables. (In other words, as the grams of fat increase, we expect to see the calories increase also.)

(1) To calculate the regression line, we first assume that there is a linear relationship between the variables. The scatterplot should show points roughly along a "line," and the correlation coefficient r should be close to 1 in absolute value. (If the sample size is large, smaller correlation coefficients might still represent a statistically significant relationship.)

For any line we need a slope. The slope here is given by `m=r(s_y/s_x)` .

Thus `m=0.947(144.2/9.8)=13.934`

The slope indicates the rate of change in the dependent variable versus the independent variable. Here we would expect the number of calories to go up by 14 for each additional gram of fat.

(2) In the equation y=mx+b, m is the slope, and b is the y-intercept. One way to write the equation of a line is to find these elements. We have the slope, so we calculate the y-intercept.

`b=bar(y)-mbar(x)`

Thus `b=472.7-13.934(20.6)=185.66`

If a sandwich had no fat (x=0) we would expect about 186 calories (presumably from the proteins and carbohydrates.)

(3) The equation of the regression line is `y'=13.934x+185.66`

(Different texts use different notations. Here, y' is the y-coordinate on the regression line for a given input x. This point may or may not be a data point. y' is the predicted value for a given x.)

The difference between y and y' (the actual and predicted values) is called the residual. Positive residuals indicate the actual value is above the predicted value and negative residuals indicate the actual value was below the predicted value.

** In order to compare chicken to beef, you would need to know about the burgers. **

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