1 Reproduce the results of this document and fill in the blanks.

Mr. Perera who lives in Soratha Mawatha - Wijerama wants to sell his house. He wants to decide a price for his house to list it in the market. He believes that the size of the house is one likely determinant of price. He asked from 200 homes in the neighbourhood “what price should you ask for your home when you put it on the market?” and square feet. Collected data are available to you in the file house.csv.

  1. Fit a suitable simple linear regression model for the data and carry out complete diagnostic tests for the model to check the validity of the regression assumptions.

  2. Comment on each and every output received in part (1). You should clearly states corresponding hypotheses as appropriate.

  3. What price should you suggest for Mr. Perera’s home? Square feet of Mr Perera’s House is 2000.

2 Load packages

library(tidyverse)
library(broom)

3 Load dataset

house <- read_csv("house.csv")
attach(house)

3.1 Overview of the dataset

glimpse(house)
Rows: 200
Columns: 2
$ price     <dbl> 68.92162, 63.97395, 69.51606, 68.86582, 69.24784, 68.54808,…
$ houseSize <dbl> 2503.489, 2362.630, 2522.676, 2491.663, 2511.292, 2476.032,…
summary(house)
     price         houseSize   
 Min.   :59.89   Min.   :2208  
 1st Qu.:65.79   1st Qu.:2437  
 Median :67.28   Median :2504  
 Mean   :67.45   Mean   :2502  
 3rd Qu.:69.27   3rd Qu.:2579  
 Max.   :74.02   Max.   :2769

4 Visualise data

qplot(data=house, x=price, geom=c("histogram"))+
  geom_histogram(color="black", fill="#d95f02")

Interpretation:

qplot(data=house, x=houseSize, geom=c("histogram"))+
  geom_histogram(color="black", fill="#1b9e77")

Interpretation:

qplot(data=house, y=price, x=houseSize) + 
  xlab("House size (sqft)") + 
  ylab("Price") + 
  labs(title="Price vs House size (sqft)")

cor(house$price, house$houseSize)
[1] 0.9331807

Interpretation:

5 Fit a regression model

slrfit <- lm(price ~ houseSize, data=house)
slrfit

Call:
lm(formula = price ~ houseSize, data = house)

Coefficients:
(Intercept)    houseSize  
    5.15772      0.02489  
qplot(data=house, y=price, x=houseSize) + 
  xlab("House size (sqft)") + 
  ylab("Price") + 
  labs(title="Price vs House size (sqft)") + 
   geom_abline(intercept = 5.15772, slope = 0.02489, colour="forestgreen", lwd=2)

Write the fitted regression model:

6 Model adequacy checking

6.1 Compute fitted values and residuals

slrfit_values <- augment(slrfit)
slrfit_values
# A tibble: 200 x 8
   price houseSize .fitted   .resid .std.resid    .hat .sigma     .cooksd
   <dbl>     <dbl>   <dbl>    <dbl>      <dbl>   <dbl>  <dbl>       <dbl>
 1  68.9     2503.    67.5  1.45       1.48    0.00500  0.975 0.00552    
 2  64.0     2363.    64.0  0.00442    0.00455 0.0145   0.981 0.000000152
 3  69.5     2523.    68.0  1.56       1.60    0.00520  0.974 0.00670    
 4  68.9     2492.    67.2  1.68       1.73    0.00506  0.973 0.00757    
 5  69.2     2511.    67.7  1.58       1.62    0.00504  0.974 0.00662    
 6  68.5     2476.    66.8  1.76       1.80    0.00534  0.973 0.00869    
 7  64.8     2471.    66.7 -1.86      -1.91    0.00547  0.972 0.0100     
 8  71.6     2634.    70.7  0.840      0.864   0.0134   0.979 0.00508    
 9  70.2     2581.    69.4  0.808      0.830   0.00799  0.979 0.00277    
10  65.9     2424.    65.5  0.414      0.425   0.00796  0.980 0.000725   
# … with 190 more rows

6.2 Residuals vs fitted values

Write the code to plot residual vs fitted values

qplot(data=slrfit_values, y=.resid, x=.fitted)

6.3 Normality test of residuals

qplot(data=slrfit_values, x=.resid, geom=c("histogram"))+
  geom_histogram(color="black", fill="lightblue")

ggplot(slrfit_values, 
       aes(sample=.resid))+
  stat_qq() + stat_qq_line()+labs(x="Theoretical Quantiles", y="Sample Quantiles")

shapiro.test(slrfit_values$.resid)

    Shapiro-Wilk normality test

data:  slrfit_values$.resid
W = 0.99755, p-value = 0.9897

H0:

H1:

Decision:

Conclusion:

7 Interpretation of the model output

summary(slrfit)

Call:
lm(formula = price ~ houseSize, data = house)

Residuals:
     Min       1Q   Median       3Q      Max 
-2.67119 -0.63937  0.00189  0.67506  2.79721 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 5.1577228  1.7063810   3.023  0.00284 ** 
houseSize   0.0248925  0.0006813  36.535  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.9781 on 198 degrees of freedom
Multiple R-squared:  0.8708,    Adjusted R-squared:  0.8702 
F-statistic:  1335 on 1 and 198 DF,  p-value: < 2.2e-16

7.1 Coefficient of determination

\(R^2 = 87.02\%\)

Interpretation:

7.2 Point estimate of \(\beta_0\)

Interpretation:

7.3 Intervals estimate of \(\beta_0\)

Interpretation:

7.4 Point estimate of \(\beta_1\)

Interpretation:

7.5 Intervals estimate of \(\beta_1\)

Interpretation:

8 Hypothesis testing on the Slope and Intercept

9 Prediction of New Observations

newhouseSize <- data.frame(houseSize = c(2000, 2500, 2300, 2400, 2760))
predict(slrfit, newdata=newhouseSize , interval="predict")
       fit      lwr      upr
1 54.94275 52.89457 56.99092
2 67.38900 65.45527 69.32274
3 62.41050 60.45773 64.36326
4 64.89975 62.96113 66.83837
5 73.86105 71.89660 75.82551