Chapter 12: Multiple Linear Regression and Certain Nonlinear Regression ModelsMINITAB Project

STATISTICS EXPLORATION # 3:CORRELATION AND REGRESSION

PURPOSE - to use MINITAB to

• observe correlation and patterns through scatter plots
• model sets of data through functions and curve fitting
• make a visual estimate of negative, low, moderate, or high linear correlation
• develop the mathematical background for least squares regression
• use the Pearson product moment correlation coefficient r for measuring goodness of fit of raw data to a least squares model
• compute the coefficient of determination

BACKGROUND INFORMATION

Some terms and background information that are associated with correlation and regression are explained below.

• A graphical display that shows the relationship between two real world quantities on a coordinate plane is called a scatter plot. A scatter plot is a graph of ordered pairs (x, y) of numbers consisting of the independent variable, x, and the dependent variable, y.

• Example: A scatter plot for the two variables diameter and volume is shown below in the diagram. Observe the approximate linear pattern in the plot.

Scatter Plot of Diameter versus Volume

• Correlation analysis is a statistical method used to determine whether a relationship between variables exists.
• When one variable increases as the other decreases, the correlation between the variables is said to be negative.
• When both variables increase or decrease together, the correlation is said to be positive. (This is illustrated in the above scatter plot).
• When there is no apparent relationship between the variables, there is no correlation.
• Example: The scatter plot for diameter versus height below shows an example of little or no correlation.

Scatter Plot Displaying Little or No Correlation

• Regression is a statistical method used to describe the nature of the relationship between variables, i.e. positive or negative, linear or nonlinear.
• Coefficient of correlation, r, is a statistical measure of how closely data fits a line. That is, it measures the strength and direction of a relationship between two variables.
• The range of the correlation coefficient is from ?1 to +1.
• If there is a strong positive linear relationship between the variables, the value of r will be close to +1.
• If there is a strong negative linear relationship between the variables, the value of r will be close to ?1.
• When there is no linear relationship between the variables or there is only a weak relationship, the value of r will be close to 0.
• The formula that is used to compute the coefficient of correlation (correlation coefficient) is given by the following formula:

• The Coefficient of Determination, denoted by or , is a measure of the variation of the dependent variable that is explained by the regression line and the independent variable.
• The value of or is usually expressed as a percentage and so it ranges from 0% to 100%. Thus, the closer the value is to 100%, the better the model is representing the data.
• As the value of r approaches 0, or decreases more rapidly. For example, if r = 0.5, then or = 0.25 which means that only 25% of the variation in the dependent variable can be attributed to the variation in the independent variable.
• The line that approximates a trend for the data on a scatter plot is called the line of best fit, or regression line. After the scatter plot is drawn, the next steps are to compute the value of the correlation coefficient and to test the significance of the relationship. If the relationship is significant, the next step is to determine the equation of the regression line.
• We use the least-squares criterion to find the equation for the line of best fit. This equation says that the line we fit to the data points must be such that the sum of the squares of the vertical distances from the points to the line is made as small as possible.
• Note: It might be useful to briefly review some algebra concepts needed to understand correlation and regression.
• A linear equation is an equation in which no variable has an exponent greater than one.
• One form in which a linear equation can be written is y = mx+b. This is called the slope-intercept form, where m is the slope of the line and b is the y-intercept.
• The slope is interpreted to be amount by which the y-value will increase for a one-unit increase in the x-value.
• Example: The scatter plot below shows a linear regression line superimposed on it. It also displays the value of and the equation of the line. Observe that the y-intercept value is 36.9435 and the value of the slope is 5.06586.

Scatter Plot Displaying a superimposed Linear Regression Line

• A quadratic equation is an equation in which no variable has an exponent greater than two.
• Example: The scatter plot below displays a quadratic pattern with the quadratic regression equation. Note, this was the same data that was used for the linear plot above. However, observe that the value has increased from 93.5% to 96.2%. The quadratic regression equation is also given.

Scatter Plot Displaying a superimposed Quadratic Regression Line

• In most research problems where regression analysis is applied, more than one independent variable is needed in the regression model. The complexity of most scientific mechanisms is such that in order to be able to predict an important response, a multiple regression model is needed. When this model is linear in the coefficients, it is called a multiple linear regression model.
• The estimated response for the linear regression model with k independent variables is given by

PROCEDURES

First, load the MINITAB (windows version) software as described in Exploration #0.

NOTE: The procedures presented in these explorations may not be the only way to achieve the end results. Also, whenever graphs are presented, only the MINITAB graphics features will be used.

1. OBSERVING CORRELATION AND PATTERNS THROUGH SCATTER PLOTS

In this section we will present examples that will enable you to get an understanding of the concept correlation through scatter plots.

Example 1: Consider the following table, which contains measurements on two variables for ten people: the number of hours the person spent riding a bicycle in the past week and the number of months the person has owned the bicycle. Present a scatter plot for this information with the number of hours along the vertical axis and the number of months owned along the horizontal axis.

 Person 1 2 3 4 5 6 7 8 9 10 Hours Exercised 5 2 8 3 8 5 5 7 10 3 Months Owned 5 10 4 8 2 7 9 6 1 12

Enter the number of hours in column C1 and the number of months owned in column C2. Rename column C1 as HOURS and column C2 as MONTHS. Next, we will present a scatter plot with the values in C1 along the vertical axis and the values in C2 along the horizontal axis. To achieve this, select Graph® Plot and the Plot dialog box will be displayed. Enter the appropriate Y and X variables as shown in Figure 3.1. Note that the Display option that was selected is Symbol.

Figure 3.1: Display of the plot selections

Click on the OK button and the plot will be displayed. Figure 3.2 shows the resulting plot. Since this is a plot of the ordered pairs (MONTHS, HOURS), this will represent a scatter plot for the two variables.

Figure 3.2: Display of the Scatter plot of the number of hours vs. the number of months owned

From Figure 3.2, you can see a definite trend. The points appear to form a line that slopes from the upper left to the lower right of the screen. As you move along that (imaginary) line from left to right, the values on the vertical axis (hours riding) get smaller, while the values on the horizontal axis (months owned) get larger. Another way to express this is to say that the two variables are inversely related: the longer the bike was owned, the less the person tends to ride it.

We say that these two variables are correlated. More than that, they are correlated in a particular negative direction.

Example 2: Consider the following table, which contains the noise levels as measured by two different instruments. Present a scatter plot for this information with the variable NOISE1 along the y-axis and NOISE2 along the horizontal axis.

 Noise1 Noise2 0.97299 0.98150 1.93680 2.13277 3.04045 3.13164 1.71018 2.06533 3.92119 4.46499 5.92306 6.20214 0.78743 1.24267 1.98965 2.18766 2.92915 3.35408 1.49930 2.10889 3.61674 4.47986 5.68941 5.72906

The plot is shown in Figure 3.3. The pattern for this scatter plot suggests a positive correlation.

Figure 3.3: Display of scatter plot with positive correlation

Sometimes there may be little or no relationship between the variables. For example, in Figure 3.4 the display is a scatter plot of a person?s cholesterol after two days on a special diet and with control diet. Observe that the scatter plot displays no particular pattern. That is, there is little or no correlation between these two variables.

Figure 3.4: Display of scatter plot with little or no correlation

1. LINEAR REGRESSION

In this section, we will compute the strength of the association between variables. That is, we will compute the correlation coefficient. However, we will first observe the scatter plots before computing the correlation coefficient.

Example 3: The table below shows the average weight, by height, of American men between the ages of 20 and 24.

 Height (inches) Weight (pounds) 62 130 64 139 66 148 68 157 70 167 72 176

Source: Grossman, Stanley. Applied Calculus. 2nd Ed Wm.C. Brown Publishers.

1. Use MINITAB to present a scatter plot with the weight being the dependent variable (y) and height being the independent variable (x).

Using the MINITAB procedures presented earlier in the exploration, the scatter plot is constructed and displayed in Figure 3.5.

Figure 3.5: Display of scatter plot with almost a perfect positive correlation

Observe from Figure 3.5, that the points are almost on a straight line with positive slope. Hence, one would expect a strong positive correlation value.

2. Use MINITAB to compute the correlation coefficient r.

To compute the correlation between the two variables, select Stat® Basic Statistics® Correlation. The Correlation dialog box will appear. Select the two variables for the Variables box as shown in Figure 3.6.

Figure 3.6: Display of the correlation (coefficient) dialog box

Click on the OK button and the correlation coefficient will be computed and displayed in the Session window. Figure 3.7 shows the output in the Session window.

Figure 3.7: Correlation value for the variables Height and Weight

Observe that the computed correlation coefficient is +1. We observed from the scatter plot that the points are almost on a straight line with positive slope. Thus, for all practical purposes, there is a perfect positive correlation between these two variables. Thus, r = +1.

Example 4: Determine a linear regression model for the data given in Example 3.

In order to get the equation for the linear regression model, select Stat® Regression ® Regression and the Regression dialog box will appear. In the dialog box, the Response variable corresponds to the dependent (y) variable and the Predictors variable corresponds to the independent variables. Here we have only one independent (x) variable, which is HEIGHT, and the response variable is WEIGHT. Complete the dialog box as shown in Figure 3.8.

Figure 3.8: Regression dialog box for the dependent variable Weight and the independent variable Height

Click on the OK button and the analysis for the regression will be displayed in the Session window. Figure 3.9 displays the output. From the output we see that the regression equation (other terms are predictor equation, line of best fit, and least squares regression line) that relates WEIGHT to HEIGHT is given as WEIGHT = -156 + 4.61´ HEIGHT. Other information is also given in the Session window but we will ignore those for the mean time.

Example 5: What is the predicted WEIGHT for a person whose HEIGHT is 69 inches?

We can use the regression equation to predict WEIGHT values for a given HEIGHT value. For instance, in this example, HEIGHT = 69 and so by substituting this value into the regression equation, we have the predicted WEIGHT = -156 + 4.61´ 69 = 162.09 pounds. That is, based on this model, the predicted weight for a person who is 69 inches (5 feet 9 inches) tall is approximately 162 pounds.

NOTE: This model will work well for independent values within the observed range of values. The range of values for the HEIGHT variable was from 62 inches to 72 inches. Thus, one should not rely on the model to make accurate predictions outside this range of values for the independent variable HEIGHT.

Figure 3.9: Regression Analysis session window output for Example 4

Example 6: What is the coefficient of determination for the model in Example 4?

Recall that the Coefficient of Determination, denoted by or , is a measure of the variation of the dependent variable that is explained by the regression line and the independent variable. This value lies between 0 (0%) and 1 (100%). Thus, the closer the value is to 100%, the better the model is fitting the data. From Figure 3.9, R2 = 100.0%. Thus, from a practical standpoint, the model has captured all the variation in the dependent variable.

NOTE: We can use MINITAB to superimpose the regression line onto the scatter plot. To achieve this, select Stat® Regression ® Fitted Line Plot and the dialog box will be displayed. Fill out as in Figure 3.10 and select the OK button.

Figure 3.10: Fitted Line Plot dialog box for Example 4

The resulting plot is shown in Figure 3.11. Observe that the regression equation is given on the output as well as the coefficient of determination R-sq.

Figure 3.11: Fitted Line Plot output for Example 4

1. NONLINEAR REGRESSION

In this section we will investigate patterns and models that are non-linear in nature.

Example 7: Alcohol absorption and the risk of having an accident have been studied for years. Extensive research has provided the following data relating the risk of having an automobile accident to the blood alcohol level. Use MINITAB to present a scatter plot for the data. We will assume that the independent variable (x) is blood alcohol level and the dependent variable (y) is relative risk of accident.

 Blood Alcohol Level (%) Relative Risk of Accident (%) 0.00 1.00 0.05 2.90 0.10 8.50 0.15 24.8 0.20 72.2 0.21 89.5

First, we need to enter the data values into MINITAB. Follow the procedure in Example 3 to present a scatter plot. The scatter plot is presented in Figure 3.12.

Figure 3.12: Scatter Plot for Example 7

The plot indicates that the pattern is non-linear. The next example will allow us to determine a model for the data.

Example 8: Fit an appropriate model for the data in Example 7. The two other options we have in the Fitted Line Plot are Quadratic and Cubic. See Figure 3.10. Using the procedure for the NOTE in Example 6, select Quadratic for the Fitted Line Plot procedure. The quadratic model superimposed on the scatter plot is shown in Figure 3.13.

Figure 3.13: Quadratic Fitted Line Plot output for Example 8

The equation for the quadratic model is

Relative Risk = 4.34362 - 309.293 Blood Alcohol + 3295.02 Blood Alcohol**2

If we let y = Relative Risk and x = Blood Alcohol, then we can write the equation as

y = 4.34362 - 309.293x + 3295.02x2

Observe that because of the square term in the equation, this will be a quadratic model. The R-Sq = 98.2%. Thus, the model explains 98.2% of the variability of the Relative Risk variable. Since this number is close to 100%, we can assume that the model is quite appropriate to describe the pattern of the scatter plot.

If we use the Cubic option, the fitted line plot as shown in Figure 3.14 will be generated.

Figure 3.14: Cubic Fitted Line Plot output for Example 8

The equation for the cubic model can be written in terms of x and y as

Relative Risk = 0.688590 + 171.944x + - 3119.78x2 + 20415.2x3

Observe that because of the cubic term in the equation, this will be a cubic model. The R-Sq = 99.9%. Thus, the model explains 99.9% of the variability of the Relative Risk variable. Since this number is closer to 100%, we can conclude that the cubic model is more appropriate to describe the pattern of the scatter plot.

Note: One can use these models to predict the Relative Risk of an Accident for a given Blood Alcohol Level. Again these input values for the Blood Alcohol Levels should be within the range of observed values since the model was derived from this range of values.

NOTE: Here we have two models ? the quadratic and the cubic. The cubic model is a better fit for the data with an R-sq value (99.9%) that is slightly greater than that for the quadratic model (98.2%). From a practical standpoint, such a slight improvement in the R-sq value may not compensate for the increase in the complexity (the addition of the cubic term) of the model. Thus, when modeling data, one should look at all aspects and give a rational why the model was chosen.

1. MULTIPLE LINEAR REGRESSION MODELS

In this section we will investigate multiple linear regression models

Example 9: An experiment was conducted to determine if the weight of an animal can be predicted after a given period of time on the basis of the initial weight of the animal and the amount of feed that was eaten. The following data, measured in kilograms, were recorded

 Final weight, y Initial weight, x1 Feed weight, x2 95 42 272 77 33 226 80 33 259 100 45 292 97 39 311 70 36 183 50 32 173 80 41 236 92 40 230 84 38 235

Use MINITAB to display scatter plots for the dependent variable versus the two independent variables. We will assume that the independent variables are x1 (initial weight) and x2 (feed weight) and the dependent variable is y (the final weight). First, we need to enter the data values into MINITAB. Follow the procedure in Example 3 to present a scatter plot for both independent variables. The scatter plots are presented in Figure 3.15 and Figure 3.16.

Figure 3.15: Scatter Plot for Example 9

Figure 3.16: Scatter Plot for Example 9

The plots indicate an approximate linear association between the dependent and the independent variables. The next example will allow us to determine a model for the data.

Example 10: Fit an appropriate model for the data in Example 9. Select Stat® Regression and fill in the dialog box as shown in Figure 3.17. Observe that in Figure 3.17, we have two Predictors or independent variables.

Figure 3.17: Regression Dialog box for the Multiple Regression model for Example 10

Select the OK button and the resulting Session window display will be shown as in Figure 3.18.

The equation for the quadratic model is

Final weight y = - 23.0 + 1.40 Initial weight x1 + 0.218 Feed weight x2

Observe that because of the square term in the equation, this will be a quadratic model. The R-Sq = 87.3%. Thus, the model explains 87.3% of the variability of the Final weight variable. Since this number is rather close to 100%, we can assume that the model is quite appropriate to describe the relationship between the variables.

Note: One can use the model to predict the Final weight of an animal for a given initial weight and feed weight. Again these input values for the independent variables should be within the range of observed values since the model was derived from these ranges.

Example 10: Select the Graphs option in the Regression dialog box as shown in Figure 3.17 to display residual plots and normality plot for the data given in Example 9. Click on the Graphs option and in the resulting dialog box, select the options as shown in Figure 3.18.

Figure 3.18: Regression-Graphs Dialog box for the Multiple Regression model for Example 9

Two of the resulting graphs are shown in Figure 3.19 and 3.20.

Figure 3.19 shows the normality plot for the residuals for the model. Observe that the plot displays a linear pattern, which indicates that the normality assumption for the regression model has not been violated.

Figure 3.20 shows the plot for the residuals versus time order for the model. This plot is usually used to help determine visually whether the independence assumption for the model has been violated. Observe that the plot displays no apparent pattern, which indicates that the independence assumption for the regression model has not been violated.

Figure 3.19: Normality Plot for the Residuals for the Multiple Regression model in Example 9

Note: Refer to your text for detailed discussions on the assumptions for a regression model.

Figure 3.20: Residual Plot versus Order of Observation for the Residuals for the Multiple Regression model in Example 9

NOTES

EXPLORATION #3: HOMEWORK ASSIGNMENT

Name: _____________________ Date: ______________________

Course #: ___________________ Instructor: _________________

1. State in each of the following cases whether you would expect the relationship between the given variables to be positive or negative or neither.
1. An individual?s height and weight. ________________________
2. The number of hours that a runner practices and his time for a 1-mile race.
____________________
3. A college student?s cumulative grade point average and the number of hours per week that she works at a job. _____________________
4. The number of automobile accidents and the amount of insurance premiums that the driver has to pay. ___________________
5. The percentage of nitrogen in a fertilizer and the height to which a treated plant will grow. ______________________
6. The amount of alcohol consumed by an individual and the length of time in which he responds to a given stimulus. ____________________
7. The amount of alcohol consumed by an individual and the number of teachers at that individual?s previous high school. _____________________
1. The table below gives the chirping frequency for the striped ground cricket and the corresponding temperature.

 Chirps / second Temperature in degrees F 20.0 88.6 16.0 71.6 19.8 93.3 18.4 84.3 17.1 80.6 15.5 75.2 14.7 69.7 17.1 82.0 15.4 69.4 16.2 83.3 15.0 79.6 17.2 82.6 16.0 80.6 17.0 83.5 14.4 76.3

1. Produce a scatter plot for the data. Let chirps/second be your x values (independent) and temperature in degrees Fahrenheit be your y values (dependent). Label appropriately and turn in a hard copy with your work.

Note: You may want to review for directions on how to produce a scatter plot.

2. Based on the scatter plot, what type of correlation, if any, do you observe? Discuss.
3. What is the value of the correlation coefficient between these two variables?
r = __________________________
4. Interpret the value of the correlation coefficient. Discuss.
5. What is the coefficient of determination (R2) for the correlation type you chose?
R2 = ___________________%
6. Interpret the value of the coefficient of determination. Discuss.
7. What is the regression equation? _____________________________________
8. Provide a print out of the graph of the scatter plot and the regression equation together.
9. Use your graph to estimate the number of chirps/second a striped ground cricket would make at a temperature of 98° Fahrenheit.
Estimated number of chirps: __________________________
10. Using the regression equation, what is the predicted temperature, in Fahrenheit, if the number of chirps/second made by a striped ground cricket was 19?
Predicted temperature: __________________________
11. Using the regression equation, what is the predicted temperature, in Fahrenheit, if the number of chirps/second made by a striped ground cricket was 50?
Predicted temperature: __________________________
12. Discuss the results obtained in part (i).
1. The following Table shows the resident population, in thousands, of 85+ year olds for the United States from 1960 to 1996.

 Year (t) Population ( in thousands) 1960 (0) 567 1961 (1) 592 1962 (2) 608 1963 (3) 627 1964 (4) 653 1965 (5) 684 1966 (6) 718 1967 (7) 760 1968 (8) 800 1969 (9) 848 1970 (10) 969 1971 (11) 977 1972 (12) 1020 1973 (13) 1069 1974 (14) 1141 1975 (15) 1227 1976 (16) 1290 1977 (17) 1365 1978 (18) 1443 1979 (19) 1521 1980 (20) 1559 1981 (21) 1649 1982 (22) 1720 1983 (23) 1786 1984 (24) 1848 1985 (25) 1906 1986 (26) 1963 1987 (27) 2025 1988 (28) 2075 1989 (29) 2137 1990 (30) 2180 1991 (31) 2279 1992 (32) 2347 1993 (33) 2467 1994 (34) 2542 1995 (35) 2661 1996 (36) 2692

1. Construct of scatter plot for the population table. Use the t values (0, 1, 2, ?, 36) for the independent x values and the population values for the dependent y values. Label appropriately and present a hard copy with your work.
2. What type of pattern, if any exist, do you think best fits the data? Discuss.
3. What is the coefficient of determination for the regression that you chose?
R2 = __________________%
4. Interpret the value of the coefficient of determination. Discuss.
5. What is an appropriate regression equation for the data?
Regression equation: ____________________________________________
6. Justify why you choose the model in part (e). Discuss.
7. Estimate, using the regression equation in part (e), the resident population of 85+ year olds in the year 1999. (Here, t = 39).
Estimated population: _____________________________
1. The following table contains data points representing the height of a model rocket at various times during its flight after its rocket motor has burned out.

 Seconds since rocket launched Height of rocket in feet 1 230 2 310 3 350 4 360 5 350 6 300 7 220

1. Use MINITAB to construct a scatter plot of the data. Let the x values be the values for the number of seconds since the rocket was launched and let the y values be the values for the height above ground of the rocket in feet. Label appropriately and present a hard copy with your work.
2. Discuss the shape of the scatter plot and whether there seems to be a correlation between the two variables.
3. What is the best regression equation for the data?
Regression equation: ________________________________________________
4. What is the value of the coefficient of determination, R2 ?
R2 = _____________________%
5. Interpret the value of the coefficient of determination. Discuss.
6. Discuss why you chose the model in part (c).
7. Construct a fitted line plot for the data and use the curve to estimate the following when the rocket will be at various heights.
• When did the rocket reach its maximum height? ____________________
• What was the rocket?s maximum height? ____________________
• When will the rocket hit the ground? ___________________
1. The following table shows the Length (in inches) and Weight (in pounds) of alligators.
1. Create a scatter plot for the length and weight of the alligators. Let the values for the length be along the x-axis and the values for the weight be along the y-axis. Include a hard copy of the plot with your work.
2. What type of correlation best fits the scatter plot?
___________________________

Note: To answer this question you may want to check several different model possibilities and choose the one with the best coefficient of determination. Discuss your reasoning

 Length (in inches) Weight (in pounds) 94 130 74 51 147 640 58 28 86 80 94 110 63 33 86 90 69 36 72 38 128 366 85 84 82 80 86 83 88 70 72 61 74 54 61 44 90 106 89 84 68 39 76 42 114 197 90 102 78 57

3. What is the value of the coefficient of determination for the type of regression that you chose?
: _____________________%
4. What is the equation of the best fitting model?
__________________________________________________________
5. Discuss the strengths and weaknesses of the model.
6. Present a hard copy of the fitted line plot superimposed on the scatter plot for your model from part (d) with your work.
7. Using your fitted line plot, what is the approximate weight of an alligator that is 100 inches long?
Weight: ____________________pounds
8. Using your regression equation, what is the predicted weight of an alligator that is 100 inches long?
Weight: ____________________pounds
1. The following table gives the percent of refillable soft-drink containers sold out of the total soft drinks sold from 1960-1990 in five year increments.

 Year (t) Percent of Total Sold 1960 (0) 96 1965 (5) 84 1970 (10) 65 1975 (15) 57 1980 (20) 34 1985 (25) 23 1990 (30) 7

Source: Prentice Hall, Algebra (1998)

1. Create a scatter plot for the percent of refillable soft drinks sold out of the total soft drinks sold from 1960-1990 in five year increments. Turn in a hard copy of the plot with your work.
2. What type of correlation, if any, exists between the percent of refillable soft drinks sold and the years 1960-1990?
____________________________
3. What is the value of the coefficient of determination R2?
: ____________________________%
4. What is the regression equation? __________________________________
5. Graph the scatter plot and the regression curve on the same display. Sketch the display below. Turn in a hard copy with your work.
6. From the regression equation, estimate the percent of refillable soft drinks sold for the year 1995 (t = 35).
____________________________%
1. The following table shows the population (in millions) of 15-19 year olds from 1960-1996.

 Year (t) Population (in millions) 15-19 year olds 1960 (0) 6586 1961 (1) 6794 1962 (2) 7376 1963 (3) 7647 1964 (4) 8008 1965 (5) 8386 1966 (6) 8842 1967 (7) 8836 1968 (8) 9013 1969 (9) 9234 1970 (10) 9437 1971 (11) 9740 1972 (12) 9988 1973 (13) 10193 1974 (14) 10349 1975 (15) 10465 1976 (16) 10582 1977 (17) 10581 1978 (18) 10555 1979 (19) 10498 1980 (20) 10413 1981 (21) 10096 1982 (22) 9809 1983 (23) 9515 1984 (24) 9287 1985 (25) 9174 1986 (26) 9206 1987 (27) 9139 1988 (28) 9029 1989 (29) 8840 1990 (30) 8709 1991 (31) 8371 1992 (32) 8324 1993 (33) 8410 1994 (34) 8580 1995 (35) 8779 1996 (36) 9043

1. Create a scatter plot for the population (in millions) of 15-19 year olds from Let t be the x values be year and the y-values be population. Provide a hard-copy of the plot.
2. What is the value of the coefficient of determination for the best model?
:_______________________%
3. What is the regression equation? ______________________________________
4. Discuss why you chose the meodel in part (c).
5. Graph the scatter plot and the regression curve on the same window and present a hard copy with your work.
6. Using your regression equation, what is the estimated number of 15-19 year olds in the year 2000 (t = 40)?
___________________________________
7. Using your fitted line plot, in what year did the population of 15-19 year olds reach its maximum thus far?
(Remember to transfer the time value back into the appropriate year).
Year: ___________________________
8. Using your fitted line plot, in what year did the population of 15-19 year olds reach its maximum thus far?
(Remember to transfer the time value back into the appropriate year).
Year: ___________________________
1. The following table displays temperature data for the years 1988-1990 taken from the middle of the mouth of the Chesapeake Bay.

 Month 1988 1989 1990 Jan 1.56 5.76 5.28 Feb 4.68 5.28 7.2 Mar 7.2 5.88 9.72 Apr 11.4 11.5 12.5 May 17.3 16.7 18.4 Jun 21.8 22.5 21.2 Jul 24.7 26 25.1 Aug 22.8 25.1 26.3 Sep 21.6 22.7 24 Oct 18 18.2 21.1 Nov 12.8 13.6 13 Dec 9.72 7.7 8.64

1. Construct scatter plots with the months along the horizontal (x) axis and the temperature along the vertical (y) axis for the three different years. Note: you should use dummy values for the months along the x-axis. That is, you can recode Jan = 1, Feb = 2, Mar = 3, etc. and let these values (1, 2, 3, ? ) be the values along the x-axis.
2. Describe the shape of the graphs.

Plot for 1988

Plot for 1989

Plot for 1990

3. Discuss any observations about these scatter plots that you have made.
4. What are the best fitting regression equations for the plots? Discuss why you made your choice.

Equation 1988: ___________________________________________________

Equation for 1989: ___________________________________________________

Equation for 1990: ___________________________________________________

1. The following table shows the funding for support technology for the Ballistic Missile Defense Organization (BMDO) from 1985 to 1999.

 Year (t) Budget for Support Technology (in millions) 1985 (1) 748 1986 (2) 1606 1987 (3) 2025 1988 (4) 2005 1989 (5) 1865 1990 (6) 1857 1991 (7) 1431 1992 (8) 1194 1993 (9) 718 1994 (10) 529 1995 (11) 382 1996 (12) 381 1997 (13) 393 1998 (14) 408 1999 (15) 637

Source: BMDOFACTSHEETPO-99-02

1. Create a scatter plot for the budget from 1985 to 1999. Let t be the x values be year and the y-values be amount of the budget in millions. Provide a hard copy of the plot.
2. What is the value of the coefficient of determination for the best model?

:_______________________%

3. What is the best regression equation for the data?

_____________________________________________________________

4. Discuss why you chose the model in part (c).
5. Graph the scatter plot and the regression curve on the same window and present a hard copy with your work.
6. Using your regression equation, what is estimated budget for technology support for BMDO for the year 2000?

Estimated budget (\$ millions) :___________________________________

1. Twenty-three student teachers took part in an evaluation program designed to measure teacher effectiveness and determine what factors are important. Eleven female instructors took part. The respopnse measure was a quantitative evaluation made on the cooperating teacher. The independent variables were scores on four standardized tests given to each instructor. The data were as follows:

 Y X1 X2 X3 X4 410 69 125 59.00 55.66 569 57 131 31.75 63.97 425 77 141 80.50 45.32 344 81 122 75.00 46.67 324 0 141 49.00 41.21 505 53 152 49.35 43.83 235 77 141 60.75 41.61 501 76 132 41.25 64.57 400 65 157 50.75 42.41 584 97 166 32.25 57.95 434 76 141 54.50 57.90

1. Use MINITAB to draw scatter plots for Y versus the independent variables. Provide hard copies of these plots.
2. Discuss any observations from these graphs.
3. Use MINITAB to fit an appropriate multiple linear regression model. Discuss why you think this is the most appropriate model.
4. Present residual plots to determine whether the assumptions for the model were violated. (Note: Check your text for the assumptions for a multiple linear regression model). To obtain the residual plots you need to select the Graphs option in the Regression dialog box. See Figure 3.17.

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