Exam Template: Statistical Inference
21014927
Sep21 run RESIT
Instructions to students
You should only use the file Exam_template.Rmd provided on blackboard and you should load this file from your scripts folder / directory.
Save this template as your studentID.Rmd; you will upload this file as your submission. Change the information on line 3 of this file – changing the author information to your student ID. Do not change the authorship to your name.
Ensure that you save your data into your data folder (as discussed in class). You may use the files mypackages.R and helperFunctions.R from blackboard. If you use these files, do not alter them. If you wish to create additional files for custom functions that you have prepared in advance, make sure that you upload these in addition to your .Rmd file and your compiled output file.
Your should knit this file to a document Word format.
Any changes that you make to the data (e.g. variable name changes) should be made entirely within R.
The subsubsections labelled Answer: indicate where you should put in your written Answers. The template also provides blank code chunks for you to complete your Answers; you may choose to add additional chunks if required.
load required libraries / additional files
#install.packages(“tidyverse”) #install.packages(“dplyr”) #install.packages(“ggplot2”) #install.packages(“summarytools”) #install.packages(“psych”) #install.packages(“performance”) #install.packages(“corrplot”) #install.packages(“lsr”) #install.packages(“lessR”) #install.packages(“boot”) Having installed all these packages, we shall call the following libraries:
# load dataset
mydata = read.csv("Jan_2022_Exam_Data.csv")
attach(mydata)The following object is masked from package:ggplot2:
mpg#View(mydata)
names(mydata) [1] "brand" "model" "year" "price" "transmission"
[6] "mileage" "fuelType" "tax" "mpg" "engineSize" Data description
This dataset is part of a larger dataset that has been collected to help to estimate the price of used cars.
It contains the following variables:
- brand (manufacturer)
- model (of car)
- year (of registration of the car)
- price (in GB pounds)
- transmission (type of gearbox)
- mileage (total distance covered by the car)
- fuelType (type of fuel used by the car)
- tax (annual cost of vehicle tax)
- mpg (miles per gallon - a measure of fuel efficiency)
- engineSize (size of the engine in litres)
Question 1: Data Preparation (11 marks)
You are interested in modelling the price of vehicles that have all of the following properties:
- mileage less than 90000
- Manual transmission
- Diesel engine (fuelType)
- Costing less than £300 in annual Vehicle Tax.
Once you have selected the rows of data with these properties, then you must use the last 4 digits of your studentID to select a random sample of 2000 rows of the data to perform the rest of your analysis with.
You should remove any redundant variables (where only one value remains in that variable).
This subset of the data is what you should use for the rest of this assessment.
- Explain what data preparation is required in order for the data in Jan_2022_Exam_Data.csv to be suitable for this analysis.
(4 marks)
Answer:
(aa) Call the libraries of all the installed packages so as to enable all the computions.
(bb) Reading the csv file to the R Studio
- Viewing it to know what types of variables available
(dd) Filtering it using mileage less than 90000, manual transmission, diesel engine (fuelType), and costing less than 300 Euro in annual Vehicle Tax
(ee) Saving the filtered dataset to another name, such as mydata, mydata2, etc, as the case may be.
(ff) Selecting a random sample of 2000 rows from the filtered dataset while setting our seed to be the last four digits of my student’s ID, which is 4927.
(gg) Removing the variables with redundant values
(hh) Converting the variables to factors.
- Implement the required data preparation in the code chunk below:
(7 marks)
Answer:
mydata2 = subset(mydata, transmission=="Manual" & mileage < 90000 & fuelType=="Diesel" & tax < 300, select = -c(fuelType, transmission))
#View(mydata2)
write.csv(mydata2, "DATA3.csv") # Just to be rest assured that the needed datasets are captured.
#Seeting our seed to be:
set.seed(4927)
mydata3 = mydata2 %>% sample_n(2000, replace = FALSE)
View(mydata3)Question 2: Exploratory Data Analysis (22 marks)
Descriptive Statistics
- What descriptive statistics would be appropriate for this dataset? Explain why these are useful in this context.
(2 marks)
Answer:
We shall consider two major descriptive statistics: measure of central tendency and measure of variability. The measure of central tendency is useful to represent the centre point of our dataset, while measure of variability is used to measure the degree of variation in our dataset.
- Produce those descriptive statistics in the code chunk below:
(4 marks)
Answer:
mydata4 = mydata3[ , c(4:8)]
#View(mydata4)
# To find the measure of central tendency, run the following command:
mean(mydata4[,1]); mean(mydata4[, 2]); mean(mydata4[, 3]); mean(mydata4[, 4]); mean(mydata4[, 5])[1] 13933.86[1] 32538.84[1] 84.9725[1] 64.1641[1] 1.78605median(mydata4[,1]); median(mydata4[, 2]); median(mydata4[, 3]); median(mydata4[, 4]); median(mydata4[, 5])[1] 13486[1] 29666.5[1] 125[1] 64.2[1] 2#install.packages("lsr")
library(lsr)
modeOf(mydata4[,1]); modeOf(mydata4[,2]); modeOf(mydata4[,3]); modeOf(mydata4[,4]); modeOf(mydata4[,5])[1] 16000 11000[1] 10[1] 145[1] 74.3[1] 2# To find the measure of variability, run the following command:
range(mydata4[,1]); range(mydata4[, 2]); range(mydata4[, 3]); range(mydata4[, 4]); range(mydata4[, 5])[1] 2395 32995[1] 5 89936[1] 0 260[1] 30.1 88.3[1] 0.0 2.2IQR(mydata4[,1]); IQR(mydata4[, 2]); IQR(mydata4[, 3]); IQR(mydata4[, 4]); IQR(mydata4[, 5])[1] 6423.75[1] 29308[1] 125[1] 13.275[1] 0.5var(mydata4[,1]); var(mydata4[, 2]); var(mydata4[, 3]); var(mydata4[, 4]); var(mydata4[, 5])[1] 22931232[1] 445497989[1] 4263.894[1] 101.5388[1] 0.06297188sd(mydata4[,1]); sd(mydata4[, 2]); sd(mydata4[, 3]); sd(mydata4[, 4]); sd(mydata4[, 5])[1] 4788.657[1] 21106.82[1] 65.2985[1] 10.07665[1] 0.250942- What have those descriptive statistics told you – and how does this inform the analysis that you would undertake on this data or any additional data cleaning requirements?
(4 marks)
Answer:
In the analysis using measure of central tendency, we observe that the average price (in GB pounds), mileage (total distance covered by the car), annual cost of vehicle tax, miles per gallon, and size of the engine (in litres) are 13933.86, 32538.84,84.97,64.16 and 1.79 respectively, while their median values are 13486, 29666.5, 125, 64.2, and 2. Not only these but also, the modes are 16000 and 11000 for only price (this means that there are bimodal values for the price of the car), 10, 145, 74.3, and 2 are the mode for mileage, tax, mpg, and engineSize. All these talk about measure of central tendency.
Come to think of the measure of variability as used here, the ranges for price (in GB pounds), mileage (total distance covered by the car), annual cost of vehicle tax, miles per gallon, and size of the engine (in litres) are (2395 - 32995), (5 - 89936), (0 - 260), (30.1 - 88.3), and (0.0 - 2.2) respectively. We got 6423.75, 29308, 125, 13.275, and 0.5 as interquartile range values for price (in GB pounds), mileage (total distance covered by the car), annual cost of vehicle tax, miles per gallon, and size of the engine (in litres). Also, the values for variance and standard deviation for each of these variables are (22931232, 4788.657), (445497989, 21106.82), (4263.894, 65.2985), (101.5388, 10.07665), and (0.06297, 0.25094).
Exploratory Graphs
- What exploratory graphs would be appropriate for this dataset? Explain why these are useful in this context.
(2 marks)
Answer:
We shall consider BOX-WHISKER PLOT and HISTOGRAM. BOX-WHISKER PLOT helps to determine the nature of the distribution in the dataset by checking the normality assumption, while HISTOGRAM is the best choice for visualizing central tendency of data.
- Now produce those exploratory graphs in the code chunk below:
(4 marks)
Answer:
# To obtain the box plot, run the command below:
win.graph(width=4, height=4, pointsize=8)
boxplot(mydata4, main="Box Plot", col=c(2:6), col.main="purple", pch=19, las=1, sub="Figure I", col.sub="coral3")
# To obtain the histogram, run the commands below:
win.graph(width=4, height=4, pointsize=8)
h=hist(mydata4[ , 1], main="Histigram for the Price of Vehicle", sub="Figire II", col.sub="green4",col=2:9, xlab="Price (in GB Pounds)", col.main="brown")
win.graph(width=4, height=4, pointsize=8)
h=hist(mydata4[ , 2], main="Histigram for Mileage", sub="Figire III", col.sub="green4",col=2:9, xlab="Mileage (less than 90000)", col.main="brown")
win.graph(width=4, height=4, pointsize=8)
h=hist(mydata4[ , 3], main="Histigram for Tax", sub="Figire IV", col.sub="green4",col=2:9, xlab="Annual Cost of Vehicle Tax", col.main="brown")
win.graph(width=4, height=4, pointsize=8)
h=hist(mydata4[ , 4], main="Histigram for Miles Per Gallon", sub="Figire V", col.sub="green4",col=2:9, xlab="mpg (Miles per Gallon - a measure of fuel efficiency)", col.main="brown")
win.graph(width=4, height=4, pointsize=8)
windows()
win.graph(width=4, height=4, pointsize=8)
h=hist(mydata4[ , 5], main="Histigram for Size of the Engine (in litres)", sub="Figire VI", col.sub="green4",col=2:9, xlab="engine Size", col.main="brown")
- Interpret these exploratory graphs. How do these graphs inform your subsequent analysis? (4 marks)
Answer:
In Box Plot, it appears that only mileage is normally distributed while the rest are far away from normality. As a result, it is advisable to go for test of normality to be double sure and also perform some statistical analysis to correct the abnorlity in the dataset.
In the Histogram, the graph shows for the price is not normal, also the graph for the mileage is skewed to the right, the one for tax shows some evidence of bimodal as depicted in the descriptive statistics of measure of central tendency, the one for mpg shows a kind of normality, while the last histogram shows that the engine Size is completely abnormal.
Correlations
- What linear correlations are present within this data? (2 marks)
Answer:
cor(mydata4) price mileage tax mpg engineSize
price 1.0000000 -0.6690729 0.5231055 -0.5381233 0.3234885
mileage -0.6690729 1.0000000 -0.3696081 0.1870735 0.1001805
tax 0.5231055 -0.3696081 1.0000000 -0.5990075 0.2571670
mpg -0.5381233 0.1870735 -0.5990075 1.0000000 -0.5261717
engineSize 0.3234885 0.1001805 0.2571670 -0.5261717 1.0000000Comparing price with mileage, the value indicates that there is a negative correlation of approximately 67%, which means that when price increases, mileage decreases and, vice-versa. Also, the correlation between price and tax shows that there is a positive correlation between them to the tone of 52%, which means that when price increases, tax also increases, and vice-versa. Also, the correlation between price and mpg shows that there is a negative correlation between them (-54%), which means that as price increases, mpg decreases , and vice-versa. The last aspect of comparison show that the correlation between price and enginesize is approximately 32%, which means that as the price increases, the enginesize also increases, and vice-versa.
Question 3: Bivariate relationship (14 marks)
- Which of the potential explanatory variables has the strongest linear relationship with the dependent variable? (1 mark)
Answer:
The potential explanatory variables are mileage and tax, while between the two, the one with the strongest linear relatiosnhip is mileage, which is negatively linearly related with approximately 67% (-0.67). This value is negatively strongest among all.
- Create a linear model to model this relationship. (2 marks)
Answer:
model_1 = lm(price ~ mileage, data = mydata4)
summary(model_1)
Call:
lm(formula = price ~ mileage, data = mydata4)
Residuals:
Min 1Q Median 3Q Max
-9014.3 -2466.4 7.5 2186.2 14214.7
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 18873.168634 146.294585 129.01 <0.0000000000000002 ***
mileage -0.151797 0.003772 -40.24 <0.0000000000000002 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 3560 on 1998 degrees of freedom
Multiple R-squared: 0.4477, Adjusted R-squared: 0.4474
F-statistic: 1619 on 1 and 1998 DF, p-value: < 0.00000000000000022#install.packages("lessR")
library("lessR")
Plot(mileage, price, fit="lm", main="Scatter Plot showing the Relationship between Price and Mileage", xlab="Mileage", ylab="Price of the Vehicle", col.main="blue3", col.lab="red3", enhance=T, sub="Figure VII")[Ellipse with Murdoch and Chow's function ellipse from their ellipse package]
>>> Suggestions
Plot(mileage, price, color="red") # exterior edge color of points
Plot(mileage, price, out_cut=.10) # label top 10% from center as outliers
>>> Pearson's product-moment correlation
Number of paired values with neither missing, n = 52533
Sample Correlation of mileage and price: r = -0.466
Hypothesis Test of 0 Correlation: t = -120.820, df = 52531, p-value = 0.000
95% Confidence Interval for Correlation: -0.473 to -0.460 >>> Outlier analysis with Mahalanobis Distance
MD ID
----- -----
214.05 9823
185.32 45614
172.81 49459
159.75 39420
146.96 4784
140.50 45801
139.85 39423
135.24 48152
133.71 45317
133.58 48548
132.96 50759
131.80 47279
131.00 48236
129.56 2256
128.41 4180
123.23 3368
121.50 48088
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120.25 52319
113.99 14307
112.35 45625
111.50 1647
107.34 10469
103.75 44174
103.49 42062
102.20 45638
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101.67 3360
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100.45 42044
94.74 7446
92.52 4743
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85.49 20088
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83.47 46549
81.70 3940
81.33 48574
75.93 4955
75.16 41395
74.31 48160
73.35 52475
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66.15 10226
65.77 43502
65.50 43624
63.97 43393
62.48 18982
62.01 4401
60.19 45077
58.56 50996
58.01 40101
57.85 48656
57.68 16031
56.63 20083
54.67 3712
54.46 49696
52.89 8631
51.88 38553
51.11 5304
50.63 46543
50.59 38396
49.99 44194
49.70 4392
48.39 48087
48.27 50797
47.28 9691
46.85 50222
46.19 51343
45.20 20673
45.02 51355
44.85 20223
44.67 46727
44.58 5141
44.45 3377
44.10 43012
43.82 15445
43.76 40380
43.73 13578
43.68 50793
43.31 43468
42.19 48690
42.15 12482
41.56 20365
40.99 52063
40.90 4672
40.50 48085
40.49 4330
40.48 48086
40.28 19885
39.71 50667
39.51 46550
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39.00 19594
38.95 7188
38.73 51432
38.47 49981
38.39 4711
37.67 11390
37.63 4055
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37.25 1108
37.20 45027
37.16 697
36.69 8421
36.52 49598
36.08 52125
36.02 37700
35.71 7405
35.38 38389
34.94 317
34.93 20318
34.77 20469
34.45 20422
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34.11 48097
33.76 10113
33.44 50518
33.23 5981
33.10 7847
32.76 47259
32.76 16546
32.69 18116
32.59 41324
32.26 42578
32.15 43273
31.88 45991
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31.82 10073
31.80 45654
31.40 16784
31.33 19860
31.26 49578
31.26 49614
31.25 47439
31.10 39192
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31.05 35958
30.91 47914
30.84 41469
30.76 37159
30.58 13271
30.53 17781
30.52 45427
30.44 45076
30.32 9435
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30.21 13607
30.17 13211
30.06 10910
29.92 17115
29.90 20364
29.60 50550
29.59 1839
29.39 38103
29.37 5476
29.27 44681
29.27 49590
29.18 36004
29.15 4946
29.14 20545
29.12 42049
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28.89 18063
28.78 42726
28.63 45595
28.61 48894
28.59 38397
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28.37 19110
28.13 12373
28.03 48316
27.92 49663
27.92 48352
27.71 50566
27.62 11478
27.60 37136
27.42 20080
27.37 14350
27.20 9865
27.19 45718
27.17 37521
27.12 48693
27.01 46933
27.00 47865
27.00 35817
26.98 7442
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26.90 36005
26.84 50567
26.79 15842
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26.70 49577
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26.50 19812
26.45 35965
26.45 20430
26.43 8639
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26.15 35952
26.05 35044
26.05 19095
25.82 51422
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25.53 19575
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25.34 12298
25.34 17027
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25.26 10554
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25.06 20630
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25.03 50661
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24.72 17550
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24.62 36007
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24.53 12627
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24.47 12665
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24.29 39491
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24.01 35029
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24.01 35999
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23.93 21383
23.85 38144
23.83 8297
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23.78 19987
23.72 43845
23.67 11029
23.66 10340
23.61 16117
23.58 48256
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23.54 38400
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22.00 8557
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22.00 13278
21.99 15183
21.91 9962
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21.77 11388
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21.73 2365
21.66 10099
21.63 9544
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21.55 19916
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21.53 17654
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21.50 17528
21.46 8268
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21.21 49021
21.17 48543
21.17 8509
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21.13 52303
21.02 18010
21.01 11305
21.01 20089
20.99 47148
20.95 20541
20.94 9897
20.94 51433
20.88 12765
20.88 49005
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20.82 17415
20.81 9545
20.71 47061
20.68 52453
20.60 4593
20.60 10428
20.58 37754
20.55 50796
20.50 11403
20.45 5493
20.40 12023
20.39 51419
20.38 5877
20.35 13736
20.35 50267
20.30 4653
20.28 9898
20.28 17275
20.27 20543
20.26 13212
20.25 9884
20.23 9277
20.23 39428
20.20 15444
20.15 19551
20.14 9517
20.07 10598
20.06 47603
20.03 8251
20.03 8281
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19.96 42471
19.95 20648
19.94 41901
19.91 51279
19.91 38158
19.87 4956
19.87 11544
19.74 40453
19.73 11043
19.71 35973
19.71 8561
19.68 9372
19.67 4477
19.67 49029
19.66 38354
19.64 12695
19.64 20652
19.60 37856
19.60 49485
19.59 44405
19.58 10650
19.56 9941
19.55 43485
19.54 8510
19.50 2739
19.48 19893
19.45 41072
19.42 3998
19.42 4049
19.42 5000
19.37 47440
19.34 21329
19.30 42801
19.29 19814
19.29 11088
19.29 47869
19.26 9728
19.22 42873
19.19 48983
19.15 50545
19.14 51271
19.14 47803
19.12 5042
19.08 20085
19.08 19615
19.07 6471
19.05 4904
19.05 50529
19.04 37703
19.03 49589
19.03 49604
19.03 46528
19.01 6087
19.00 40089
18.95 20405
18.95 48100
18.94 50530
18.92 45993
18.92 47488
18.90 52009
18.88 38119
18.88 21393
18.87 51185
18.82 12340
18.81 7884
18.78 16550
18.75 45611
18.73 16257
18.69 51575
18.69 11873
18.68 47596
18.67 20421
18.66 48070
18.64 9245
18.61 47129
18.60 18896
18.58 37966
18.53 46681
18.51 52181
18.51 15466
18.49 34952
18.49 35979
18.48 2783
18.44 45592
18.44 7297
18.43 18419
18.41 16725
18.35 20587
18.31 42606
18.28 47151
18.26 10064
18.23 17407
18.23 8248
18.22 19094
18.20 7476
18.19 14586
18.17 7751
18.15 37974
18.08 48904
18.05 50273
17.98 45194
17.97 10225
17.94 9272
17.91 20720
17.90 41347
17.80 11582
17.78 34930
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17.77 9279
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17.72 51423
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17.72 49594
17.72 19237
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17.70 8514
17.70 37730
17.69 12440
17.67 4430
17.66 9259
17.66 19862
17.63 7690
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17.62 7705
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17.57 49879
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17.55 38917
17.53 13786
17.49 16848
17.49 10030
17.48 38893
17.47 41689
17.47 42371
17.45 14214
17.44 13127
17.43 19972
17.42 14818
17.42 44142
17.40 51022
17.37 38975
17.36 9273
17.33 12019
17.32 36000
17.30 20461
17.29 17953
17.29 38107
17.27 2647
17.24 17311
17.24 452
17.23 20470
17.22 16737
17.22 15195
17.21 16804
17.21 300
17.19 38897
17.17 8232
17.17 20406
17.17 19853
17.16 2774
17.15 14579
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6.00 13852
6.00 13408
5.99 27753
5.99 15880
... ... - Explain and interpret the model: (3 marks)
Answer:
The estimated coefficient of mileage (-0.151797) shows that there is a negative correlation between price and mileage, which equally confirms the result obtained previously under correlation matrix. The p-values for mileage (0.0000000) shows that the null hypothesis is rejected indicating that the model is statistically significant, when we set our level of significance to be 0.05 respectively. The value of Adjusted R-squared (0.4474) shows that only 44% variations in price can be explained by mileage, while the rest cannot be explained. This suggests that there are still other variables that are supposed to be included in the model, but not captured here.
- Comment on the performance of this model, including comments on overall model fit and the validity of model assumptions. Include any additional code required for you to make these comments in the code chunk below. (4 marks)
Answer:
library(performance)
model_performance(model_1)# Indices of model performance
AIC | BIC | R2 | R2 (adj.) | RMSE | Sigma
---------------------------------------------------------------
38389.597 | 38406.400 | 0.448 | 0.447 | 3558.024 | 3559.805shapiro.test(mydata4[, 1])
Shapiro-Wilk normality test
data: mydata4[, 1]
W = 0.97235, p-value < 0.00000000000000022shapiro.test(mydata4[, 2])
Shapiro-Wilk normality test
data: mydata4[, 2]
W = 0.96004, p-value < 0.00000000000000022The estimated coefficient of mileage (-0.151797) shows that there is a negative correlation between price and mileage, which equally confirms the result obtained previously under correlation matrix. The p-values for mileage (0.0000000) shows that the null hypothesis is rejected indicating that the model is statistically significant, when we set our level of significance to be 0.05 respectively. The value of Adjusted R-squared (0.4474) shows that only 44% variations in price can be explained by mileage, while the rest cannot be explained. This suggests that there are still other variables that are supposed to be included in the model, but not captured here.
However, the result for the model performance shows that R2 is 0.45 (by approximation) indicating that thereare only 45% variations in price that can be explained by mileage. The results obtained from using Shapiro-wilk test for normality shaow that none of the two variables are statistically normally distributed.
Bootstrap
- Use bootstrapping on this model to obtain a 95% confidence interval of the estimate of the slope parameter. (4 marks)
Answer:
model_2 = step(model_1)Start: AIC=32711.84
price ~ mileage
Df Sum of Sq RSS AIC
<none> 25319073503 32712
- mileage 1 20520460250 45839533752 33897#Set up the bootstrap as follows:
# function to obtain R-Squared from the data:
set.seed(4927)
library(boot)
#define function to calculate R-squared
rsq_function = function(formula, data, indices) {
d = data[indices,] #allows boot to select sample
fit = lm(formula, data=d) #fit regression model
return(summary(fit)$r.square) #return R-squared of model
}
#perform bootstrapping with 2000 replications
reps = boot(data=mydata4, statistic=rsq_function, R=2000, formula=price~mileage)
#view results of boostrapping
reps
ORDINARY NONPARAMETRIC BOOTSTRAP
Call:
boot(data = mydata4, statistic = rsq_function, R = 2000, formula = price ~
mileage)
Bootstrap Statistics :
original bias std. error
t1* 0.4476586 0.00004506277 0.01378635#plot(reps)
#We can also use the following code to calculate the 95% confidence interval for the estimated R-squared of the model:
#calculate adjusted bootstrap percentile (BCa) interval
boot.ci(reps, type="bca")BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
Based on 2000 bootstrap replicates
CALL :
boot.ci(boot.out = reps, type = "bca")
Intervals :
Level BCa
95% ( 0.4217, 0.4761 )
Calculations and Intervals on Original ScaleQuestion 4: Multivariable relationship (10 marks)
Create a model with all of the appropriate remaining explanatory variables included: The remaining explanatory variables are tax, mpg, and engineSize:
model_3 = lm(price ~ tax + + mpg + engineSize - 1, data = mydata4)
summary(model_3)
Call:
lm(formula = price ~ tax + +mpg + engineSize - 1, data = mydata4)
Residuals:
Min 1Q Median 3Q Max
-16590.4 -2286.1 71.5 2314.2 15820.4
Coefficients:
Estimate Std. Error t value Pr(>|t|)
tax 34.103 1.518 22.460 <0.0000000000000002 ***
mpg 2.600 5.800 0.448 0.654
engineSize 6044.215 242.987 24.875 <0.0000000000000002 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 4015 on 1997 degrees of freedom
Multiple R-squared: 0.9259, Adjusted R-squared: 0.9257
F-statistic: 8313 on 3 and 1997 DF, p-value: < 0.00000000000000022anova(model_3)Analysis of Variance Table
Response: price
Df Sum Sq Mean Sq F value Pr(>F)
tax 1 316268751564 316268751564 19622.20 < 0.00000000000000022 ***
mpg 1 75715160476 75715160476 4697.58 < 0.00000000000000022 ***
engineSize 1 9972929338 9972929338 618.75 < 0.00000000000000022 ***
Residuals 1997 32187462035 16117908
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1- Explain and interpret the model: (4 marks)
Answer:
From the results of the model, it is pobvious that NOT all the three remaining explanatory variables are statistically significant at 0.05 level of significance; only two of them (tax and engineSize) are significant (p-values are less than 0.05) while mpg is not significant (p-value > 0.05). However, the results for the multiple R-squared and Adjusted multiple R-squared (0.9259 and 0.9257) tend towards 0.93 by approximation, meaning that 93% variations in the price of the vehicle can be jointly explained by these explanatory variables while only 7% cannot be explained.
However, the results of the analysis of variance show that all the three of them are statistically significant which is contrary to what we have in the model estimation. There is a suspect of a particular problem in the datasets.
- Comment on the performance of this model, including comments on overall model fit and the validity of model assumptions. Include any additional code required for you to make these comments in the code chunk below.
(4 marks)
Answer:
library(performance)
model_performance(model_3)# Indices of model performance
AIC | BIC | R2 | R2 (adj.) | RMSE | Sigma
---------------------------------------------------------------
38871.635 | 38894.038 | 0.926 | 0.926 | 4011.699 | 4014.711# Using Normal Q-Q Plot to detect the normality assumption:
res=resid(model_3)
qqnorm(res, col=3,lwd=1, pch=19, col.main="blue", col.lab="purple")
## Breusch-Pagan Test for Heteroscedasticity Assumption
# H0: Homoscedasticity is present verse H1: Heteroscedasticity is present
library(lmtest)Loading required package: zoo
Attaching package: 'zoo'The following objects are masked from 'package:base':
as.Date, as.Date.numericbptest(model_3, studentize=FALSE)
Breusch-Pagan test
data: model_3
BP = 464.56, df = 2, p-value < 0.00000000000000022From the model performance results, it is established that multiple R-squared is approximately 93% which shows that 93% variations in the price of the vehicle can be explained by all the remaining explanatory variables. From the Normal Q-Q plot, the behaviour of the model is somehow normal as a straight line is almost formed. From the results obtained from Breusch-Pagan test, it is established, since the p-value (0.0000) is less than 0.05, that the null hypothesis should be rejected, indicating that the datasets do not have equal variances.
- What general concerns do you have regarding this model? (2 marks)
Answer:
So far, we observe that the datasets used for this analysis have some kinds of issues that need to be addresed before any conclusions can be made. The high value of R-squared shows that the model is fit and adequate but at the same time, repports from the normality and homoscedasticity do not support this model adequacy.
Question 5: Model simplification (8 marks)
- What approaches for model simplification would you consider implementing and why? (4 marks)
Answer:
Though there are about three model simplification approaches (maximal model, minimum adequate model, and null model), we will be selecting maximal model. This is because of the following reasons: - We are dealing with multiple regression cases - It is more efficient - It is more reliable - It is more dependable
- What are the potential advantages of simplifying a model? (2 marks)
Answer:
- Efficiency
- Dependability
- Reliability
- What are the potential disadvantages of simplifying a model? (2 marks)
Answer:
- High cost
- Time consuming
- Apathy
Question 6: Reporting (35 marks)
A client is looking to purchase a used Skoda Superb (registration year either 2018 or 2019, manual transmission, diesel engine) and wants to understand what factors influence the expected price of a used car, (and how they influence the price).
Write a short report of 300-500 words for the client.
Furthermore, include an explanation as to which statistical model you would recommend, and why you have selected that statistical model.
Comment on any suggestions for alterations to the statistical model that would be appropriate to consider.
Highlight what may or may not be directly transferable from the scenario analysed in Questions 1 to 5.
Answer:
A client who wants to buy to a used Skoda Superb (registration year either 2018 or 2019, manual transmission, diesel engine) car would need to understand that some basic factors such as brand, model, fuel type, and tax would influence the expected price of a used car. In the first instance, someone who needs Audi will know that its price will surely be different (either less or more than) from the price of another brand, say BMW. Also, the model of the car will determine its price. It is expected that Audi with model A4 will be different in price while comparing it with BWM of model X3. The type of the fuel a car uses determines the price of such a car. In this scenario, you cannot expect a car using diesel to be at the same price with that of a car using petrol. Also, annual cost of vehicle tax influences its price.
I would like to recommend that a multiple linear regression model of price connecting with brand, model, fuel type, and tax be used because of the roles played by all these explanatory variables in explaining the price of this used car.
However, it is a good idea to suggest that some modifications should be made to our previous models such that the new model can be captured or incorporated into our analysis.
From the scenario analyzed in Questions 1 - 5, mileage less tha 90000 may not be transferable while manual transmission, diesel engine, and cost less than 300 Euro are transferable.
Session Information
Do not edit this part. Make sure that you compile your document so that the information about your session (including software / package versions) is included in your submission.
sessionInfo()R version 4.2.0 (2022-04-22 ucrt)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 22000)
Matrix products: default
locale:
[1] LC_COLLATE=English_United States.utf8
[2] LC_CTYPE=English_United States.utf8
[3] LC_MONETARY=English_United States.utf8
[4] LC_NUMERIC=C
[5] LC_TIME=English_United States.utf8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] lmtest_0.9-40 zoo_1.8-10 boot_1.3-28 lessR_4.2.2
[5] lsr_0.5.2 corrplot_0.92 performance_0.9.1 psych_2.2.5
[9] summarytools_1.0.1 forcats_0.5.1 stringr_1.4.0 dplyr_1.0.9
[13] purrr_0.3.4 readr_2.1.2 tidyr_1.2.0 tibble_3.1.7
[17] ggplot2_3.3.6 tidyverse_1.3.1
loaded via a namespace (and not attached):
[1] nlme_3.1-158 matrixStats_0.62.0 fs_1.5.2
[4] lubridate_1.8.0 RColorBrewer_1.1-3 insight_0.18.0
[7] httr_1.4.3 tools_4.2.0 backports_1.4.1
[10] bslib_0.3.1 utf8_1.2.2 R6_2.5.1
[13] DBI_1.1.3 colorspace_2.0-3 withr_2.5.0
[16] tidyselect_1.1.2 mnormt_2.1.0 compiler_4.2.0
[19] cli_3.3.0 rvest_1.0.2 xml2_1.3.3
[22] sass_0.4.1 scales_1.2.0 checkmate_2.1.0
[25] DEoptimR_1.0-11 robustbase_0.95-0 digest_0.6.29
[28] rmarkdown_2.14 jpeg_0.1-9 base64enc_0.1-3
[31] pkgconfig_2.0.3 htmltools_0.5.2 highr_0.9
[34] dbplyr_2.2.1 fastmap_1.1.0 rlang_1.0.4
[37] readxl_1.4.0 rstudioapi_0.13 pryr_0.1.5
[40] jquerylib_0.1.4 generics_0.1.3 jsonlite_1.8.0
[43] zip_2.2.0 magrittr_2.0.3 rapportools_1.1
[46] leaps_3.1 interp_1.1-3 Rcpp_1.0.9
[49] munsell_0.5.0 fansi_1.0.3 lifecycle_1.0.1
[52] stringi_1.7.8 yaml_2.3.5 MASS_7.3-57
[55] plyr_1.8.7 grid_4.2.0 parallel_4.2.0
[58] crayon_1.5.1 deldir_1.0-6 lattice_0.20-45
[61] haven_2.5.0 pander_0.6.5 hms_1.1.1
[64] magick_2.7.3 knitr_1.39 pillar_1.7.0
[67] tcltk_4.2.0 reshape2_1.4.4 codetools_0.2-18
[70] reprex_2.0.1 glue_1.6.2 evaluate_0.15
[73] latticeExtra_0.6-30 modelr_0.1.8 png_0.1-7
[76] vctrs_0.4.1 tzdb_0.3.0 cellranger_1.1.0
[79] gtable_0.3.0 assertthat_0.2.1 xfun_0.31
[82] openxlsx_4.2.5 broom_1.0.0 viridisLite_0.4.0
[85] ellipse_0.4.3 ellipsis_0.3.2