Auction selling price of antique grandfather clocks

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 Price given Age and Bidders.

library(erikmisc)

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loading 'erikmisc'

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dat_auction <-read_csv("ADA2_CL_03_auction.csv")

Rows: 32 Columns: 3
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
dbl (3): Age, Bidders, Price
ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

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

(1 p) Scatterplot matrix

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.

library(ggplot2)library(GGally)

Registered S3 method overwritten by 'GGally':
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+.gg ggplot2

p <-ggpairs(dat_auction)print(p)

Solution

(1 p) Correlation matrix

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

Solution

(1 p) Plot interpretation

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.

Attaching package: 'gridExtra'

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`geom_smooth()` using formula = 'y ~ x'

Warning: The following aesthetics were dropped during statistical transformation: label
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`geom_smooth()` using formula = 'y ~ x'

Warning: The following aesthetics were dropped during statistical transformation: label
ℹ This can happen when ggplot fails to infer the correct grouping structure in
the data.
ℹ Did you forget to specify a `group` aesthetic or to convert a numerical
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Solution

(2 p) Multiple regression assumptions (assessing model fit)

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.

# fit the simple linear regression modellm_p_a_b <-lm(Price ~ Age + Bidders, data = dat_auction)

Warning in plot.xy(xy.coords(x, y), type = type, ...): "add" is not a graphical
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Solution

From the diagnostic plots above,

(1 p) Added variable plots

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.

Solution

(1 p) Multiple regression hypothesis tests

State the hypothesis test and conclusion for each regression coefficient.

# fit the simple linear regression modellm_p_a_b <-lm(Price ~ Age + Bidders, data = dat_auction)# use summary() to get t-tests of parameters (slope, intercept)summary(lm_p_a_b)

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

Solution

(1 p) Multiple regression interpret coefficients

Interpret the coefficients of the multiple regression model.

Solution

(1 p) Multiple regression \(R^2\)

Interpret the Multiple R-squared value.

Solution

(1 p) Summary

Summarize your findings in one sentence.

Solution

## 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)