Scatter diagram. correlation coefficient (ungrouped data), Fitting of line of regression, residual plot

#Q.1). Draw a scatter diagram for the data given below and comment on the correlation.

Husband_Age <- c(24,27,28,28,29,30,32,33,35,35,40);
Wife_Age <- c(18,20,22,25,22,28,28,30,27,30,22);
data_1 <- data.frame(Husband_Age, Wife_Age);data_1
plot(data_1);
cat("Also calculate Karl Pearson's coefficient of correlation between age of husband and
that of wife and comment on it.");
covariance <- (sum(Husband_AgeWife_Age)/length(Husband_Age)) - (mean(Husband_Age)mean(Wife_Age));
SD_Husb <- sqrt(var(Husband_Age)(length(Husband_Age) - 1)/length(Husband_Age));
SD_Wife <- sqrt(var(Wife_Age)(length(Wife_Age) - 1)/length(Wife_Age));
r <- covariance/(SD_Husb*SD_Wife);

#Q.2). Given the following information about production and demand of a commodity

Y = c(85,5); #Production
X = c(90,6); #Demand
Parameter = c("mean","sd");
data_2 <- data.frame(Parameter,Y,X);data_2
cat("Coefficient of correlation between X and Y is r = 0.65.");
r = 0.65;
cat("a) obtain the regression line of Y on X.");
cat("Regression line eq: y - Y = byx(x - X)");
byx <- r*(Y[2]/X[2]);
y

cat("Also estimate the production when demand is 100.");

#Q.3). The following data gives the sales and expenses of 10 firms

Firm_nu <- c(1:10);
Sales <- c(45,70,65,30,90,40,50,75,85,60);
Expenses <- c(35,90,70,40,95,40,60,80,80,50);
data_3 <- data.frame(Firm_nu,Sales,Expenses);data_3

cat("a) Obtain the least squares regression line of expenses on sales.");
covariance <- (sum(SalesExpenses)/length(Sales)) - (mean(Sales)mean(Expenses));
sd_Sales <- sqrt(var(Sales)(length(Sales) - 1)/length(Sales));
sd_Expenses <- sqrt(var(Expenses)(length(Expenses) - 1)/length(Expenses));
r <- covariance/(sd_Salessd_Expenses);
byx <- r(sd_Sales/sd_Expenses);
cat("Putting this in the eq of regression line");
y <- byx*(Sales - mean(Sales)) + mean(Expenses);
cat("New data frame;");
data_3a <- data.frame(Sales, Expenses, y);
plot(y,Sales, col = "orange", pch = 25, main = "least squares regression line of expenses on sales");

cat("b) Estimate expenses if sales are Rs. 75000.");
x <- 75000; #Because sales are 75000
y <- byx*(x - mean(Sales)) + mean(Expenses);

cat("c) Calculate the coefficient of correlation between expenses and sales.");
cat("Already calculated in question a)");

cat("d) Find mean residual sum of squares.");
yhat <- byx*(Sales - mean(Sales)) + mean(Expenses);yhat
c <- (Expenses - yhat)^2;
sum(c)
MRSS <- sum(c)/(length(Sales) - 2);
cat("e) Draw residual plot.");
plot(yhat, Sales, pch = 25, main = "Residual plot");