• Data set examines effect of different does of Vitamin C on tooth length in Guinea Pigs
• Orange Juice (OJ) and Vitamin C as Ascorbic Acid (VC) were used as the supplement types
• A Regression Model and Hypothesis Test were conducted

We model the Linear relationship between tooth length and dose as

\[ y_i = \beta_0 + \beta_1 x_i + \epsilon_i \]

where

\[ \epsilon_i \sim N(0, \sigma^2). \]

The ordinary least squares (OLS) estimator is

\[ \hat{\beta} = (X^T X)^{-1} X^T y. \]

To test whether dose significantly affects tooth length, we test

\[ H_0 : \beta_1 = 0 \] versus \[ H_A : \beta_1 \ne 0. \]

The test statistic is

\[ t = \frac{\hat{\beta}_1}{SE(\hat{\beta}_1)} \]

which follows a \(t_{n-p}\) distribution under \(H_0\).

A 95% confidence interval for the slope is

\[ \hat{\beta}_1 \pm t_{n-p,\,0.975}\, SE(\hat{\beta}_1). \] If the interval does not include 0, we reject \(H_0\)

library(ggplot2)
ggplot(ToothGrowth, aes(x = dose, y = len, color = supp)) + 
  geom_point() + 
  geom_smooth(method = 'lm', level = .99, se = TRUE) + 
  labs(title = "Linear Regression: Dose vs Tooth Length",
       x = "Dose (mg)",
       y = "Tooth Length (mm)")