Question

im having a hard time with this question would someone please be avle to explain to me this answer in detail

1 Problem 8 (20 marks) A sales manager for an advertising agency believes there is a relationship between the number of contacts and the amount of the sales. To verify this belief, the following data was collected: Salesperson Number of Contacts Sales (in thousands) 14 24 2. 12 14 3 20 28 4 16 30 5 46 80 6 23 30 7 48 90 8 50 85 55 120 10 50 110 Assume normality of variables. a) Calculate the coefficient of correlation r. Provide an interpretation of the computed value of r. (9 marks) b) Calculate the coefficient de determination r square (R2). Provide an interpretation of the computed value of r square (R2). (3 marks) e) Determine the least squares line. Provide an interpretation of your results. (8 marks)

Answer 1

Coefficient of determination: R² = 0.9513 Interpretation: 95.13% of variance in y can be explained by x. Data summary: number of points: 10 ý (mean of y) 61.1 regression line: 2.19x - 12.2 Summary of Ra calculation: Variable Value Formula SST: 14248.9 2yi - y) SSR: 13555.4248 Sỹ - y)^ SSE: 693.4752 Elyi - ġo) R2: 0.9513 SSR/SST


Details of SST calculation: iyi ў (yi - y)^ 0 24.0000 61.1000 1376.41 1 14.0000 61.1000 2218.41 28.0000 61.1000 1095.61 N 3 30.0000 61.1000 967.21 4 80.0000 61.1000 357.21 5 30.0000 61.1000 967.21 6 90.0000 61.1000 835.21 7 85.0000 61.1000 571.21 8 120.0000 61.1000 3469.21 9 110.0000 61.1000 2391.21 Summing the last column, we get SST = 14248.9


Details of SSR calculation: i ġi ў (ỳi - y)^ 0 18.5239 61.1000 1812.7202 1 14.1347 61.1000 2205.7427 2 31.6918 61.1000 864.8423 3 22.9132 61.1000 1458.2293 4 88.7525 61.1000 764.66 5 38.2757 61.1000 520.9475 6 93.1418 61.1000 1026.6751 7 97.5311 61.1000 1327.2217 8 108.5043 61.1000 2247.1642 9 97.5311 61.1000 1327.2217 Summing the last column, we get SSR = 13555.4248

a)
r = $\sqrt{R},\:$ with the sign same as that of $\widehat{\beta_1}$ . Therefore r = +0.9753
Interpretation: The value is quite high to its maximum value of 1 and hence this indicates that the relation between x and y is highly linear and the relation is strong and is positive(i.e increase in one of them will cause an increase in the other, or they both decrease or increase together).

b)
R² = 0.9513
Interpretation has been given in the first image itself. Adding to it is that the linear regression model on y on x explains 95.13% of the variability in y. It's a measure of the goodness of fit of the linear model on the variable y.

c)
$\widehat{y}$ = 2.19*x - 12.2. Therefore one unit increase in the No. of contacts will increase sale by 2.19 thousands.
Workings hae been posted in the images above

Thank You :)