Statistics
Rameshpradhan
assign2_sol.pdf
TSTA602 Assignment 2 Solutions
June 14, 2020
Instructions. This assignment has a total of 30 marks and worth 30% of the final grade of TSTA602. The detailed marks allocation are provided at the beginning of each question. You are encouraged to discuss the assignment with your peers, but must write down your own solution. This assignment is due at 5pm on Friday of Week 11 (12th June, 2020). Please submit your assignment on Moodle before the deadline.
Scenario. You, as a property investor, are interested in understanding which factor (or factors) drives the prices of investment properties. A dataset is collected which contains the prices (in thousand dollars, as denoted by apart price) for 50 one- bedroom apartments in city X, their corresponding rents per week (in dollars, as denoted by rent) and the costs to hold each of these properties per week (in dollars, as denoted by cost of property). Following the procedures below to analyse the dataset ’assign2 data.csv’ by using Rstudio. Please only include relevant outputs from Rstudio in your solution and attach the R codes as appendice.
(a). (2 marks) Import the data into Rstudio, draw two scatter plots: apart price versus rent and apart price versus cost.
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Price vs Rent
Rent
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Price vs Cost
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(b). (4 marks) Fit the following two linear models:
Model 1: apart price = b0 + b1 × rent
Model 2: apart price = c0 + c1 × cost
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Write down the equations of the two models with correct coefficients.
Model 1: apart price = 259.7 + 0.80rent
Model 2: apart price = −6.30 + 1.14cost
(c). (4 marks) Comment on the significance of all coefficients obtained from (b) based on the p-values (from the outputs of Rtudio). The significance level is 0.05.
p− values are
< 2e−16 (significant)
< 2e−16 (significant)
0.0765 (not significant)
< 2e−16 (significant)
(d). (6 marks) Produce residual plots for each model in (b), comment on each plot.
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Residual Plot − Rent
Rent
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Residual Plot − Cost
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Linearity: There are no obvious pattern in each residual plots; linear relation- ship seems to be reasonable.
Constant Variance: Both plots show a rough band shape; constant variance assumption did not violate.
(e). (4 marks) Produce normal qq plots for each model in (b), and comment on each plot.
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QQ Plot − Rent
Theoretical Normal Quantiles
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u a ls
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QQ Plot − Cost
Theoretical Normal Quantiles R
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Each figure above shows a rough linear relationship in the qq plot which suggests normality assumption on error are appropriate.
(f). (3 marks) Fit the following linear model:
Model 3: apart price = d0 + d1rent + d2cost
Write down the equation of the model with correct coefficients.
Model 3: apart price = 186.23 + 0.58rent + 0.32cost
(g). (3 marks) Comment on the significance of all coefficients obtained from (f) based on the p-values (from the outputs of Rtudio). The significance level is 0.05.
p− values are
2.35e−10 (significant)
8.11e−11 (significant)
0.00268 (significant)
(h). (2 marks) Compare Model 1 and Model 3, explain which one is better.
From ANOVA, we see that the p-value is 0.002683 < 0.05 which suggests Model 3 is better.
(i). (2 marks) Given rent = 810 and cost = 800, predict prices under Model 1 and Model 3.
Model 3: apart price = 259.7 + 0.80rent = 907.7
Model 3: apart price = 186.23 + 0.58rent + 0.32cost = 912.03
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Marking Guidelines
(a). 1 mark for scatter plot apart price versus rent; 1 mark for scatter plot apart price versus cost.
(b). 1 mark for each correct coefficient.
(c). 1 mark for each correct conclusion.
(d). 1 mark for each residial plot; 1 mark for each comment on linearity; 1 mark for each comment on constant variacne.
(e). 1 mark for each correct normal qq plot; 1 mark for each comment on normality.
(f). 1 mark for each correct coefficient.
(g). 1 mark for each correct conclusion.
(h). 1 mark for using ANOVA; 1 mark for correct conclusion.
(h). 1 mark for each correct prediction.
Note: if no codes are attached, 3 marks are taken away.
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