LINEAR REGRESSION WORKSHEET:

Purpose

This assignment will assess your ability to complete the regression analysis, given business situations, identify the independent and dependent variables, based on the regression analysis, determine the relationship between the variables and use the information to make predictions.

Action Items

1. Meet with your team to analyze the data in Worksheet 5-5. This data is real data that was collected by a doctor who specialized in leg and knee injuries. He gathered data from 43 people, indicated by their initials. Each person was required to perform three anterior (frontward) lunges with the right leg followed by three with the left leg. The distances were recorded. The person was then required to do three lateral (sideways) lunges with the right leg followed by three with the left leg. To use these lunges you will need to average the three trials to get a mean. You can use the Excel "=average(cellsReferenced)" computation to do this. By copying the formula down the column the task you can make short work of this. Not all of the tests below use the lunge information.
2. Complete a correlation and regression analysis on the data.
3. Answer the following questions:

1. Is there a difference in leg length (right $v$. left)? Use a paired $t$-test to determine the answer. Show all steps.
2. Is there a difference in lateral lunge (right $v$. left)? Use paired t-test to determine the answer. Show all steps.

Problem 31: The following crosstabulation shows household income by educational level of the head of household (Statistical Abstract of the United States: 2002)

1. Compute the row percentages and identify the percent frequency distributions of income for households in which the head of household is a high school graduate and in which the head of household holds a bachelor's degree.
2. What percentage of households headed by high school graduates earn $75,000 or more? What percentage of households headed by bachelor's degree recipients earn $75,000 or more?
3. Construct percent frequency histograms of income for households headed by persons with a high school degree and for those headed by persons with a bachelor's degree. Is any relationship evident between household income and educational level?

Pr oblem 37: The National Football League rates prospects by position on a scale that ranges from 5 to 9. The ratings are interpreted as follows: 8-9 should start the first year; 7.0-7.9 should start; 6.0-6.9 will make the team as a backup; and 5.0-5.9 can make the club and contribute. Table 2.15 shows the position, weight, speed (seconds for 40 yards), and ratings for 40 NFL prospects (USA Today, April 14, 2000).

1. Prepare a crosstabulation of the data on Position (rows) and Speed (columns). Use classes of 4.00-4.49,4.50-4.99,5.00-5.49, and 5.50-5.99 for speed.
2. Comment on the relationship between Position and Speed based upon the crosstabulation developed in part (a).
3. Develop a scatter diagram of the data on Speed and Rating. Use the vertical axis for Rating.
4. Comment on the relationship, if any, between Speed and Rating.

45. Five observations taken for two variables follow.

1. Develop a scatter diagram with $x$ on the horizontal axis.
2. What does the scatter diagram developed in part (a) indicate about the relationship between the two variables?
3. Compute and interpret the sample covariance.
4. Compute and interpret the sample correlation coefficient.

Problem 47 : Nielsen Media Research provides two measures of the television viewing audience: a television program rating, which is the percentage of households with televisions watching a program, and a television program share, which is the percentage of households watching a program among those with televisions in use. The following data show the Nielsen television ratings and share data for the Major League Baseball World Series over a nine-year period (Associated Press, October 27, 2003).

1. Develop a scatter diagram with rating on the horizontal axis.
2. What is the relationship between rating and share? Explain.
3. Compute and interpret the sample covariance.
4. Compute the sample correlation coefficient. What does this value tell us about the relationship between rating and share?

5. Technological advances helped make inflatable paddlecraft suitable for backcountry use These blow-up rubber boats, which can be rolled into a bundle not much bigger than a golf bag, are large enough to accommodate one or two paddlers and their camping gear. Canoe & Kayak magazine tested boats from nine manufacturers to determine how they would perform on a three-day wilderness paddling trip. One of the criteria in their evaluation was the baggage capacity of the boat, evaluated using a 4-point rating scale from 1 (lowest rating) to 4 (highest rating). The following data show the baggage capacity rating and the price of the boat (Canoe & Kayak, March 2003).

1. Develop a scatter diagram for these data with baggage capacity rating as the independent variable.
2. What does the scatter diagram developed in part (a) indicate about the relationship between baggage capacity and price?
3. Draw a straight line through the data to approximate a linear relationship between baggage capacity and price.
4. Use the least squares method to develop the estimated regression equation.
5. Provide an interpretation for the slope of the estimated regression equation.
6. Predict the price for a boat with a baggage capacity rating of 3 .

7. Would you expect more reliable cars to cost more? Consumer Reports rated 15 upscale sedans. Reliability was rated on a 5 -point scale: poor (1), fair (2), good (3), very good (4), and excellent (5). The price and reliability rating for each of the 15 cars are shown (Consumer Reports, February 2004).

1. Develop a scatter diagram for these data with the reliability rating as the independent variable.
2. Develop the least squares estimated regression equation.
3. Based upon your analysis, do you think more reliable cars cost more? Explain.
4. Estimate the price for an upscale sedan that has a good reliability rating.

7. A sales manager collected the following data on annual sales and years of experience.

1. Develop a scatter diagram for these data with years of experience as the independent variable.
2. Develop an estimated regression equation that can be used to predict annual sales given the years of experience.
3. Use the estimated regression equation to predict annual sales for a salesperson with nine years of experience.

17. The data from exercise 3 follow.

The estimated regression equation for these data is \(\hat{y}=7.6+.9 x\). What percentage of the total sum of squares can be accounted for by the estimated regression equation? What is the value of the sample correlation coefficient?

19. The data from exercise 7 follow:

The estimated regression equation for these data is \(\hat{y}=40,639-1301.2 x\). What percentage of the total sum of squares can be accounted for by the estimated regression equation? Comment on the goodness of fit. What is the sample correlation coefficient?

21. An important application of regression analysis in accounting is in the estimation of cost. By collecting data on production volume (units) and cost and using the least squares method to develop an estimated regression equation relating production volume and cost, an accountant can estimate the cost associated with a particular production volume. Consider the following sample of production volume and total cost data for a manufacturing operation.

1. Use these data to develop an estimated regression equation that could be used to predict the total cost for a given production volume.
2. What is the variable cost per unit produced?
3. Compute the coefficient of determination. What percentage of the variation in total cost can be explained by production volume?
4. The company's production schedule shows 500 units must be produced next month. What is the estimated total cost for this operation?

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