Here Police is the independent variable and Crime is the dependent variable. Scatter Plot Police

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1 5. a. If we want to estimate crimes on the basis of the number of police, which variable is the dependent variable and which is the independent variable? Here Police is the independent variable and Crime is the dependent variable b. Draw a scatter diagram. Scatter Plot 5 Number of Crimes Police c. Determine the coefficient of correlation. We have, the coefficient of correlation ( X X ( r ( s s X ( X X ( ( X X ( Here, n 8, X 146/8 18.5, 95/ ( X X s (41.5/

2 ( (90.875/ s and ( X X ( Hence r (8 1(5.8737(6.446 d. Determine the coefficient of determination. The coefficient of determination, R r ( ^ e. Interpret these statistical measures. Does it surprise ou that the relationship is inverse? Coefficient of correlation is a measure of the strength of the linear relationship between two variables. Here the correlation coefficient is This suggests a strong inverse relationship. The negative sign in the correlation coefficient indicates that the correlation between number of police and the number of crimes is negative, which is quite unsurprising because the crime decreases as the number of police increases. The coefficient of determination is the ratio of the eplained variation to the total variation. Here the coefficient of determination is This means that 76.46% of the variation in the dependent variable, number of Crimes( is eplained b the variation in the independent variable, number of Police (X. The remaining variation remains uneplained. 15. a. Determine the regression equation. Let X denote the number of rooms and denote Kilowatt-hours, in thousands. Assume that X and are linearl related. Let α + β X be the suggested linear relationship. B the method of least squares the estimates of β and α are given b, b s r s and a bx X ( X X ( ( X X (

3 Here, n 10, X 91/10 9.1, 74/ ( X X s (66.9/9.764 ( s (36.4/ ( X X ( 44.6 and ( X X ( 44.6 r ( s s (10 1(.764(.0111 Therefore, b *(.0111/ and a * Thus the fitted least square regression line is ˆ X or, Kilowatt-Hours * Number of rooms b. Determine the number of kilowatt-hours, in thousands, for a si-room house. The number of kilowatt-hours, in thousands, for a si-room house is given b ˆ * a. Determine the regression equation. Let α + β X be the suggested linear relationship. B the method of least squares the estimates of β and α are given b, b s r s and a bx

4 X ( X X ( ( X X ( Here, n 5, X 8/5 5.6, 9/5 5.8 ( X X s (9./ ( s (6.8/ ( X X ( 10.6 and ( X X ( 10.6 r 0.75 ( s s (5 1(.7019( Therefore, b 0.75*(1.3038/ and a * Thus the fitted least square regression line is ˆ X b. Determine the value of when X is 7. The value of ˆ when X 7 is given b, ˆ * c. Determine the standard error of estimate. The standard error of estimate is given b, ( ˆ S. n ˆ ( ˆ

5 Here, ( ˆ.951 Thus, S d. Suppose a large sample is selected (instead of just five. About 68 percent of the predictions would be between what two values? ˆ ± The director of marketing at Reeves Wholesale Products is studing monthl sales. Three independent variables were selected as estimators of sales: regional population, per capita income, and regional unemploment rate. The regression equation was computed to be (in dollars: 64, X X - 11,600X3 a. What is the full name of the equation? Multiple regression equation b. Interpret the number 64,100. The number 64,100 is the - intercept. This is the value of the dependent variable when X1 X X3 0. c. What are the estimated monthl sales for a particular region with a population of 796,000, per capita income of $6,940, and an unemploment rate of 6.0 percent? The estimated monthl sales for a particular region with a population of 796,000, per capita income of $6,940, and an unemploment rate of 6.0 percent is given b, 64, *796, *6,940-11,600*6.0 $374, A sample of General Mills emploees was studied to determine their degree of satisfaction with their present life. A special inde, called the inde of satisfaction, was used to measure satisfaction. Si factors were studied, namel, age at the time of first marriage (X1, annual income (X, number of children living (X3, value of all assets (X4, status of health in the form of an inde (X5 and the average number of social activities per week such as bowling and dancing (X6 Suppose the multiple regression equation is: X X + 4X X4 +.19X X6 a. What is the estimated inde of satisfaction for a person who first married at 18, has an annual income of $6,500, has three children living, has assets of $156,000, has an inde of health status of 141, and has.5 social activities a week on the average? The estimated inde of satisfaction for a person who first married at 18, has an annual income of $6,500, has three children living, has assets of $156,000, has an inde of health status of 141, and has.5 social activities a week on the average is given b, * *6, * *156, * *

6 b. Which would add more to satisfaction, an additional income of $10,000 a ear or two more social activities a week? An additional income of $10,000 a ear added onl 0.008*10,000 8 to the inde. Two more social activities a week added 6.8* 53.6 to the inde. So two more social activities a week would add more to satisfaction. 5. The following table lists the annual amounts of glass cullet produced b Kimble Glass Works, Inc. Scrap ear Code (tons Determine the least squares trend equation. Estimate the amount of scrap for the ear 008. The least squares trend equation is ˆ a + bt Here, n 5, t 15/5 3, 0/5 4 ( t t st (10/ ( s (10/ ( t t ( 9 and t ( t t ( ( t t ( ( t t ( 9 r 0.9 ( s s (5 1(1.5811( t Therefore, b 0.9*(1.5811/ and a 4 0.9*3 1.3 Thus the fitted least squares trend equation is ˆ t The amount of scrap for the ear 008 is obtained b substituting t 7 in the least square trend equation and is given b, ˆ *7 7.6

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