2024-06-05
This presentation explores the state.x77 dataset provided in the R Datasets Package, version 4.5.0. This presentation showcases the demographic and socioeconomic statistics for 50 US states in the 1970’s.
Using various statisical tools, we will study the relationships between following variables:
If we were to plot the two variables ‘HS Grad’ and ‘Illiteracy’ against each other, we get the following ggplot. The data suggests that there is a negative relationship between illiteracy rate and high school graduation ration. (Higher illiteracy rates tend to have lower high school graduation rates.)
What else is there? Another approach to determine the the relationship (or lack thereof) between variables is to find the R-Squared value–a number between 0-1, with 1 meaning a strong relationship–which indicates the proportion of the variance in the dependent variable that is predictable from the independent variable(s) in a linear regression model.
\[ R^2 = 1 - \frac{\sum(y_i - \hat{y}_i)^2}{\sum(y_i - \bar{y})^2} \]
Applied to our data, we get that our R-squared value is 0.4318969, which indicates that, still, about 43% of the variation in high school graduation rate can be explained by the illiteracy rate, suggesting that addressing illiteracy rates may be an important step in improving high school graduation rates.
We could also make simpler graphs if we’d simply like to get a brief overview of the some of the data. For example, what if we wanted to know an estimate of in what range, on average, were most people’s income?
ggplot(as.data.frame(state.x77), aes(x = Income)) +
geom_histogram(bins = 20) +
labs(x = "Income", y = "Frequency") +
ggtitle("Average Income")
\[ \bar{I} = \frac{\sum_{i=1}^{n} I_i}{n} \]
Now, we can also keep it really simple. What if I simply wanted to know the average income in the US in the 1970’s? We can use the formula (as seen above) to find the average. Or, we can keep it simpler and utilize the functions given to us in the R library!
data <- as.data.frame(state.x77)
avg_income <- mean(data$Income)
cat("Average income:", avg_income)
Average income: 4435.8
What else can we tell from our data? We can do a confidence interval test to compare the mean life expectancy of the 50 states to a known mean value of 70.8786 years. Here are our findings from a sample size of 17 random states:
Overall, we find that that the sample of 17 states is representative of the population of all 50 states, and that the mean life expectancy of the sample is not significantly different from the known population mean.
Using point estimation, we find that the correlation coefficient between Income and Life Expectancy: r = 0.3402553
Our conclusions suggest that as Income increases, Life Expectancy also tends to increase; states with higher incomes tend to have higher life expectancies, and vice versa. However, the moderate strength of the correlation implies that there may be other factors at play that influence Life Expectancy, and that Income is not the sole determining factor.