Implementing Welch’s t-test in R
We will use a hypothetical example involving two groups of students and their exam scores. Let’s assume Group A consists of 30 students, while Group B consists of 25 students.
# Generate sample data
set.seed(123) # for reproducibility
group_a <- rnorm(30, mean = 75, sd = 10) # Group A scores
group_b <- rnorm(25, mean = 80, sd = 12) # Group B scores
# Perform Welch's t-test
result <- t.test(group_a, group_b, alternative = "two.sided", var.equal = FALSE)
result
Output:
Welch Two Sample t-test data: group_a and group_b t = -2.9025, df = 51.646, p-value = 0.005432 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -12.873584 -2.348193 sample estimates: mean of x mean of y 74.52896 82.13985
The t-value is -2.9025, which indicates the size of the difference between the means of the two groups relative to the variability within the groups.
- The degrees of freedom (df) are approximately 51.646, adjusted for unequal variances.
- The p-value is 0.005432, indicating that the difference in means between the groups is statistically significant at a 95% confidence level (since the p-value is less than 0.05).
- The alternative hypothesis suggests that the true difference in means is not equal to zero, meaning there is a real difference between Group A and Group B.
- The 95% confidence interval for the difference in means is from -12.873584 to -2.348193, providing a range of plausible values for the true difference.
- The sample estimates show that the mean score for Group A (mean of x) is approximately 74.53, while the mean score for Group B (mean of y) is around 82.14.
These results indicate that there is a significant difference in the means of the two groups, with Group B having a higher average score compared to Group A.
Implementing Welch’s t-test on mtcars dataset
# Load the dataset
data(mtcars)
# Subset the data for automatic and manual transmission cars
automatic <- subset(mtcars, am == 0)$mpg
manual <- subset(mtcars, am == 1)$mpg
# Perform Welch's t-test
result <- t.test(automatic, manual, var.equal = FALSE)
# Print the test result
print(result)
Output:
Welch Two Sample t-test data: automatic and manual t = -3.7671, df = 18.332, p-value = 0.001374 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -11.280194 -3.209684 sample estimates: mean of x mean of y 17.14737 24.39231
The t-statistic is -3.7671, indicating a difference in means between automatic and manual transmission cars.
- The degrees of freedom (df) are approximately 18.332, calculated using the Welch-Satterthwaite equation.
- The p-value is 0.001374, which is less than the typical significance level of 0.05, suggesting that there is a statistically significant difference in the mean mpg between automatic and manual transmission cars.
- The 95% confidence interval for the difference in means ranges from -11.280194 to -3.209684. This interval gives a range of plausible values for the true difference in mean mpg between the two transmission types.
- The sample estimates show that the mean mpg for automatic transmission cars is 17.14737, while the mean mpg for manual transmission cars is 24.39231.
Welch’s t-test in R
In statistical analysis, comparing the means of two groups is a common task. However, traditional methods like the Student’s t-test assume equal variances between groups, which may not hold true in real-world data. Welch’s t-test, named after its developer B. L. Welch, provides a robust solution for comparing means when dealing with unequal variances or sample sizes between groups in the R Programming Language.
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