QAF Assignment_Nitin Lahoti

QAF Assignment_Nitin Lahoti - Transcript

1 Mr French Horn is the business manager for a symphony orchestra located in the metropolitan area He is preparing to make a presentation to a civic group that has pledged to make a large contribution As a part of his presentation he would like to show that there is a relationship between the number of full time employees and the number of performances He would also like to be able to estimate the number of performances based on the size of the orchestra He selected a sample of 19 cities throughout the US and Canada and determined the number of performances given last year and the number of full time employees The sample information is reported below City 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Performances Last year 137 149 157 164 166 170 173 175 176 179 179 181 182 183 190 192 195 196 200 Size of Orchestra Full time employees 52 54 52 63 52 48 64 47 55 62 75 57 53 53 57 87 81 84 88

i Plot the scatter diagram and comment ii State the model Mr French Horn is attempting to estimate iii Assuming a linear relationship use the ordinary least squares method to find the regression coefficients b0 and b1 Correct to 4 decimal places Write the estimated model

iv Interpret the meaning of the Y intercepts b0 and the slope b1 in this problem v Use the regression model developed in iii to predict the number of performances in a city with 50 full time employees vi Compute the correlation coefficient r

Solution
i The Scatter Diagram

Comment There is an increasing movement growing from lower left hand corner to the upper right hand corner so the correlation is positive The points are around the same line It proves that there is a linear positive relationship between the number of full time employees and the performances The points are not scattered between each other so a moderate correlation is expected between each points ii State the model Mr French Horn is attempting to estimate Mr French Horn is founding a relation between the performance of the orchestra and the size of the orchestra We can expect that Performance of the orchestra is a dependent variable and depends upon the size of the orchestra which is an independent variable iii Assuming a linear relationship use the ordinary least squares method to find the regression coefficients b0 and b1 Correct to 4 decimal places Write the estimated model The formula for the regression coefficients is Here Y is dependent on X so Y is the performances and X is the size of the orchestra because we know that the performance is dependent on the size of the orchestra Using method of least squares

2 Nitin Lahoti MBA 14 C

X 52 54 52 63 52 48 64 47 55 62 75 57 53 53 57 87 81 84 88 1184

Y 137 149 157 164 166 170 173 175 176 179 179 181 182 183 190 192 195 196 200 3344

XY 7124 8046 8164 10332 8632 8160 11072 8225 9680 11098 13425 10317 9646 9699 10830 16704 15795 16464 17600 211013

X2 2704 2916 2704 3969 2704 2304 4096 2209 3025 3844 5625 3249 2809 2809 3249 7569 6561 7056 7744 77146

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After putting the values in the prescribed equations we get

We multiply equation 1 with 1184 and equation 2 with 19 then 22496a 1401856b 3959296 22496a 1465774b 4009247 After subtracting these two equations and solving we get b 0 7814 Substituting the value of b in 2 and solving we get a 127 3064 Because the value of b0 is the same as the value of a so b0 127 3064 Because the value of b1 is the same as the value of b so b1 0 7814 Here the equation model is Coefficients B0 is equal to 127 3064 and the B1 is 0 7814 because the coefficient of X is positive the correlation will also be positive iv Interpret the meaning of the Y intercepts b0 and the slope b1 in this problem Value of the slope B0 is 127 3064 while the value of slope B1 is 0 7814 so if the value of the slope B1 is zero we can mean that without any employees in the orchestra so the minimum of performances will be approximately 127 4 Nitin Lahoti MBA 14 C

v

Use the regression model developed in iii to predict the number of performances in a city with 50 full time employees If the full time employees are 50 given then we can calculate the number of performances in a city by using the equation already derived

Here the value of X employees 50 so we get the value of Y which is the number of performances

For 50 full time employees there will be about 166 performances vi Compute the correlation coefficient r Equation of Karl Pearson s Coefficient of correlation

X 137 149 157 164 166 170 173 175 176 179 179 181 182 183 190

Y 52 54 52 63 52 48 64 47 55 62 75 57 53 53 57

XY 7124 8046 8164 10332 8632 8160 11072 8225 9680 11098 13425 10317 9646 9699 10830

X2 18769 22201 24649 26896 27556 28900 29929 30625 30976 32041 32041 32761 33124 33489 36100

Y2 2704 2916 2704 3969 2704 2304 4096 2209 3025 3844 5625 3249 2809 2809 3249 5

Nitin Lahoti MBA 14 C

192 195 196 200 Total 3344

87 81 84 88 1184

16704 15795 16464 17600 211013

36864 38025 38416 40000 593362

7569 6561 7056 7744 77146

Putting the values in equation from the table we get

Comment The performances and number of employees are highly positively correlated 6 Nitin Lahoti MBA 14 C

7 Nitin Lahoti MBA 14 C