Assignment A questions.qmd

---
title: "Assignment A"
format: html
editor: visual
---

## Q1

When asked to state the simple linear regression model, a students wrote it as follows: $E(Y_i) = \beta_0 + \beta_1X_1 + \epsilon_i$. Is this correct? Justify your answer.

Multiple choice: 1.11, 1.18, 1.40

To obtain a point estimate of the regression coefficients, the error term must be assumed to be normally distributed

Short answer 1.5, 1.7, 1.37

application 1.19, 1.43c

Proofs: MLEs of regression coefficients

Properties of fitted regression lines

Derive the Least squares estimator of no-intercept model, and check if it is biased. Assume X to be constant. (1.41)

```{r}
x = c(1,2,3)
set.seed(1)
y = 3*x + rnorm(3)
data = data.frame(x = x, y = y)
model = lm(y~x, data = data)
model$coefficients
ypred = predict(model,newdata = list(x=1.4))
data2 = rbind(data, c(1.4, ypred))
model2 = lm(y~x, data = data)
model2$coefficients
library(ggplot2)
ggplot(data = data, aes(x = x, y = y))+
  geom_point()+
  geom_smooth(method = "lm", color = "blue", se = FALSE)+
  geom_point(data = data2)+
  geom_smooth(data = data2, method = "lm", color = "red", se = FALSE)
```