1. Consider the dataset given by x=c(0.725,0.429,-0.372 ,0.863). What value of mu minimizes sum((x - mu)ˆ2)?
#Load the data set

x <- c(0.725,0.429,-0.372 ,0.863)

The value of mu that minimizes the sum((x - mu)ˆ2) is the mean of the data.

mu <- mean(x)

mu
## [1] 0.41125

Hence, the value that minimizes sum((x - mu)ˆ2) is 0.41125.

  1. Reconsider the previous question. Suppose that weights were given, w = c(2, 2, 1, 1) so that we wanted to minimize sum(w * (x - mu) ˆ 2) for mu. What value would we obtain?

The value that minimizes sum(w * (x - mu) ˆ 2) is is the weighted mean. We can use the function weighted.mean().

w <- c(2, 2, 1, 1)
mu <- weighted.mean(x, w)
mu
## [1] 0.4665

Thus, the value of mu that minimizes sum(w * (x - mu) ˆ 2) is 0.4665.

  1. Take the Galton and obtain the regression through the origin slope estimate where the centered parental height is the outcome and the child’s height is the predictor.

First, we center the parent’s height by subtracting from it their mean.

library(HistData)

# Load the Galton dataset
data("Galton")

# Center the parental height
Galton$parent_centered <- Galton$parent - mean(Galton$parent)

# Fit the regression model through the origin
fit <- lm(parent_centered ~ child + 0, data = Galton)

# Obtain the slope estimate
slope_estimate <- coef(fit)
slope_estimate
##        child 
## 0.0004442533
Hence, the regression egression through the origin slope estimate where the centered parental height is the outcome and the child’s height is the predictor is 0.0004442533 .