The data include the selling price in pounds sterling at auction of 32 antique grandfather clocks, the age of the clock in years, and the number of people who made a bid. In the sections below, describe the relationship between variables and develop a model for predicting selling
Classes 'spec_tbl_df', 'tbl_df', 'tbl' and 'data.frame': 32 obs. of 3 variables: $ Age : num 127 115 127 150 156 182 156 132 137 113 ... $ Bidders: num 13 12 7 9 6 11 12 10 9 9 ... $ Price : num 1235 1080 845 1522 1047 ... - attr(*, "spec")= .. cols( .. Age = col_double(), .. Bidders = col_double(), .. Price = col_double() .. )
Age Bidders Price Min. :108.0 Min. : 5.000 Min. : 729 1st Qu.:117.0 1st Qu.: 7.000 1st Qu.:1053 Median :140.0 Median : 9.000 Median :1258 Mean :144.9 Mean : 9.531 Mean :1327 3rd Qu.:168.5 3rd Qu.:11.250 3rd Qu.:1561 Max. :194.0 Max. :15.000 Max. :2131
In a scatterplot matrix below interpret the relationship between each pair of variables. If a transformation is suggested by the plot (that is, because there is a curved relationship), also plot the data on the transformed scale and perform the following analysis on the transformed scale. Otherwise indicate that no transformation is necessary.
Below is the correlation matrix and tests for the hypothesis that each correlation is equal to zero. Interpret the hypothesis tests and relate this to the plot that you produced above.
Age Bidders Price Age 1.00 -0.25 0.73 Bidders -0.25 1.00 0.39 Price 0.73 0.39 1.00 n= 32 P Age Bidders Price Age 0.1611 0.0000 Bidders 0.1611 0.0254 Price 0.0000 0.0254
Below are two plots. The first has \(y =\) Price, \(x =\) Age, and colour = Bidders, and the second has \(y =\) Price, \(x =\) Bidders, and colour = Age. Interpret the relationships between all three variables, simultaneously. For example, say how Price relates to Age, then also how Price relates to Bidders conditional on Age being a specific value.
Below the multiple regression is fit. Start by assessing the model assumptions by interpretting what you learn from the first six plots (save the added variable plots for the next question). If assumptions are not met, attempt to address by transforming a variable and restart at the beginning using the new transformed variable.
From the diagnostic plots above,
Use partial regression residual plots (added variable plots) to check for the need for transformations. If linearity is not supported, address and restart at the beginning.
State the hypothesis test and conclusion for each regression coefficient.
Call: lm(formula = Price ~ Age + Bidders, data = dat_auction) Residuals: Min 1Q Median 3Q Max -207.2 -117.8 16.5 102.7 213.5 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -1336.7221 173.3561 -7.711 1.67e-08 *** Age 12.7362 0.9024 14.114 1.60e-14 *** Bidders 85.8151 8.7058 9.857 9.14e-11 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 133.1 on 29 degrees of freedom Multiple R-squared: 0.8927, Adjusted R-squared: 0.8853 F-statistic: 120.7 on 2 and 29 DF, p-value: 8.769e-15
Interpret the coefficients of the multiple regression model.
Interpret the Multiple R-squared value.
Summarize your findings in one sentence.
## Aside: I generally recommend against 3D plots for a variety of reasons. ## However, here's a 3D version of the plot so you can visualize the surface fit in 3D. ## I will point out a feature in this plot that we wouldn't see in other plots ## and it would typically only be detected by careful consideration ## of a "more complicated" second-order model that includes curvature. # library(rgl) # library(car) # scatter3d(Price ~ Age + Bidders, data = dat_auction)