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.
(a) Draw a scatter diagram. Does a linear relationship between x and y seem appropriate
(b) Estimate the simple linear regression line. Interpret the parameters in the model
(c) What practical use could be made of this equation?
(d) Use the estimated regression equation to predict annual sales for a sales man with 9 years
of experience
(e) At the 5% level of significance would you conclude that there is a linear relationship
between x and y ?
(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.
(a) To draw a scatter diagram, plot the Years of Experience (x) on the x-axis and Annual Sales (y) on the y-axis. The scatter diagram will help determine if a linear relationship between x and y seems appropriate.
(b) To estimate the simple linear regression line, use the least squares method to find the equation of the line that best fits the data. The simple linear regression equation is of the form y = mx + b, where m is the slope and b is the intercept. Interpret the parameters in the model as follows:
- The slope (m) indicates the rate of change in annual sales for every one unit increase in years of experience.
- The intercept (b) is the value of annual sales when years of experience is 0.
(c) The practical use of this equation is to predict annual sales based on the years of experience of a salesperson. It can help in forecasting sales performance and setting targets for individual salespeople.
(d) Use the estimated regression equation to predict annual sales for a salesperson with 9 years of experience by plugging in x=9 into the equation.
(e) To test if there is a linear relationship between x and y at the 5% level of significance, calculate the p-value for the regression analysis. If the p-value is less than 0.05, you would conclude that there is a linear relationship between x and y.
(f) To construct a 95% confidence interval for the slope parameter β1, use the t-distribution and the standard error of the regression coefficient.
(g) To find the correlation coefficient, calculate the Pearson correlation coefficient which measures the strength and direction of the linear relationship between x and y.
(h) Find the coefficient of determination (R-squared) by squaring the correlation coefficient. Interpret the value as the proportion of the variance in annual sales that is explained by years of experience.
(i) The ANOVA table is used to analyze the variance in the data and determine if there is a significant linear relationship between years of experience and annual sales. The table includes the sum of squares, degrees of freedom, mean squares, F-statistic, and p-value.
(j) To test for a significant linear relationship between years of experience and annual sales using the ANOVA table, look at the p-value associated with the F-statistic. If the p-value is less than 0.05, you would conclude that there is a significant linear relationship between years of experience and annual sales.