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.
(f) To construct a 95% confidence interval for the slope parameter β1, we can use the formula:
Confidence interval = b1 ± t(alpha/2, n-2) * standard error of b1
First, let's calculate the slope parameter b1. We can use the formula:
b1 = Σ((xi - x̄)(yi - ȳ)) / Σ((xi - x̄)^2)
Where Σ represents the sum for all observations, x̄ is the mean of the years of experience, and ȳ is the mean of annual sales.
x̄ = (1+3+4+4+6+8+10+10+11+13) / 10 = 6.0
ȳ = (80+97+92+102+103+111+119+123+117+136) / 10 = 107.0
b1 = ((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)) / ((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) = 2.85
Next, let's calculate the standard error of b1 using the formula:
standard error of b1 = sqrt((Σ(yi - ŷi)^2) / (n-2)) / sqrt(Σ(xi - x̄)^2)
Where ŷi is the predicted value of yi using the estimated regression equation.
ŷi = b0 + b1xi
Let's first calculate b0 using the formula:
b0 = ȳ - b1x̄ = 107 - 2.85*6 = 89.1
Now we can calculate ŷi and the standard error of b1:
ŷ1 = 89.1 + 2.85*1 = 91.95
ŷ2 = 89.1 + 2.85*3 = 95.8
ŷ3 = 89.1 + 2.85*4 = 97.8
ŷ4 = 89.1 + 2.85*4 = 97.8
ŷ5 = 89.1 + 2.85*6 = 102.6
ŷ6 = 89.1 + 2.85*8 = 107.4
ŷ7 = 89.1 + 2.85*10 = 112.2
ŷ8 = 89.1 + 2.85*10 = 112.2
ŷ9 = 89.1 + 2.85*11 = 115.05
ŷ10 = 89.1 + 2.85*13 = 120.6
Σ(yi - ŷi)^2 = (80-91.95)^2 + (97-95.8)^2 + (92-97.8)^2 + (102-97.8)^2 + (103-102.6)^2 + (111-107.4)^2 + (119-112.2)^2 + (123-112.2)^2 + (117-115.05)^2 + (136-120.6)^2 = 616.995
standard error of b1 = sqrt(616.995 / 8) / sqrt(70) = 0.0349
t(0.025, 8) = 2.306
Therefore, the confidence interval for the slope parameter β1 is:
Confidence interval = 2.85 ± 2.306 * 0.0349 = (2.81, 2.89)
(g) To find the correlation coefficient, we can use the formula:
r = Σ((xi - x̄)(yi - ȳ)) / sqrt(Σ(xi - x̄)^2 * Σ(yi - ȳ)^2)
r = Σ((xi - 6)(yi - 107)) / sqrt(Σ(xi - 6)^2 * Σ(yi - 107)^2)
r = ((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)) / sqrt(((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) * ((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))
r = 316.30 / sqrt(70 * 1204)
r = 0.920
Therefore, the correlation coefficient is 0.920.
(h) To find the coefficient of determination, we can use the formula:
R^2 = 1 - Σ(yi - ŷi)^2 / Σ(yi - ȳ)^2
R^2 = 1 - 616.995 / 2534
R^2 = 0.757
This means that 75.7% of the variability in annual sales can be explained by the linear relationship with years of experience.
(i) The ANOVA table is used to assess the statistical significance of the regression model. It splits the total variability in the dependent variable (annual sales) into different sources.
(j) Use the ANOVA table to test for a significant linear relationship between years of experience and annual sales. This can be done by performing a hypothesis test where the null hypothesis is that there is no significant linear relationship between years of experience and annual sales, and the alternative hypothesis is that there is a significant linear relationship. The ANOVA table will provide the F-statistic and p-value for this hypothesis test.