Description

Yaniv Bronshtein
#Question 1
(1)Create a column of y that includes 100 random sample from a normal distribution with mean=10 and sd=1
set.seed(1)
y <- matrix(rnorm(n=100,mean=10, sd=1))
#(2) Create a column of x that includes numbers 1 to 100 x <- matrix(1:100)
#(3) Draw a scatter plot using R. Based on the figure, do you think #x and y are correlated?
plot(x,y)

x
corr <- cor(x,y) corr
## [,1]
## [1,] -0.01456896
Based on the plot and the result of cor(), I do not believe that x and y are correlated. The value is -0.01456896 which is very close to 0
(4) Fit a regression model y = B_0 + B_1 * x + epsilon and obtain the Rˆ2 value.
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lm_model <- lm(formula=y~x) summary <- summary(lm_model) r_sq <- summary$r.squared r_sq
## [1] 0.0002122545
(5)Fit a regression model y=??x + epsilon and obtain the Rˆ2 value.*
Compare this R2 value with the one you have in (4) and** explain which one is more reasonable. Which model you will recommend?
lm_model_2 <- lm(formula = y~x+0) summary2 <- summary(lm_model_2) r_sq2 <- summary2$r.squared r_sq2
## [1] 0.7467852
The new Rˆ2 is 0.7467852. The new Rˆ2 is much higher and according to standards, is in a good range. If the Rˆ2 had been significantly higher, it would signal likely over-fitting. I would recommend the second model as the first one suffers from under-fitting which explains why the Rˆ2 is so low. The sum of squared distance between the points and the regression line is very high.
(6) Based on the model that you recommend in (5), is x an important factor to y? Is this conclusion consistent with the **one you have in (3)?
plot(x,y)
#Better model abline(lm_model_2, col=’red’)
#Worse model
abline(lm_model, col=’blue’)

x Based
on the model I recommended in (5), we still cannot tell if x is an important factor to y because Rˆ2 for the model is high while the correlation calculated mathematically and observed visually in the plot shows inconsistency between (3) and (5)
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