Element of economics - Essay Example

Question: An insurance manager believes that the number of new policies written annually by his agents is related to the number of years of selling experience these agents have. A random sample of 12 agents revealed the following data:No. of years of experienceNo. of new policies written annually21752273463712509415132048133942010352063a) Find the least-squares regression line of number of new policies on number of years of experience.b) Find the coefficient of determination for the line in (a) and comment.c) Estimate the number of new policies written by an agent with 15 years of experience.d) What can be said about the slope of a least-squares regression line if the two variables involved have a correlation coefficient of r=0?e) In the exercise 9.

1, can we reliably estimate the number of policies written annually by an agent with 1 year experience? Explain.f) Interpret the slope of the least-squares regression line in exercise 9.1.ANSWER:No. of yrs. of Experience (x)No. of new policies (y)x2y2xiyi217428934522254841107344911562386373613692221250144250060094181168136951325169652048400230496013391691521507420164008010351001225350206340039691260 1134191449170674795 N = 12a) The best fit line associated with the n points (x1, y1), (x2, y2), . . .

, (xn, yn) has the form y = b0 + b1x where slope = b1=n(xy)  (x)(y)n(x2)  (x)2intercept = b0=y  m(x)nSo using our calculated values, we put them in formulae: b1 =12(4795)  (113)(419)12(1449)  (113)2 b1 =101934619=2.2068And b0 =419-2.2068(113)12=14.136So the least-squares regression line isy = 14.136 + 2.2068x b) Coefficient of determination is calculated as r 2 =b1y + b0xy - (y)2/ny2  (y)2/nSo we calculate it using the given data,r2 =14.136(419)+2.2068(4795)-(419)/1217067-(419)2/12 r2 =1874.512436.917 r2 = 0.

7692So, r2 = 0.7692 indicates that 76.92% of the variability in y is explained by its linear relationship with the independent variable x and 23.08% of the variation is due to chance or other factors.c) The estimated number of new policies by an agent having 15 years of experience is y = 2.2068*15+14.136 =47.2804 d) If r = 0, i.e., the coefficient of correlation is zero, than there is no linear relationship between the dependent and the independent variable.

This means that the slope of least-squares regression line will change from point to point and will not be constant.e) The coefficient of correlation is r = 0.887, this means that there is a strong positive relationship between the independent and dependant variable. Hence the least-squares regression line fitted is very good. So if we estimate the number of policies written by an agent with one year experience, it will be reliable.f) The slope of the line, i.e., b1=2.2068, indicates that the values of y increase by 2.

2068 units for a unit increase in x.BibliographyChatterjee, Samprit and Hadi, Ali S. Regression Analysis by Example. 4th Edition. Wiley Series in Probability and Statistics. 2006.