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Confusion behind Student's t-test example in R

I thought I understood the t-test. It turns out, I don’t.

If you look at the documentation for the t.test function to perform t-tests, there is this example:

t.test(1:10, c(7:20, 200))
#         Welch Two Sample t-test
# data:  1:10 and c(7:20, 200)
# t = -1.6329, df = 14.165, p-value = 0.1245
# alternative hypothesis: true difference in means is not equal to 0
# 95 percent confidence interval:
#  -47.242900  6.376233
# sample estimates:
# mean of x mean of y
#   5.50000  25.93333

This compares the means of a set of numbers between 1 through 10 and 7 through 20 with 200.

I found this odd because intuitively, this doesn’t make sense. I expected this to have a significant difference in means with the 200 in there. I felt perplexed by this example because I was showing this to a student and found myself questioning my own assumptions about this test.

With some searching around, I’m glad to find out that I’m not the only one confused by this.

The explanations from others suggest that the addition of the 200 value increases the variance used in the t-statistic. Thus, it changes the t-statistic to not perceive a difference in means.

Looking at the documentation once more, var.equal is set to FALSE. With the help of Wikipedia, there are various ways to calculate the t-statistic, depending on your data.

For our toy example, R tells us we are doing a “Welch Two Sample t-test.” The t-statistic for this is define as

\[t = \frac{\bar X_1 - \bar X_2}{s_{\bar \delta}}\]


\[s_{\bar \delta} \sqrt{\frac{s^2_1}{n_1} + \frac{s^2_2}{n_2}}\]

Adding 200 into the mix, it radically changes the variance.

var(c(7:20, 200))
# [1] 2335.067
# [1] 17.5

This higher variance is pooled into the demoniator of the t-statistic. Thus, it pulls down the statistic to be lower and to no longer be significant.

So in the end, it appears I’ve forgotten my assumptions about how a t-test works: equal sample sizes, equal variances. These are the kinds of learning opportunities I love: easy looking questions, but give you unintuitive answers. Lastly, the review lesson here is to check your assumptions.

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