Applied Statistical Methods - Solution 3

Problem 1: Simple Linear Regression

Use the data on Width and Height that we collected during the class to fit a linear regression model to the data. Use the formula derived in class to compute estimates ${\hat{b}}_0$ and ${\hat{b}}_1$ for the intercept ($b_0$) and the slope ($b_1$) that define the linear regression line. Verify the obtained results using the function lm() in R.

Tasks

• Read data from https://charlotte-ngs.github.io/asmasss2024/data/asm_width_height.csv using the function read.table()

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# read data to data.frame

s_data_url_p1 <- "https://charlotte-ngs.github.io/asmasss2024/data/asm_width_height.csv"

df_width_height <- read.table(s_data_url_p1, header = T, sep = ",")

df_width_height

• Compute estimate ${\hat{b}}_1$ for the slope $b_1$ using the formula from class given as

$${\hat{b}}_1=\frac{(xy).-N\overline{x}.\overline{y}.}{(x^2).-N\overline{x}{.}^2}$$

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# estimate of slope

N <- nrow(df_width_height)

x_y_dot <- sum(df_width_height$Width * df_width_height$Height)

x_dot_bar <- mean(df_width_height$Width)

y_dot_bar <- mean(df_width_height$Height)

x2_dot <- sum(df_width_height$Width^2)

b_1_hat <- (x_y_dot - N * x_dot_bar * y_dot_bar) / (x2_dot - N * x_dot_bar^2)

b_1_hat

• Compute estimate ${\hat{b}}_0$ for the slope $b_0$ using the formula from class given as

$${\hat{b}}_0=\overline{y}.-{\hat{b}}_1\overline{x}.$$

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# estimate intercept

b_0_hat <- y_dot_bar - b_1_hat * x_dot_bar

b_0_hat

• Validation with R

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# validation in R using lm()

lm_hei_wid <- lm(Height ~ Width, data = df_width_height)

summary(lm_hei_wid)