Means

To determine if one sample mean is equal to another sample mean.

To calculate the exact difference between two population means without error.

To prove that one population mean is larger than the other.

To estimate the difference between two population means based on sample data.

The precision of the difference estimate increases, leading to a narrower confidence interval for the difference.

The increased sample size leads to greater statistical power but does not affect the mean difference's confidence interval width.

There is no change in the precision or width of the confidence interval since variances are similar.

Decreasing the small sample size to match the larger one provides a more conservative estimate with a broader confidence interval width.

198

99

200

50

Both samples must be less than 30 and come from distributions with similar shapes.

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

The variances of both populations should differ significantly.

The samples must come from populations with known variances.

Twin study findings imply a strong genetic component that definitively determines academic skill outcomes.

Consistent patterns only indicate correlation, not causation, without further experimental evidence to establish direct genetic links.

Incremental changes over time in twin study score patterns can only arise from environmental factors not related to heritability.

There is no value in looking at twins data as it doesn't provide any insight into general population behaviors regarding skills development.

Sample sizes ($n_{1}$ and $n_{2}$)

Maximum values in both samples

Combined variance

Total sum across both samples

Normal distributions will result in a smaller confidence interval width regardless of sample size.

Non-normal distributions can only be used when there is no variability in sample data points.

A non-normal distribution implies that both populations must have equal variances.

Small sample sizes do not benefit from the Central Limit Theorem, hence normality ensures accurate estimation.

One sample size is large and the other is small.

Both populations have equal standard deviations.

Both populations have equal means.

Both sample sizes are large (Central Limit Theorem applies).

Verifying that both populations are normally distributed.

Using the Central Limit Theorem (n ≥ 30).

Checking the 10% condition for both samples

Constructing a box plot of both samples.

Larger sample sizes result in narrower confidence intervals

Larger sample sizes result in wider confidence intervals

Sample size does not affect the width of the confidence interval

The relationship depends on the shape of the population distributions