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Researchers wanted to study physical activity in children, so they gave 74 randomly chosen children a "FitBit" to recorded how many steps they took each day. Additionally, the scientists recorded each child’s Body Mass Index (BMI) and grade in school. Grade was coded as middle or high school. The results of their analysis are below:
Factors Contributing to Daily Steps Taken
Estimate SE
Intercept 6,350.02 316.10
BMI -217.90 54.99
Grade_middle 2,702.12 788.02
BMI*Grade_middle 601.44 429.65

a. Interpret the estimate of BMI in context.
b. Calculate the slope of BMI on steps taken for students in middle school and for students in high school. What do the model results tell you about these two slopes?
c. Suppose a middle school student with a BMI of 20 took 7,000 in a day. Calculate this student’s residual.

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Answer:

Step-by-step explanation:

The estimated regression equation to predict the daily steps taken is:

Daily step= 6350.02 - 217.90BMI + 2702.12Grade_middle + 601.44BMI * Grade_middle

where Grade_middle=1 if the student is in middle school, and Grade_middle=0 if the student is in high school

b) The slope of BMI is -217.90. The negative value of the slope indicates that BMI and Daily steps taken move in opposite direction.  

We can say that for 1 unit increase in the BMI, the predicted value of Daily Steps taken by the student decreases by 217.90, while keeping other variables unchanged.

c) The regression equation for middle school students can be obtained by setting Grade_middle=1

Daily step= 6350.02 - 217.90BMI + 2702.12 * 1 + 601.44BMI * 1

               = 6350.02 - 217.90BMI + 2702.12 + 601.44BMI

              = 9052.14 + 383.54BMI

Therefore, the slope of BMI on steps taken for students in middle school is 383.54

The regression equation for high school students can be obtained by setting Grade_middle=0

Daily Steps = 6350.02 - 217.90BMI + 2702.12 x 0 +601.44BMI*0 = 6350.02 - 217.90BMI

Therefore, the slope of BMI on steps taken for students in high school is -217.90

The model says that variables, the daily steps taken and the BMI of a middle school student are positively correlated (move in the same direction) and the daily steps taken and the BMI are negatively correlated (move in opposite direction) for a high school student. The above slopes say that for 1 unit increase in the BMI for students in middle school increases the predicted daily steps taken by 383.54 and for 1 unit increase in the BMI for students in high school decreases the predicted daily steps taken by 217.90

d) The predicted value of daily steps taken for a BMI=20 and Grade_middle=1 is

Daily Steps = 6350.02 – 217.90 x 20 + 2702.12 x1 +601.44 x 20 x 1 = 16722.94

The actual value of the daily steps taken by a middle school student with a BMI of 20 is 7000

The residual is calculated as

[tex] \text{residual}=\text{Actual}-\text{predicted}=7000-16722.94=-9722.94 [/tex]

Therefore, this student’s residual is -9722.94

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