HW1_Applied Linear Regression_BANA7030002
2023-10-22
Members of the group: Keerthi Chereddy, Devashish Kulkarni, Shashank Tippa, Trishla Thakur, Akash Rachewad
1a. Start with a basic exploratory data analysis. Show summary statistics of the response variable and predictor variable.
summary_X <- summary(alumni$'percent_of_classes_under_20')
summary_X## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 29.00 44.75 59.50 55.73 66.25 77.00summary_Y <- summary(alumni$'alumni_giving_rate')
summary_Y## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 7.00 18.75 29.00 29.27 38.50 67.001b. What is the nature of the variables X and Y? Are there outliers? What is the correlation coefficient? Draw a scatter plot. Any major comments about the data?
From the boxplot, we can see that there are no outliers in the data of both variables X and Y.
The correlation coefficient is 0.6578, the positive coefficient idicating that as the percentage of “classes with fewer than 20 students” increases, the “Alumni Giving Rate” tends to increase.
As the p-value is very low (7.23e-07), the coefficient is statistically significant.
par(mfrow=c(1,2), mar=c(2,2,2,2), oma=c(0,0,0,0))
boxplot(alumni$'percent_of_classes_under_20')
hist(alumni$'percent_of_classes_under_20', main = 'percent_of_classes_under_20')
boxplot(alumni$'alumni_giving_rate')
hist(alumni$'alumni_giving_rate', main = 'alumni_giving_rate')
plot(alumni$`percent_of_classes_under_20`, alumni$`alumni_giving_rate`,
xlab="percent_of_classes_under_20", ylab="alumni_giving_rate", pch = 20)
fit <- lm(alumni_giving_rate ~ percent_of_classes_under_20, data = alumni)
#print(fit)
summary(fit)##
## Call:
## lm(formula = alumni_giving_rate ~ percent_of_classes_under_20,
## data = alumni)
##
## Residuals:
## Min 1Q Median 3Q Max
## -21.053 -7.158 -1.660 6.734 29.658
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -7.3861 6.5655 -1.125 0.266
## percent_of_classes_under_20 0.6578 0.1147 5.734 7.23e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 10.38 on 46 degrees of freedom
## Multiple R-squared: 0.4169, Adjusted R-squared: 0.4042
## F-statistic: 32.88 on 1 and 46 DF, p-value: 7.228e-071c. Fit a simple linear regression to the data. What is your estimated regression equation?
Deducing from the output, the value of b0 is -7.3861 and b1 is 0.6578. Hence the estimated regression equation is Y = -7.3861 + 0.6578 * X
1d. Interpret your results.
In addition to the above mentioned analysis, it is worth noting that
The R-squared value (0.4169) suggests that approximately 41.69% of the variability in Alumni Giving Rate can be explained by the percentage of classes under 20.
The F-statistic (32.88) and its associated p-value (7.23e-07) suggest that the model as a whole is statistically significant.