StatisticsLinear Regression AnalysisInterpretation and Prediction
Question Four
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.
(i) Give the ANOVA table
(j) Use the ANOVA table to test for a significant linear relationship between years of experience and annual sales.
(i) Calculating the ANOVA table:
Step 1: Calculate the sum of squares total (SST)
SST = Σ(y - ȳ)²
SST = (80 - 108.8)² + (97 - 108.8)² + (92 - 108.8)² + (102 - 108.8)² + (103 - 108.8)² + (111 - 108.8)² + (119 - 108.8)² + (123 - 108.8)² + (117 - 108.8)² + (136 - 108.8)²
SST = 776.4
Step 2: Calculate the sum of squares regression (SSR)
SSR = Σ(ŷ - ȳ)²
First, calculate ŷ (predicted sales):
ŷ = b₀ + b₁x
ŷ = 63.1 + 7.4x
Calculate ŷ for each data point and then SSR:
SSR = (63.1 + 7.4(1) - 108.8)² + (63.1 + 7.4(3) - 108.8)² + ... + (63.1 + 7.4(13) - 108.8)²
SSR = 621.8
Step 3: Calculate the sum of squares error (SSE)
SSE = SST - SSR
SSE = 776.4 - 621.8
SSE = 154.6
Step 4: Calculate the degrees of freedom:
df total = n - 1 = 10 - 1 = 9
df regression = 1
df error = n - 2 = 10 - 2 = 8
Step 5: Calculate the mean squares:
MSR = SSR / df regression
MSR = 621.8 / 1 = 621.8
MSE = SSE / df error
MSE = 154.6 / 8 = 19.325
(ii) Testing for a significant linear relationship:
Step 1: Calculate the F-statistic
F = MSR / MSE
F = 621.8 / 19.325
F ≈ 32.17
Step 2: Look up the critical value of F for α = 0.05 with df regression = 1 and df error = 8
F_critical = 5.318
Step 3: Compare the F-statistic and F_critical
Since F > F_critical, we reject the null hypothesis.
Therefore, there is a significant linear relationship between years of experience and annual sales.