Evaluating Study Time Differences Across Schools

Table of Contents

1. Data
2. Results
3. Analysis Approach
4. Conclusion

Data

The dataset consists of the time (in hours) that students from three different schools spent studying or doing homework during an exam period:

• School 1: 25 observations
• School 2: 23 observations
• School 3: 20 observations

Results

Given the limited sample sizes, typical frequentist methods may not yield significant results. Therefore, we adopt a Bayesian approach. Using a normal model with a conjugate prior, we calculate the posterior means for each school, which are approximately:

• School 1: 9.29 hours
• School 2: 6.95 hours
• School 3: 7.81 hours

To construct 95% confidence intervals for these means, we implement a Monte Carlo procedure, drawing 1,000 random samples from the distributions. The confidence interval for School 1 does not overlap with those of the other schools, indicating a significantly higher average study time. In contrast, the confidence intervals for Schools 2 and 3 overlap, prompting further analysis of probabilities concerning which school's mean is greater.

The probability that the mean study time for School 3 is less than that of School 2, which is in turn less than 1, is approximately 0.4814 (the most likely outcome). The probability that the mean for School 2 is less than that of School 3 is 0.4719. Additionally, the probability that School 3 has a lower mean than School 2 is slightly greater than the complementary probability based on all outcomes.

Analysis Approach

1. Posterior Means Calculation:
   • Use Bayesian methods to calculate the posterior means for each school based on prior distributions and observed data.

# Using monte carlo simulation for all three schools
# Mu0, var0, k0, v0 given
mu0 <- 5
var0 <- 4
k0 <- 1
v0 <- 2
#posterior mean for s1
n1 <- length(s1)
y_bar1 <- mean(s1)
var1 <- var(s1)
mu_1 <- (k0 * mu0 + n1 * y_bar1)/(k0 + n1)

2. 95% Confidence Intervals:

   • Employ Monte Carlo simulations to derive confidence intervals for the means and standard deviations of study times for each school.

3. Probability Comparisons:

   • Assess the probabilities of the relationships between the means of different schools to determine which school has a statistically greater average study time.

Conclusion

The Bayesian analysis reveals distinct differences in average study times among the schools, with School 1 significantly outpacing the others. Further investigation into the overlapping confidence intervals for Schools 2 and 3 highlights the nuanced relationships between their average study times, suggesting that while School 1 is clearly superior, the competition between Schools 2 and 3 remains closely matched.