Can t-tests be 1 sided? This example has confused what I thought I knew.

Question

We first take a random sample of $8$ students, record their final score in $2018$, and check their final scores again in $2019$.

$2018: 76, 73, 66, 95, 75, 78, 96, 93$

$2019: 75, 80, 70, 93, 81, 90, 88, 79$

We're testing whether the grades have improved for these 8 students.

Since we're looking to see if the grades have "improved", aren't we looking for evidence that they have increased and not decreased?

These are the null and alternative hypotheses I wrote up to convey my reasoning:

$H$_$0$ : $μ$₁ $-$ $μ$₂ ≥ $0$

$H$_$1$ : $μ$₁ $-$ $μ$₂ < $0$

From what I know, the alternative hypothesis always has to hold the remaining outcomes dictated by the null hypothesis, hence my confusion.

I was told the correct hypotheses tests were:

$H$_$0$ : $μ$₁ $=$ $μ$₂

$H$_$1$ : $μ$₁ $≠$ $μ$₂

I'm sure the word 'improved' has to affect the direction of the hypotheses tests of this t-test but I don't know anymore.

Thanks

Answer 1

One-sided $t$-tests absolutely do exist and are valid hypothesis tests, just as we also have one-sided $z$-tests.

I disagree with the claimed hypothesis and agree with yours. If $\mu_1$ is the mean score in 2018 and $\mu_2$ is the mean score in $2019$, then your hypothesis is structured correctly for testing whether the scores improved in from 2018 to 2019. The other hypothesis could potentially conclude that a difference exists but without examining the sign of the test statistic, it would not tell you whether that difference is due to improvement. In other words, if all I saw is the $p$-value from that test, I could not tell you whether the scores improved or not.

Another subtlety here is that this is a paired $t$-test, so we are not really testing whether the mean scores have improved, but rather, whether the mean difference in scores improved. Each student is their own control, and what is following a $t$-distribution is the difference between their 2018 score and 2019 score. As such, the scores themselves do not need to be $t$-distributed, but their differences do.