16.1 The term regression was originally used in 1885 by Sir Francis Galton in his analysis of the relationship between the heights of children and parents. He formulated the “law of universal regression,” which specifies that “each peculiarity in a man is shared by his kinsmen, but on average in a less degree.” (Evidently, people spoke this way in 1885.) In 1903, two statisticians, K. Pearson and A. Lee, took a random sample of 1,078 father-son pairs to examine Galton’s law (“On the Laws of Inheritance in Man, L Inheritance of Physical Characteristics,” Biometrika 2:457-462). Their sample regression line was:
Son’s height = 33.73 + .516 x Father’s 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?
16.7:- Florida condominiums are popular winter retreats for many North Americans. In recent years, the prices have steadily increased. 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 price of the apartment. He recorded the price (in $1,000’s) of 1,200 sq. ft. condominiums in several buildings in the same location that have sold recently and the floor number of the condominium.
Floors: x = 13.68, s2x = 59.32
Price: y = 210.42, s2y = 496.41;
n = 50, sxy = 86.93
a. Determine the regression line
b. What do the coefficients tell you about the relationship between the variables?
16.28
In television’s early years most commercials were 60 seconds long. Now, however, commercials can be any length. The objective of commercials remains the same – to have as many viewers as possible remember the product in a favorable way and eventually buy it. 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, …, 60 seconds. The essential content of the commercials were the same. After the show, each person was given a test to measure how much he or she remembered about the product. The commercial times and test scores (on a 30-point test) were recorded.
Lengths: x = 38.00, s2x = 193.90
Test: y = 13.80, s2y = 47.96
n = 60, sxy = 51.86
a. What is the standard error of estimate? Interpret it’s 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
16.100: -The president of a company that manufactures car seats has been concerned about the number and cost of machine breakdowns. The problem is that the machines are old and becoming quite unreliable. However, the cost of replacing then is quite high, and the president is not certain that the cost can be made up in today’s slow economy. To help make a decision about replacement, he gathered data about last month’s costs for repairs and the ages (in months) of the plant’s 20 welding machines.
Age: x = 113.35, s2x = 378.77
Repairs: y = 395.21, s2y = 4,094.79
n = 20, sxy = 936.82
a. Find the sample regression line
b. Interpret the coefficients
c. Determine the coefficient of determination, and discuss what this statistic tells you
d. Conduct a test to determine whether the age of a machine and its monthly cost of repair are linearly related
e. Is the fit of the simple linear model good enough to allow the president to predict the monthly repair cost of a welding machine that is 120 months old? If so, find a 95% prediction interval. If not, explain why not.
Exercise 17.2:
Pat Studded, a student ranking near the bottom of the statistics class, decided that a certain amount of studying could actually improve final grades. However, too much studying would not be warranted, since Pat’s ambition (if that’s what one could call it) was to ultimate graduate with the absolute minimum level of work. Pat was registered in a statistics course, which had only 3 weeks to go before the final exam, and where the final grade was determined in the following way:
Total mark = 20% (Assignment)
+ 30% (Midterm test)
+ 50% (Final exam)
To determine how much work to do in the remaining 3 weeks, Pat needed to be able to predict the final exam mark on the basis of the assignment work (worth 20 points) and the midterm mark (worth 30 points). Pat’s marks on these were 12/20 and 14/30, respectively. Accordingly, Pat undertook the following analysis. The final exam mark, assignment mark, and midterm test mark for 30 students who took the statistics course last year were collected.
Exercise 17.5:
When one company buys another company; it is not unusual that some workers are terminated. The severance benefits offered to the laid-off workers are often the subject of dispute. Suppose that the Laurier Company recently bought the Western Company and subsequently terminated 20 of Western Company and subsequently terminated 20 of Western employees. As part of the buyout agreement, it was promised that the severance packages offered to the former Western employees would be equivalent to those offered in Laurier employees who had been terminated in the past year. Thirty-six-year-old Bill Smith, a Western employee for the past 10 years, earning $32,000 per year, was one of those let go. His severance package included an offer of 5 weeks’ severance pay. Bill complained that this offer was less than that offered to Laurier’s employees when they were laid off, in contravention of the buyout agreement. A statistician was called in to settle the dispute. The statistician was told that severance is determined by three factors: age, length of service with the company, and pay. To determine how generous the severance package had been, a random sample of 50 Laurier ex-employees was taken. For each, the following variables were recorded: