Chi–Squares

Less than the expected frequency distribution.

Significantly different from the expected frequency distribution.

The same as the expected frequency distribution.

Greater than the expected frequency distribution.

Multi-stage sample

Judgmental sampling

Convenience sampling

Snowball sampling

Suggests a lack of power to detect actual effects, which necessitates increasing the overall alpha level to enhance sensitivity in detecting true positives.

Implies the presence of outliers that must be removed to ensure robust performance in the final goodness-of-fit analysis.

If significant, it suggests that the assumption of equal variances across compared groups is violated and needs to be addressed before proceeding with further analysis.

Indicates that normality assumptions are met, therefore proceed directly with the goodness-of-fit procedure without concern for unequal variance impact.

The large contribution of that particular category to the chi-square statistic; this suggests non-uniformity between the observed and expected frequencies.

A negative standardized residual for the category, which shows small numbers suggesting consistency with the expected distribution.

A near-zero chi-square contribution from the category, which implies that it is well aligned with the expected frequencies.

A positive residual value for the category, indicating that the observed frequency is lower than expected.

Observed counts and expected counts

Two sets of expected counts based on different hypotheses

Observed probability and theoretical probability per category

Two sets of observed counts from different samples

Binomial distribution

Chi-square distribution

Uniform distribution

Normal distribution

0.5

0.01

0.1

0.05

All expected cell frequencies must be at least five.

No more than two cells can have observed frequencies less than three.

At least half of the cells must have observed frequencies greater than ten.

Each cell's observed frequency must exceed its corresponding expected frequency by at least two units.

There's significant evidence against what was expected; teens' brand preferences differ markedly from assumed distributions.

This indicates that additional data collection is needed because our existing dataset does not provide clarity on brand preference patterns.

There isn't enough evidence to reject the null hypothesis; brand popularity may follow expected distribution.

The calculations probably contain errors since statistical tests generally result in small p-values when they're correct.

Samples must be collected randomly from less than ten percent of all bags produced.

The sample size must be over thirty.

Expected count in each category should be at least five.

All categories must have an equal number of observed cases.