PYB350 Chap.1 T-Tests, Effect Size and Bootstrapping
T-Tests, Effect Size and Bootstrapping
The unit opens with a Week 1 refresher because every later analysis reuses its logic. This chapter rebuilds the single-sample and independent-groups t-tests, the standard error, Cohen's d and the conventions for calling an effect small, medium or large.
It then explains why the normality assumption belongs to the sampling distribution of the mean rather than to the raw scores, and how the bootstrap builds an empirical sampling distribution by resampling your own data with replacement, which is the tool the unit uses whenever a small or skewed sample makes the Central Limit Theorem a risky bet.
The chapter's worked example tests nine screening scores against a published norm and carries the result through the interval and the effect size, and the practice cards check the pooled-variance formula and how to read a bootstrap that moves between runs.
What this chapter covers
- 01
Notation for scores, group means, the grand mean, n and N
- 02
The single-sample t-test when the population SD is unknown
- 03
Cohen's d and the small, medium and large conventions
- 04
The Central Limit Theorem and the normality of the sampling distribution
- 05
Bootstrapped standard errors and confidence intervals
- 06
Independent groups, pooled variance and Levene's test
- 07
Reading a bootstrapped interval against zero, and why reruns move it slightly
Worked example · free
A single-sample t-test with an interval and an effect size
- 1Describe the sample: the scores sum to 405, so M = 45.00; squared deviations sum to 126, so s = 3.97.
- 1Standard error: 3.97 divided by the square root of 9 = 1.323.
- 2Test: t(8) = 5 / 1.323 = 3.78, beyond the critical 2.306, so p is below .05 (p = .005).
- 2Interval and size: 5 plus or minus 2.306 times 1.323 gives 1.95 to 8.05 points; d = 5 / 3.97 = 1.26, a large effect.
Key terms
- Standard error
- The standard deviation of the sampling distribution of a statistic, estimated for a mean as s divided by the square root of n.
- Cohen's d
- A mean difference expressed in standard-deviation units, with 0.2, 0.5 and 0.8 marking small, medium and large effects.
- Central Limit Theorem
- The result that means of large enough random samples are approximately normally distributed whatever the population's shape.
- Pooled variance
- A single variance estimate combining two or more groups, which with equal group sizes is the average of the group variances.
T-Tests, Effect Size and Bootstrapping FAQ
Why does the normality assumption not apply to the raw scores?
The assumption concerns the sampling distribution of the mean, the distribution you would get from many samples of the same size. The raw scores are only evidence about it: if your sample is roughly symmetrical and reasonably large, about thirty scores, the sampling distribution can be treated as normal even if individual scores are not perfectly bell shaped.
When should a bootstrap be used instead of the regular test?
The lecture recommends it when the distribution is strongly skewed or the sample is small, because those are the cases where the Central Limit Theorem may not deliver a normal sampling distribution. It also notes that Wilcox (2004) treats the bootstrap as a sound option most of the time, and that it works well as a check on a parametric result.
Why do bootstrapped intervals change each time the analysis is run?
Each run draws a fresh set of random resamples from your data, so the empirical sampling distribution, and the interval cut from it, moves a little. The tutorial suggests at least 5,000 resamples with the bias corrected accelerated interval, which keeps the movement small. A conclusion that flips between runs is a sign the result sits right on the boundary.
What does a 95% confidence interval for a mean difference tell me?
It gives the range of population differences that are compatible with your data at the 95% level. If the interval excludes zero, a zero difference is not plausible and the test is significant at .05; if it includes zero, the data are compatible with no difference. The interval also shows how precise the estimate is, which a p value alone does not.
Exam move
Practise the single-sample t-test by hand until the sequence of mean, standard deviation, standard error, t, interval and d takes under five minutes, then learn to read the same numbers off SPSS output. When you check a bootstrap, compare its interval with the parametric one and write one line on whether they agree, because that comparison is the reason for running it.
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