Comparison of T-Test and Bootstrap Confidence Intervals

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1 Executive Summary

This report commpares two statistical analysis methods and determines the usefulness of the bootstrap method for the desired population mean. Our variable of interest is the percentage of protein intake in a country consumed via animal products. We can then use the provided evidence and make decisions on which method would be best beneficial for future research questions.

url="https://pengdsci.github.io/STA321/ww02/w02-Protein_Supply_Quantity_Data.csv"
protein = read.csv(url, header = TRUE)
var.name = names(protein)

2 Introduction of Data Set

This data set includes the percentage of protein intake from different types of food in countries around the world. The last couple of columns also includes counts of obesity and COVID-19 cases as percentages of the total population for comparison purposes.

This data set can be found at kaggle.com. The data has been uploaded to w02-Protein_Supply_Quantity_Data.csv for internet availability.

3 T-Test Method

The equation for the T-Test confidence interval is given by the following: \[ \bar{X} \pm t_{n-1, 1-alpha/2} \times \frac{s}{\sqrt{n}} \] Where \(\bar{X}\) is the sample mean, \(t\) is the t statistic at our 95% confidence interval, \(s\) is the standard deviation, and \(n\) is the sample size.

The calculated confidence interval is:

t <- t.test(protein$AnimalProducts, 
       conf.level = 0.95)

hist(protein$AnimalProducts,
     xlab = "Protein Intake % Consumed Via Animal Products",
     main = "Empirical Distribution")

t$conf.int

## [1] 20.03275 22.43156
## attr(,"conf.level")
## [1] 0.95

4 Bootstrap Method

The bootstrap method is a simulated sampling method. We use \(\{x_1^{(i*)}, x_2^{(i*)}, \cdots, x_n^{(i*)}\}\) to denote the \(i^{th}\) bootstrap sample. Then the corresponding mean is called bootstrap sample mean and denoted by \(\hat{\mu}_i^*\), for \(i = 1, 2, ..., n\).

set.seed(123)
bt.sample.mean.vec = NULL
for(i in 1:1000){ith.bt.sample = sample(x = protein$AnimalProducts,
                       size = 170,
                       replace = TRUE)
  bt.sample.mean.vec[i] = mean(ith.bt.sample)}

hist(bt.sample.mean.vec,
     xlab = "Protein Intake % Consumed Via Animal Products",
     main = "Bootstrap Emprical Distribution")

bt.CI = quantile(bt.sample.mean.vec, c(0.025, 0.975))
bt.CI

##     2.5%    97.5% 
## 20.06479 22.41047

5 Conclusions

Both methods are valuable for analysis. We can see that our confidence intervals are similar. With a standard confidence interval available, the bootstrap method is not necessarily pertinent to this study. With no significant difference between the two, the standard sample may be more useful for computational ease. Further decisions can be made based on project needs.