Question

For this exercise we will run a regression using Swiss demographic data from around 1888. The sample is a cross-section of French speaking counties in Switzerland This data come with the R package datasets. The first step is to load the package into your current environment by typing the command libraryldatasets) in to the R console. This loads a number of datasets including one called swiss. Type help/swiss) in the console for additional details. The basic variable definitions are as follows A data frame with 47 observations on 6 variables, each of which is in percentie, in o, 100 L.1] Fertility commen standardzed fertility measure [.2] Agriculture %0f males moted i, agrot re as occupaton [3] Examnation % drahees recevrg highest mark on army et [A] Education %education beyond prmary school fordranees 5] Catholic L6] Infant Mortality live births who Iive less than 1 year n % ,catholc. (as agposed to 'protestant'). All variables but Ferlity give proportions of the population Use the summaryl command to report the mean and median for the variables Fertility, Education, and Catholic. 2. We want to estimate the expected Fertility level in a Swiss county conditional on the county's education level. We assume the relationship is linear. So, we are interested i, estimating α and in Fertility.-α + β . Education, + te If we use Ordinary Least Squares to estimate α and β we have fol wing formulas Where y is the left hand side variable, x is the right hand side variable, the bar mean, s is the standard deviation, and r is the correlation denotes the If you encounter an error you might have to install the package onto your computer using the command Find the correlation between Education and Fertility using the cor) function, as well as the sample standard deviation for each variable using the sd0 function. Report these numbers and the code used. a. b. Use the cor) and sd) function to get an estimate for B in the equation relating Fertility to Education. Keep this value stored in a scalar called beta hat. Report this number. c. Now use the estimate beta hat, along with the function mean) to get an estimate for alpha. Keep this value stored as a scalar called alpha hat. Report this number 3. Use alpha-hat and beta-hat to predict the average ferity rate in a county where 40% of the population is educated 4. Plot the relationship between Fertility and Education using the plotD function with Education on the horizontal axis. Do this using the plotD function. Make sure to label your axis!

Answer 1

AS FOR GIVEN DATA..

1. Code used:

a<-read.csv("swiss.csv")
summary(a)

Output:

	Mean	Median
Fertility	70.14	70.4
Education	10.98	8
Catholic	41.14	15.14

2. a) Code used:

x<-cor(a$Education,a$Fertility)
y<-sd(a$Education)
z<-sd(a$Fertility)

Output:
Correlation, r = -0.66

Sx (education) = 9.62

Sy(Fertility) = 12.49

b) Code:

beta_hat <- x*z/y
Output:

Beta estimate is -0.862

c) Code:

alpha_hat <- mean(a$Fertility) - mean(a$Education)*beta_hat
alpha_hat

Alpha = 79.60

3. Code: pred<- alpha_hat + beta_hat*40

Output: 45.11

4. Code:

plot(a$Education, a$Fertility, xlab="Education", ylab="Fertility")