Homework 1 Regression | Complete Solution

You are planning to visit the UK and would like to purchase an antique grandfather clock while you’re there.  Your friend in the UK collected data on the selling price (pounds sterling) and age (100 - 200 years) of 32 antique grandfather clocks.  Because these clocks were sold at auction, there is also an extra variable - the number of bidders (5 - 15) at the auction.  

You decide to create a linear regression model to help you predict the selling price of various clocks to help you to determine how much to bid if you find a clock that interests you.

You begin by examining the relationship between price (your response variable) and the two potential explanatory variables.  You generate the following information:

•    The correlation between price and the age of the clock is 0.73.  

•    The correlation between price and the number of bidders at the auction is 0.39.

Question 1:
Which of the variables (age or number of bidders) appears to have the strongest linear relationship with price (and hence should be selected as the explanatory variable for your model)?
    a. Age
    b. Number of Bidders
    c. Neither variable has a linear relationship with price

Now, suppose that you use Excel to fit a linear regression model with price as the response variable and age as the explanatory variable.  The output from the regression analysis is shown on the next page.  Use this output to answer the remaining questions.
SUMMARY OUTPUT                            
                                
Regression Statistics                            
Multiple R    0.7302                            
R Square    0.5332                            
Adjusted R Square    0.5177                            
Standard Error    273.0284                            
Observations    32                            
                                
ANOVA                                
     df    SS    MS    F    Significance F            
Regression    1    2554859.011    2554859    34.27    0.0000021            
Residual    30    2236335.207    74544.51                    
Total    31    4791194.219                           
                                
     Coefficients    Standard Error    t Stat    P-value    Lower 95%    Upper 95%    Lower 95.0%    Upper 95.0%
Intercept    -191.66    263.89    -0.7263    0.4733    -730.59    347.27    -730.59    347.27
Age    10.48    1.79    5.8543    0.0000021    6.82    14.13    6.82    14.13
                                
                                
Question 2:
What percentage of the variability in price of clocks can be explained by the age of the clocks (e.g., what percentage of the variability in clocks is explained by this regression model)?  
    a.  53.32%
    b. 10.48%
    c.  73.02%

Question 3:
What is the form of the regression equation for this model?
    a. y = 263.89 + 1.79x
    b. y = -191.66 + 10.48x
    c. y = 10.48 – 191.66x

Question 4:
What is the p-value that you would use to test the hypothesis that age is a useful predictor of the price of antique grandfather clocks?  
    a. 0.4733
    b. 5.8543
    c. 0.0000021

Question 5:

Continuing with question 4, at the 5% significance level, can you conclude that age is a useful predictor of the price of antique grandfather clocks?

    a. Yes.  You would reject the null hypothesis and conclude that age is a useful predictor.
    b. No.  You would fail to reject the null hypothesis and conclude that age is not a useful predictor.

Question 6:

How much would you expect the price of an antique grandfather clock to increase if its age increased by 1 year?

    a. $10.48
    b. $191.66
    c. We can’t tell from the information given.
Question 7:
 Suppose that you’ve got your eye on a clock that is 125 years old.  How much would you expect (predict) the price of this clock to be?
    a.  $1118.34
    b. $1310
    c. $1250

Question 8:
Suppose that you’re also interested in buying an antique grandfather clock that is only 50 years old.  Should you use this model to predict the price of that clock?  
    a. Yes – You would predict the price of the clock to be $332.34
    b. No – This would be extrapolation.

Question 9:
Would it be correct to say that increasing age causes the price of antique grandfather clocks to increase?
    a. Yes – The slope is positive which indicates an increasing trend.
    b. No – Correlation does not imply causation.

Question 10:
Based on the following residual plots, which of the following statements is true?
    a. None of the assumptions appear to be violated.
    b. The normality assumption appears to be violated.
    c. The constant variance assumption appears to be violated.