Question

Advertisement ($'000)   Sales ($'000)

1068    4489

1026    5611

767      3290

885      4113

1156    4883

1146    5425

892      4414

938      5506

769      3346

677      3673

1184    6542

1009    5088

• Construct a scatter plot with this data.
• Do you observe a relationship between both variables?
• Use Excel to fit a linear regression line to the data. What is the fitted regression model?
• What is the slope? What does the slope tell us?Is the slope significant?
• What is the intercept? Is it meaningful?
• What is the value of the regression coefficient,r? What is the value of the coefficient of determination, r^2? What does r^2 tell us?
• Use the model to predict sales and the business spends $950,000 in advertisement. Does the model underestimate or overestimates ales?

Answer 1

Solution:

· Construct a scatter plot with this data.

The required scatter plot is given as below:

· Do you observe a relationship between both variables?

From above scatterplot we observe the positive linear relationship or association between the two variables.

· Use Excel to fit a linear regression line to the data. What is the fitted regression model?

The regression model by using Excel is given as below:

Regression Statistics
Multiple R	0.823733298
R Square	0.678536546
Adjusted R Square	0.6463902
Standard Error	592.7335727
Observations	12
ANOVA
	df	SS	MS	F	Significance F
Regression	1	7415845.784	7415845.784	21.10773517	0.00098869
Residual	10	3513330.883	351333.0883
Total	11	10929176.67
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	-25.16823406	1042.259927	-0.024147752	0.981209823	-2347.468062	2297.131593
Advertisement	4.921595798	1.071235915	4.594315528	0.00098869	2.534733447	7.30845815

The regression equation is given as below:

Sales = -25.16823406 + 4.921595798*Advertisement

· What is the slope? What does the slope tell us?Is the slope significant?

The slope is given as b = 4.921595798. The slope tell us that there is an increment of 4.92 ($’000) in the sales as there is an increment of $1000 on advertisement.

The slope is statistically significant because corresponding p-value is given as 0.00098869 which is less than alpha value 0.05.

· What is the intercept? Is it meaningful?

The value of intercept is given as -25.16823406.

The intercept is not meaningful, because corresponding p-value is given as 0.9812 which is greater than alpha value 0.05.

· What is the value of the regression coefficient, r? What is the value of the coefficient of determination, r^2? What does r^2 tell us?

The value of the regression coefficient r is given as 0.823733298.

The value of the coefficient of determination is given as r^2 = 0.678536546.

This value indicated that about 67.85% of the variation in the dependent variable sale is explained by the independent variable advertisement.

· Use the model to predict sales and the business spends $950,000 in advertisement. Does the model underestimate or overestimates ales?

We have

Sales = -25.16823406 + 4.921595798*Advertisement

Advertisement = 950 thousand dollar ( or $950,000)

Sales = -25.16823406 + 4.921595798*950

Sales = 4650.347774

The value of sales is lies within range, so it is underestimate.