Question

In the process of collecting weight and height data from 29 female and 81 male students at your university, you also asked the students for the number of siblings they have. Although it was not quite clear to you initially what you would use that variable for, you construct a new theory that suggests that children who have more siblings come from poorer families and will have to share the food on the table. Although a friend tells you that this theory does not pass the “straight-face” test, you decide to hypothesize that peers with many siblings will weigh less, on average, for a given height. In addition, you believe that the muscle/fat tissue composition of male bodies suggests that females will weigh less, on average, for a given height. To test these theories, you perform the following regression:

Studentw = –229.92 – 6.52 Female + 0.51 Sibs+ 5.58 Height,

R2=0.50, SER = 21.08

where Studentw is in pounds, Height is in inches, Female takes a value of 1 for females and is 0 otherwise, Sibs is the number of siblings.

a. Interpret the regression results.

b. Build the 95% confidence interval for female coefficient if the standard deviation is 1.25, build the 95% confidence interval for female coefficient if the standard deviation is 2.5. How different are they? How does this difference impact your analysis?

Answer 1

A).

Consider the given problem here the regression equation is given by.

=> Student W = (-229.92) + (-6.52)*Female + 0.51*Sibs + 5.58*Height.

Here “Female” is the dummy variable takes “1” for “female student” and “0 for male student”.

So, here the coefficient of “Female” is “-6.52”, => everything else same “female student” having lower weight of “6.52 pounds” compare to the “Male students”.

Now, the coefficient of “Sibs” is given by “0.51”, => if the number of siblings increases by 1, => the weight of the student will increase by “0.51 pound” regardless of “male or female student”. The coefficient of “Height” is given by “5.58”, => if height increases by 1 inch, => the weight of the student will increase by “5.58 pounds” regardless of “male or female student”.    

Now, the “R^2” value is given by “0.5”, => “50%” variation is “Student Weight” is being explained by the given regression equation and the rest is remain unexplained.

B).

Now, given the regression equation the coefficient of “Female” is “-6.52” and the standard deviation is “1.25”, => the 95% confidence interval is given by.

=> (-1.96) <= (-6.52-b1)/1.25 < = 1.96, => (-1.96)*1.25 <= (-6.52-b1) < = 1.96*1.25,

=> (-2.45)+6.52 <= (-b1) < = 2.45+6.52, => 4.07 <= (-b1) < = 8.97,

=> (-8.97) < = b1 < = (-4.07).      

Now, if the standard deviation increases to “2.5”, => the 95% confidence interval is given by.

=> (-1.96) <= (-6.52-b1)/2.5 < = 1.96, => (-1.96)*2.5 <= (-6.52-b1) < = 1.96*2.5,

=> (-4.9)+6.52 <= (-b1) < = 4.9 +6.52, => 1.62 <= (-b1) < = 11.42,

=> (-11.42) < = b1 < = (-1.62).

So, as the standard deviation increases, => the wide of the internal also increases. So, here as the standard deviation increases implied the coefficient of “Female” may become insignificant, => the probability of rejecting “H0” increases.