Discuss the below:
A. Employers sometimes seem to prefer executives who appear physically fit, despite the legal troubles that may result. Employers may also favor certain personality characteristics. Fitness and personality are related. In one study, middle-aged college faculty who had volunteered for a fitness program was divided into low-fitness groups based on a physical examination. The subjects then took the Cattell Sixteen Personality Factor Questionnaire. Here are the data for the "ego strength" personality factor:
Low Fitness 4.99 4.24 4.74 4.93 4.16 5.53 4.12 5.10 4.47 5.30 3.12 3.77 5.09 5.40
High Fitness 6.68 6.42 7.32 6.38 6.16 5.93 7.08 6.37 6.53 6.68 5.71 6.20 6.04 6.51
(a) Is the difference in mean ego strength significant at the 5% level? At the 1% level? Be sure to state H0 and Ha.
(b) You should hesitate to generalize these results to the population of all middle-aged men. Explain why.
(c) You should also hesitate to conclude that increasing fitness causes an increase in ego strength. Explain why?
B. Bear markets. Investors speak of a bear market when stock pieces drop substantially Table below give the data on all declines of at least 10% in the Standard & Poor's 500-stock index fell from its peak and how long the decline in stock prices lasted.
Year Decline (%) Duration
(months)
1940-1942 42 28
1946 27 5
1950 14 1
1953 15 8
1955 10 1
1956-1957 22 15
1959-1960 14 15
1962 26 6
1966 22 8
1968-1970 36 18
1973-1974 48 21
1981-1982 26 19
1983-1984 14 10
1987 34 3
1990 20 3
(a) Draw a scatter plot and comment on the relationship.
Calculation shows that the mean and standard deviation of the durations are = 10.73 sx = 8.20
For the declines, = 24.67 sy = 11.20
The correlation between duration and decline is r = 0.6285.
(b) Find the equation of the least squares line for predicting decline from duration.
(c) What percent of observed variation in these declines can be attributed to the linear relationship between decline and duration?
(d) One bear market has duration of 15 months but a very low decline of 14%. What is the predicted decline for a bear market with duration = 15? What is the residual for this particular bear market?
C. The Leaning tower of Pisa. (Use Minitab to do this problem.) The leaning tower of Pisa was reopened to the public late in 2001 after being closed for almost 12 years while engineers took steps to prevent the tower from collapsing. Data on the lean of the tower over time show why it was in danger of collapse. The following table gives measurements for the years 1975 to 1987. The variable "lean" represents the difference between where a point near the top of the tower would be if the tower were straight and where it actually is. The data are coded as tenths of a millimeter in excess of 2.9 meters, so that the 1975 lean, which was 2.9642 meters, appears in the table as 642. Only the last two digits of the year were entered into the computer.
Year 75 76 77 78 79 80 81
Lean 642 644 656 667 673 688 696
Year 82 83 84 85 86 87
Lean 698 713 717 725 742 757
(a) Plot the lean of the tower against time; Does the trend to be linear? That is, is the tower's lean increasing at a fixed rate?
(b) What is the equation of the least-squares line for predicting lean? What percent of the variation in lean is explained by this line?
(c) Give a 95% confidence interval for the average rate of change (tenths of a millimeter per year) of the lean.
(d) Looking into the past. In 1918 the lean was 2.9071 meters. (The coded value is 7.1). Using the least-squares equation for the years 1975 to 1987, calculate a predicted value for the lean in 1918. (Note that you must use the coded value 18 for year.)
(e) Although the least-squares line gives an excellent fit into to the data for 1975 to 1978, this pattern does not extrapolate to 1918. Write a short statement explaining why this conclusion follows from the information available. Use numerical and graphical summaries to support your explanation.
(f) Looking to the future. The engineers working on the tower were most interested in how much the tower would lean if no corrective action was taken. Use the least-squares equation to predict the towers lean in the year 2000 if no corrective action had been taken. Compute the prediction interval and comment.
D. Faculty Salaries. Data on the salaries of full professors in an engineering department at a large Midwestern university are given below. The salaries are for the academic years 1996-1997 and 1999-2000. The data also include years in rank as a full professor.
Years in rank 1996 salary ($) 1999 salary ($)
3 70,200 87,000
4 71,900 89,000
5 78,200 89,500
6 92,900 108,000
6 75,700 88,000
7 82,300 100,000
7 67,300 76,950
8 82,800 89,875
10 102,600 118,000
10 86,200 108,000
11 88,240 105,000
13 94,600 108,000
15 96,000 106,100
15 97,200 104,800
35 131,350 144,700
36 109,200 118,481
(a) Write the model that you would use for multiple regression to predict salary in 1999 from salary in 1996 and years in rank.
(b) What are the parameters of your model?
(c) Run the multiple regressions and give estimates of the model parameters. Identify which coefficients are significant via p-values.
(d) Run a simple linear regression of 1999 salary on years in rank. Comment how this model differs from (c )