Problems on simple linear regression concepts

The following problems enable students to practice important concepts from Chapter 2, without having to use a computer. Answers are provided below.

1. For simple linear regression, state the four assumptions concerning the probability distribution of the random error term. The mean of the distribution is 0 for each value of X, the variance of the distribution remains constant as X increases, the distribution is normal for each value of X, and the errors are independent.
2. A restaurant owner use the straight-line equation, E(Y) = b₀ + b₁ X, to model the relationship between mean daily costs, E(Y), and customers served, X. To test this theory, a week of data was collected and the results for n = 7 days is shown in the accompanying table followed by an SPSS printout of the regression analysis.

   	Day		   1	   2	   3	   4	   5	   6	   7
   	Costs, Y ($)	1000	2180	2240	2410	2590	2820	3060
   	Customers, X	   0	  60	 120	 133	 143	 175	 175
   
   		R Square = 0.905	Std. Error of the Estimate (s) = 224
   	
   		Unstandardized Coefficients
   
   			B		Std. Error	t	Sig.
   	(Constant)	1192		185		6.44	0.001
   	X		   9.87		  1.43		6.91	0.001
   

   1. State the set of hypotheses that you would test to determine whether costs are positively linearly related to number of customers in the situation above? NH: b₁=0 vs. AH: b₁>0
   2. In the situation above, is there sufficient evidence of a positive linear relationship between costs and number of customers? Use significance level 5%. Yes, since the p-value of the test (0.0005) is less than 0.05.
   3. Give a practical interpretation of the estimate of the slope of the least squares line in the situation above. For every additional customer, we estimate costs to increase by $9.87.
   4. Give a practical interpretation of the estimate of the Y-intercept of the least squares line in the situation above. For a day with no customers (presumably a day the restaurant is closed), we estimate costs (presumably fixed costs) to be $1192.
   5. For the situation above, give a practical interpretation of s, the estimate of the standard deviation of the random error term in the model. We expect about 95% of the observed cost values to lie within $448 of their least squares predicted values.
   6. For the situation above, give a practical interpretation of R², the coefficient of determination for the least squares model. About 90.5% of the total variation in the sample of cost values can be explained by (or attributed to) the linear relationship between cost and number of customers.
3. Is it true that a simple linear regression model allows the E(Y) values (expected or predicted Y-values) to fall around the regression line while the actual values of Y must fall on the line?
4. Is it true that the coefficient of correlation is a useful measure of the linear relationship between two variables?

Last updated: May, 2008

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© 2008, Iain Pardoe, Lundquist College of Business, University of Oregon