1) Two statisticians took a random sample of 1,078 father and son pairs to examine. Galton’s law on the laws of inheritance, their sample regression line was: Son’s height = 33.73 + .516 X Fathers height.
a) Interpret the coefficients.
b) What does the regression line tell you about the heights of sons of tall fathers?
c) What does the regression line tell you about the heights of sons of short fathers?
2) A real estate agent wanted to know why prices of similar-sized apartments in the same building vary. A possible answer lies in the floor. It may be that the higher the floor, the greater the sale of the apartment. He recorded the price (in $1,000s) of 1,200 sq. ft. condominiums in several buildings in the same location that sold recently and the floor number of the condominium.
a) Determine the regression line.
b) What do the coefficients tell you about the relation between the two variables?
3) In an experiment to determine how the length of a commercial is related to people’s memory of it, 60 randomly selected people were asked to watch a 1-hour television program. In the middle of the show, a commercial advertising a brand of toothpaste appeared. Some viewers watched a commercial that lasted for 20 seconds; others watched one that lasted for 24 seconds, 28 seconds and 60 seconds. The essential content of the commercials was the same. After the show, each person was given a test to measure how much he or she remembered about the product. The commercials times and test scores (on a 30-point test) were recorded.
a) What is the standard error of estimate? Interpret its value.
b) Describe how well the memory test scores and length of television commercial are linearly related.
c) Are the memory test scores and length of commercial linearly related? Test using a 5% significance level.
d) Estimate the slope coefficient with 90% confidence.vac
5) A student wants to graduate with the absolute minimum level of work. He was registered in a statistics course that had only 3 weeks to go before the final exam and for which the final grade was determined in the following ways:
Total mark = 20% (assignment) + 30% (midterm test) + 50% (final grade). To determine how much work to do in the remaining 3 weeks, he needed to be able to predict the final exam mark on the basis of the assignment mark (worth 20 points) and the midterm mark (worth 30 points). His marks on these were 20/20 and 14/30, respectively. Accordingly, he undertook the following analysis. The final exam mark, assignment mark, and the midterm mark for 30 students who took the statistics course last year were collected.
a) Determine the regression equation.
b) What is the standard error of estimate? Briefly describe how you interpret this statistic.
c) What is the coefficient of determination? What does this statistic tell you?
d) Test the validity of the model.
e) Interpret each of the coefficients.
f) Can the student infer that the midterm mark is linearly related to the final grade in this model?
g) Can the student infer that the midterm mark is linearly related to the final grade in this model?
h) Predict the student’s final exam mark with 95% confidence.
i) Predict the student’s final grade with 95% confidence.
6) A statistician was told that severance pay for a laid off employee is determined by three factors: age, length of service with the company, and pay. To determine how generous the severance package had been in the past a random sample of 50 ex-employees was taken. For each, the following variables were recorded: 1) Number of weeks of severance pay. 2) Age of employee. 3) Number of years with the company, 4) Annual pay (in thousands of dollars).
a) Determine the regression equation.
b) Comment on how well the model fits the data.
c) Do all the independent variables belong in the equation? Explain.