Chapter 9: Non-Parametric Statistics and Chi-Square Tests

What are the two primary requirements for using parametric tests like the t-test or F-test?

Normality (or large samples) and interval level data.

Why are tests like the t-test and F-test specifically categorized as 'parametric' tests?

They assume there are population parameters to be estimated and tested.

Statistical tests that do not require population parameter assumptions are known as __ tests.

Non-parametric

Which non-parametric test is used to determine if the frequency distribution of a single variable follows a specified (typically uniform) distribution?

One-Way Chi-Square Test.

Observed frequency (fₒ)

The actual or true frequency distribution obtained by counting.

Expected frequency (fₑ)

The anticipated frequency based on the assumption that the null hypothesis is true.

What is the most commonly used non-parametric statistic?

Chi-square (χ²).

The Chi-square test allows researchers to test the significance of the difference between which two values?

Observed frequency (fₒ) and expected frequency (fₑ).

What is the null hypothesis (H₀) for a Chi-square test regarding frequencies?

H₀: fₒ = fₑ

Formula: One-Way Chi-Square (χ²)

χ² = (fₒ - fₑ)² / fₑ

Formula: Degrees of freedom (df) for a One-Way Chi-Square test

df = k - 1

In the formula df = k - 1, what does the variable k represent?

The number of categories.

According to the Chi-square procedure, where is the critical value (χ²_c) typically found?

Table E.

Under what condition regarding the Chi-square statistic (χ²) and the critical value (χ²_c) should the null hypothesis be rejected?

If χ² > χ²_c

Under what condition regarding the Chi-square statistic (χ²) and the critical value (χ²_c) should the null hypothesis be retained?

If χ² ≤ χ²_c

What is the primary function of a Two-Way Chi-Square Test?

To determine whether two categorical variables are associated or not.

In a Two-Way Chi-Square test, what tool is used to cross-tabulate the two categorical variables?

A contingency table.

Formula: Degrees of freedom (df) for a Two-Way Chi-Square test

df = (r - 1)(c - 1)

In the Two-Way Chi-Square df formula, what do the variables r and c represent?

The number of rows (r) and the number of columns (c).

Formula: Expected frequency (fₑ) for a specific cell in a contingency table

fₑ = (row marginal total × column marginal total) / N

What is the underlying rationale for the expected frequency (fₑ) formula in a Two-Way Chi-Square test?

It assumes the two variables are not related (independent).

When using Chi-square, researchers should be wary of expected frequencies that fall below what value?

Five (5).

What is a recommended action for small samples to prevent misinterpretation due to low expected frequencies?

Collapse categories if it is logical to do so.

Under what specific range of expected values (fₑ) should Yate's correction be applied?

When fₑ is greater than five but less than ten.

Why is Yate's correction used when cell frequencies are smaller than 5?

To prevent the Chi-square test result from being erroneous.

How does the application of Yate's correction impact the final size of the Chi-square statistic?

It reduces the size of the statistic.

Yate's correction is specifically designed for use with which type of contingency table?

2 × 2 tables.

Formula: Chi-Square with Yate's correction

χ² = (|fₒ - fₑ| - 0.5)² / fₑ

What level of measurement is required for the variables in a Chi-square test of significance?

Nominal data.

What sampling method is a requirement for the use of the Chi-square test?

Random sampling.

One requirement for Chi-square is that the __ cell frequencies should not be too small.

Expected

Which non-parametric test is appropriate for use with ordinal data?

The Median Test.

Formula: Position of the Median

Position = (N + 1) / 2

In the Median Test, what values function as the observed frequencies (fₒ)?

The counts of numbers above and below the median.

Which statistical correction is typically used when calculating the Chi-square statistic for a Median Test?

Yate's correction.

What does the Median Test specifically compare between two or more samples?

Their medians.

What is the measurement level requirement for the Median Test?

Ordinal data.

How does the power of non-parametric procedures generally compare to parametric tests?

Non-parametric procedures are less powerful.

True or False: The calculation process for a Two-Way Chi-Square test remains the same even if the table has more than two columns or rows.

True.

The assumption that the joint frequency is the product of two marginal frequencies allows for the calculation of __.

Expected frequency (fₑ)

What is the first step in the general procedure for conducting a Chi-square test?

Establish the null hypothesis (H₀) and the research hypothesis (H₁).

Non-parametric tests are particularly useful when __ requirements, such as normality, cannot be met.

Parametric

Concept: fₒ

Observed frequency (the actual data counted).

Concept: fₑ

Expected frequency (the data predicted by the null hypothesis).

In a 2 × 2 table where N=40, if the Row 1 marginal is 25 and the Column 1 marginal is 20, what is the fₑ for the first cell?

12.5

If a researcher is comparing two samples with ordinal data and random sampling, which test should they use?

The Median Test.

If a researcher is comparing two nominal variables for association using a 3 × 4 table, what are the degrees of freedom?

6

Which test requires the calculation of the position of the median before counts can be used in a Chi-square formula?

The Median Test.

A One-Way Chi-Square test is used to see if a distribution follows a __ distribution.

Specified (or uniform)

Non-parametric tests are 'non-parametric' because they do not assume there are __ to be estimated.

Population parameters

If χ² = 5.2 and χ²_c = 5.2, what is the statistical conclusion?

Retain the null hypothesis.

In the context of the Median Test, 'random sampling' is listed as a __ for its use.

Requirement (or condition)