Below are the two versions of the t-test, for each of the two IQ-estimates (own and neighbor's), comparing 2025 to 2026. As you can see, the difference between Student and Welch is quite small and does not lead to qualitatively different conclusions. Based on these data, we cannot conclude that the SSR cohorts 2025 vs 2026, judge their (neighbor's) IQ differently.
| 95% CI for Cohen's d | ||||||||
|---|---|---|---|---|---|---|---|---|
| Test | Statistic | df | p | Cohen's d | SE Cohen's d | Lower | Upper | |
| own-iq | Student | -0.014 | 181.000 | .989 | -0.002 | 0.148 | -0.292 | 0.288 |
| Welch | -0.014 | 156.519 | .989 | -0.002 | 0.148 | -0.292 | 0.288 | |
| neighbor-iq | Student | 1.330 | 181.000 | .185 | 0.197 | 0.149 | -0.094 | 0.487 |
| Welch | 1.308 | 153.645 | .193 | 0.195 | 0.149 | -0.096 | 0.486 | |
| Note. Group 1: 2025 , Group 2: 2026 . | ||||||||
Adding Q-Q plots to show; here we don't need to worry about normality since we have a high enough sample size (thank you CLT). If we would have a smaller sample size (let's say, 15 in each group), these Q-Q plots indicate we should take our inferential statistics (p-value, confidence interval) with a grain of salt. For instance, if observing a p-value just below alpha and then see such a Q-Q plot, I would not be so confident in reject the null hypothesis and might want to consider a nonparametric alternative instead (these have less power, but are robust to the normality assumption, see lecture 20).
Descriptives summarize the observed data. The SD column can be used to assess the equal variances assumption. The ratio of SD's is:
The threshold mentioned in the book is that if the higher variance is more than twice as high (or, if the higher SD is more than 1.4 times as high), that it could be grounds to choose Welch over Student. As with other assumptions there are degrees of inequality (i.e., a variance can be twice as high, or 400 times as high; in the latter case it's definitely a good idea to use Welch over Student).
| Group | N | Mean | SD | SE | Coefficient of variation | |
|---|---|---|---|---|---|---|
| own-iq | 2025 | 96 | 120.281 | 20.683 | 2.111 | 0.172 |
| 2026 | 87 | 120.333 | 28.224 | 3.026 | 0.235 | |
| neighbor-iq | 2025 | 96 | 117.885 | 19.733 | 2.014 | 0.167 |
| 2026 | 87 | 113.184 | 27.756 | 2.976 | 0.245 | |
The Raincloud plot visualizes the differences in variance, between the years.