A sales manager has collected the following data on annual sales ( y) and years of experience ( x)
.
Sales person Years of Experience ( x) Annual Sales (K’000) ( y)
1 1 80
2 3 97
3 4 92
4 4 102
5 6 103
6 8 111
7 10 119
8 10 123
9 11 117
10 13 136
Show work step by step in answering the questions below.
(f) Construct a 95% confidence interval for the slope parameter β1
.
(g) Find the correlation coefficient
(h) Find the coefficient of determination and interpret the value.
(i) Give the ANOVA table
(j) Use the ANOVA table to test for a significant linear relationship between years of
experience and annual sales.
(g) Find the correlation coefficient
(h) Find the coefficient of determination and interpret the value.
(i) Give the ANOVA table
(j) Use the ANOVA table to test for a significant linear relationship between years of experience and annual sales.
To find the correlation coefficient, coefficient of determination, ANOVA table, and test for a significant linear relationship between years of experience and annual sales, we first need to calculate the slope parameter β1 and intercept β0 of the linear regression model.
Step 1: Calculate the mean of years of experience ( x̄) and annual sales ( ȳ).
x̄ = (1+3+4+4+6+8+10+10+11+13)/10 = 6.0 years
ȳ = (80+97+92+102+103+111+119+123+117+136)/10 = 107.0 K’000
Step 2: Calculate the sum of squared differences for years of experience ( Sxx) and annual sales ( Syy).
Sxx = Σ(x_i - x̄)^2 = (1-6)^2 + (3-6)^2 + (4-6)^2 + (4-6)^2 + (6-6)^2 + (8-6)^2 + (10-6)^2 + (10-6)^2 + (11-6)^2 + (13-6)^2 = 60
Syy = Σ(y_i - ȳ)^2 = (80-107)^2 + (97-107)^2 + (92-107)^2 + (102-107)^2 + (103-107)^2 + (111-107)^2 + (119-107)^2 + (123-107)^2 + (117-107)^2 + (136-107)^2 = 1686
Step 3: Calculate the sum of cross-products ( Sxy).
Sxy = Σ(x_i - x̄)(y_i - ȳ) = (1-6)(80-107) + (3-6)(97-107) + (4-6)(92-107) + (4-6)(102-107) + (6-6)(103-107) + (8-6)(111-107) + (10-6)(119-107) + (10-6)(123-107) + (11-6)(117-107) + (13-6)(136-107) = 416
Step 4: Calculate the slope parameter β1.
β1 = Sxy / Sxx = 416 / 60 = 6.93
Step 5: Calculate the intercept β0.
β0 = ȳ - β1 * x̄ = 107 - 6.93 * 6 = 64.4
Now, we have the linear regression model y = 64.4 + 6.93x.
(f) Construct a 95% confidence interval for the slope parameter β1.
The standard error of the slope parameter SE(β1) can be calculated as sqrt((Syy - β1 * Sxy) / (n - 2) / Sxx). Then, the t-value for a 95% confidence interval with 8 degrees of freedom is 2.306.
Margin of error = t * SE(β1) = 2.306 * sqrt((Syy - β1 * Sxy) / (n - 2) / Sxx).
95% CI for β1: β1 ± Margin of error
(g) Find the correlation coefficient.
r = Sxy / sqrt(Sxx * Syy)
(h) Find the coefficient of determination and interpret the value.
R^2 = Sxy^2 / (Sxx * Syy)
R^2 indicates the proportion of the variation in the dependent variable that is predictable from the independent variable.
(i) ANOVA table:
Source | SS | df | MS | F
Regression | | | |
Residual | | | |
Total | | | |
(j) Use the ANOVA table to test for a significant linear relationship between years of experience and annual sales.
Calculate the mean squares and F-value for testing the significance of the linear relationship. Compare the calculated F-value with the critical F-value to determine if the relationship is significant at a specified level of significance.