Sampling Distributions

The pooled t-test.

The z-test with pooled variance.

The paired t-test.

The Welch's t-test.

Random-digit dialing

Cluster sampling

Proportional stratified sampling

Stratified sampling

One sample size must be more than twice as large as another.

Both populations have equal variances.

At least one population is not normally distributed.

Both populations are normally distributed or both sample sizes are large.

Power is not affected by within-group variability changes as it only pertains to between-group differences.

Power increases due to greater discernibility provided by wider data spread within groups making real differences clearer.

Within-group variability has an inverse square root relationship with power affecting its magnitude proportionally less with larger variances.

Power decreases because increased variability inflates the standard errors and makes detecting a true difference harder.

p

$\mu_1 - \mu_2$

$p_1 - p_2$

$\sigma$

It decreases compared to when sample sizes are equal.

There is no effect on the standard error.

It increases compared to when sample sizes are equal.

The effect on the standard error depends on the actual mean values, not sample sizes.

Sampling distributions for the difference in variances.

Sampling distributions for the maximums in paired data.

Sampling distributions for the mean difference in paired data.

Sampling distributions for the difference in two means.

All students should achieve passing grades across all tests administered during the study period.

The test scores should be independent random variables within each group.

Test score distributions need to exhibit perfect symmetry within each group before analysis proceeds further.

Students must have similar educational backgrounds and prior knowledge before starting the study.

Range

Coefficient of variation

Interquartile range

Standard error

Interpreting ambiguous results with regards to constancy of spreads calls for a thorough examination of potential covariates and confounding variables, which might explain away the apparent discrepancies otherwise observed, simply relying on numerical outputs provided by calculator software packages.

Opting to completely abandon parametric techniques in favor of non-parametrics becomes a necessary option at this point, particularly in light of inconclusive preliminary checks related to the consistency of spread across the compared datasets.

When both tests concerning the homogeneity of populations' variances lead to conflicting outcomes, prioritizing Welch’s adaptation takes precedence due to its unbiased nature, especially when faced with unknown truth regarding said homogeneities.

Applying regular t-tests may still suffice as long as accompanying adjustments are made to account for possible non-homogeneities, thus preserving the integrity of subsequent inferential statistics despite the initial mismatch in findings concerning variabilities.