Exercise 1 (15 pts)

• State the dimension of the data set you have.

• Which variables could be suitable for the response based on some other set of features.

• Fit a simple linear regression model to predict Sales using Price. Provide an interpretation of the coefficient in the model and write out the model explicitly.

Hint: Think about the summary() function if you used lm !

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#answersQ1:

#1
dim(car_df)

## [1] 400  11

#2
numeric_carseats <- subset(Carseats, select = sapply(Carseats, is.numeric))

correlation_matrix <- cor(numeric_carseats)
print(correlation_matrix)

##                   Sales   CompPrice       Income  Advertising   Population
## Sales        1.00000000  0.06407873  0.151950979  0.269506781  0.050470984
## CompPrice    0.06407873  1.00000000 -0.080653423 -0.024198788 -0.094706516
## Income       0.15195098 -0.08065342  1.000000000  0.058994706 -0.007876994
## Advertising  0.26950678 -0.02419879  0.058994706  1.000000000  0.265652145
## Population   0.05047098 -0.09470652 -0.007876994  0.265652145  1.000000000
## Price       -0.44495073  0.58484777 -0.056698202  0.044536874 -0.012143620
## Age         -0.23181544 -0.10023882 -0.004670094 -0.004557497 -0.042663355
## Education   -0.05195524  0.02519705 -0.056855422 -0.033594307 -0.106378231
##                   Price          Age    Education
## Sales       -0.44495073 -0.231815440 -0.051955242
## CompPrice    0.58484777 -0.100238817  0.025197050
## Income      -0.05669820 -0.004670094 -0.056855422
## Advertising  0.04453687 -0.004557497 -0.033594307
## Population  -0.01214362 -0.042663355 -0.106378231
## Price        1.00000000 -0.102176839  0.011746599
## Age         -0.10217684  1.000000000  0.006488032
## Education    0.01174660  0.006488032  1.000000000

cor_with_sales <- correlation_matrix["Sales", ]
strong_correlations <- cor_with_sales[abs(cor_with_sales) > 0.5]
print(strong_correlations)

## Sales 
##     1

colnames(car_df)

##  [1] "Sales"       "CompPrice"   "Income"      "Advertising" "Population" 
##  [6] "Price"       "ShelveLoc"   "Age"         "Education"   "Urban"      
## [11] "US"

model <- lm(Sales ~ Price, data = car_df)

print(model)

## 
## Call:
## lm(formula = Sales ~ Price, data = car_df)
## 
## Coefficients:
## (Intercept)        Price  
##    13.64192     -0.05307

# fitted model:
# Sales=13.64192−0.05307×Price

# Your plotting code here

#1
par(mfrow = c(2, 2))
plot(model)

#2

conf_interval <- confint(model)

print(conf_interval)

##                  2.5 %      97.5 %
## (Intercept) 12.3978438 14.88598655
## Price       -0.0635995 -0.04254653

Exercise 3 (30 pts)

Fit a multiple regression model to predict Sales using Price, Urban, and US. Provide an interpretation of each coefficient in the model. Be careful-some of the variables in the model are qualitative!

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set.seed(27058)
model <- lm(Sales ~ Price + Urban + US, data = car_df)
summary(model)

## 
## Call:
## lm(formula = Sales ~ Price + Urban + US, data = car_df)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.9206 -1.6220 -0.0564  1.5786  7.0581 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 13.043469   0.651012  20.036  < 2e-16 ***
## Price       -0.054459   0.005242 -10.389  < 2e-16 ***
## UrbanYes    -0.021916   0.271650  -0.081    0.936    
## USYes        1.200573   0.259042   4.635 4.86e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.472 on 396 degrees of freedom
## Multiple R-squared:  0.2393, Adjusted R-squared:  0.2335 
## F-statistic: 41.52 on 3 and 396 DF,  p-value: < 2.2e-16

WARNING: WHEN I RUN YOUR .Rmd file in my computer, in case of any necessity, I need to get the same results that you mentioned in your PDF report !

Exercise 4 (30 pts)

Now consider the same process over the splited data, not the whole sample.

• What is data splitting context and why is it important?

• How can you split your data into training and testing sets in R using the tidymodels package or any other alternative? Consider the 80-20 split rule for training and testing, respectively.

Hint: You can check this website for tidymodels https://rsample.tidymodels.org/reference/initial_split.html

or feel free to use other options that we discussed before

• Once you have split your data, refit the previously considered simple linear or multiple linear regression models on training set only. Look at the summary of the model.

• Use the predict() function in R to generate predictions for the testing data using the regression models you fitted. Evaluate their performance on the testing data using metrics such as RMSE or R-squared?

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#-1 data splits is critical for ML and statistical modeling by dividing data into two or more subsets for purpose of traning models and evualting predictive models for multiple reasons.

chooseCRANmirror(ind = 1)
install.packages("rsample")

## 
## The downloaded binary packages are in
##  /var/folders/0l/0n3rw26n0p53c1lcs4jqddsr0000gn/T//RtmpFWDPHQ/downloaded_packages

library(rsample)

split <- initial_split(car_df, prop = 0.8)

train_data <- training(split)
test_data <- testing(split)

lm_model <- lm(Sales ~ Price, data = train_data)

summary(lm_model)

## 
## Call:
## lm(formula = Sales ~ Price, data = train_data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.5202 -1.8251 -0.2093  1.6791  7.5432 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 13.498128   0.720805  18.726  < 2e-16 ***
## Price       -0.051863   0.006105  -8.496  7.7e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.571 on 318 degrees of freedom
## Multiple R-squared:  0.185,  Adjusted R-squared:  0.1824 
## F-statistic: 72.18 on 1 and 318 DF,  p-value: 7.696e-16

predictions <- predict(lm_model, newdata = test_data)

rmse <- sqrt(mean((test_data$Sales - predictions)^2))

rsquared <- cor(predictions, test_data$Sales)^2

cat("RMSE:", rmse, "\n")

## RMSE: 2.373974

cat("R-squared:", rsquared, "\n")

## R-squared: 0.2556286

Review your work

After answering each of the exercise, take a moment to review your work and to check that your code is clear and understandable. A proportion of the homework assignment marks are attributable to how you present your work, both code and written. This is a good practice for getting a new habit on writing codes.

When reviewing your code, add new lines, tabs and spaces where appropriate to help aid the reader to understand what your code is doing. Even if the exercise does not explicitly ask you to write some text, it is also good practice to add a short sentence to be explicit about what is your answer to the question.

How the homework graded

The homework is grades out of a 100 where:

• Each exercise is marked out of different portions regarding the question difficulty.

• If you have any questions about the assignment, please raise your voice during online lectures and send your inquiries via email.