Problem session #5

Q.1

You have been asked by your younger sister to help her with a science fair project. During the previous years she already studied why objects float and there also was the inevitable volcano project. Having learned regression techniques recently, you suggest that she investigate the weight-height relationship of 4th to 6th graders. Her presentation topic will be to explain how people at carnivals predict weight. You collect data for roughly 100 boys and girls between the ages of nine and twelve and estimate for her the following relationship:

= 45.59 + 4.32Height4, R2= 0.55, SER= 15.69

(3.81) (0.46)

where Weight is in pounds, and Height4 is inches above 4 feet. Figures in brackets are the respective standard error of the estimated coefficient

(a)Interpret the results.

(b) You remember from the medical literature that females in the adult population are, on average, shorter than males and weigh less. You also seem to have heard that females, controlling for height, are supposed to weigh less than males. To see if this relationship holds for children, you add a binary variable (DFY) that takes on the value one for girls and is zero otherwise. You estimate the following regression function:

= 36.27 + 17.33DFY+ 5.32Height4 – 1.83 (DFY×Height4),

(5.99) (7.36) (0.80) (0.90)

R2= 0.58, SER= 15.41

Are the signs on the new coefficients as expected? Are the new coefficients individually statistically significant? Write down and sketch the regression function for boys and girls separately.

(c) The medical literature provides you with the following information for median height and weight of nine- to twelve-year-olds:

Median Height and Weight for Children, Age 9-12

Boys' Weight / Boys' Height / Girls' Weight / Girls' Height
9-year-old / 60 / 52 / 60 / 49
10-year-old / 70 / 54 / 70 / 52
11-year-old / 77 / 56 / 80 / 57
12-year-old / 87 / 58.5 / 92 / 60

Insert two height/weight measures each for boys and girls and see how accurate your predictions are.

(d)The F-statistic for testing that the intercept and slope for boys and girls are identical is 2.92. Find the critical values at the 5% and 1% level, and make a decision.

(e)Allowing for a different intercept with an identical slope results in a t-statistic for DFY of (–0.35). Having identical intercepts but different slopes gives a t-statistic on (DFYHeight4) of (–0.35) also. Does this affect your previous conclusion?

(f)Assume that you also wanted to test if the relationship changes by age. Briefly outline how you would specify the regression including the gender binary variable and an age binary variable (Older) that takes on a value of one for eleven to twelve year olds and is zero otherwise.

(g)Indicate in the table below how the estimated relationship (in (f) above) would vary between younger girls, older girls, younger boys, and older boys.

Younger / Older
Boys
Girls

Q.2

Your School District manager is considering offering subsidized lunch in school because she believes that test scores are affected by the % of students eligible for subsidized lunch. You suspect that the effect of %eligible for subsidized lunch is not linear. In fact, you conjecture that increases in this variable from 10% to 20% have little effect on test scores but that changes from 50% to 60% have a much larger effect.

  1. Describe and specify a regression model that you can use to capture your belief. (Define and describe your variables completely)
  2. How would you test whether your conjecture is better than the linear specification?

Q.3
Suppose you are reading an article concerning the effects of immigration status on utilization levels of social services among legal and illegal immigrants who have been in the Canada for less than 10 years and who have been receiving social services. You encounter the following estimated model based on a sample of 676. . (parameter standard errors are given in parentheses below each point estimate)

SERVi = 30.90 - 1.20 TIMEi + 9.30 LEGALi + 0.33 TIMEi*LEGALi
(5.2) (0.31) (5.50) (0.20)

Where
SERVi = value of social services utilized (in hundreds of dollars per year);
TIMEi = time spent in the Canada (in years);
LEGALi = 1 if legal immigrant; = 0 if illegal

a.Based on the point estimates, what is the average utilization of social services for a legal immigrant in the first year after arrival in Canada? (be careful)______

For an illegal immigrant in the first year? ______

b.Test the hypothesis that there is no difference in the average utilization of social services between the two groups?

c.Based on the point estimates, how does utilization of social services vary with time in Canada for a legal immigrant? ______

For an undocumented immigrant? ______

d.Test the hypothesis that there’s no difference in the rates at which service utilization changes over time for the two groups..

e.When will predicted utilization be the same for both groups?

Q.4 (Help is in your textbook on q4 and q5) In the equation = 607.3 + 3.85 Income – 0.0423Income2, what income level results in the maximum test score

Q.5. You have estimated the following equation:

= 607.3 + 3.85 Income – 0.0423 Income2,

If income increased from 10 to 11 ($10,000 to $11,000), then the predicted effect on testscores would be