Question

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CHAPTER 19: CHI-SQUARE TEST FOR QUALITATIVE DATA

Key Terms

One-way test – Evaluates whether observed frequencies for a single qualitative variable are adequately described by hypothesized or expected frequencies.

Expected frequency – The hypothesized frequency for each category, given that the null hypothesis is true.

Observed frequency – The obtained frequency for each category.

Two-way test – Evaluates whether observed frequencies reflect the independence or two qualitative variables.

Squared Cramer’s phi coefficient – Very rough estimate of the population of predictability between two qualitative variables

Text Review

You may recall from Chapter 1, that when observations are classified into categories, the data are (1)_____________________. The hypothesis test for qualitative data is known as chi-square. When the variables are classified along a single variable, the test is a one-way chi-square. The one-way chi-square test makes a statement about two or more population (2)_______________ that are reflected by expected frequencies.

If the null hypothesis is true, then except for the effects of chance, the hypothesized proportions should be reflected in the sample. The number of observations hypothesized is referred to as (3)___________ and is calculated by multiplying the expected proportion by the total sample size. If the discrepancies between the observed and expected frequencies are small enough to be attributed to chance, then the null hypothesis would be retained. But if the discrepancies between the observed and expected frequencies are large enough to qualify as a rare outcome, the null hypothesis would be (4)_________.

The value of chi-square can never be (5)______________________________ because of the squaring of each difference between observed and expected frequencies.

For the one-way chi-square test, the degrees of freedom always equal the number of (6)____________ minus one.

The chi-square test is non-directional because the squaring of the discrepancies always produces a (7)__________________ value. However, for the same reason, only the upper tail of the sampling distribution contains the rejection region.

It is possible to cross-classify observation along two qualitative variables. This is referred to as a (8)_________________________ chi-square test. For the two-way test, the null hypothesis makes a statement about the lack of relationship between the two qualitative variables. In the two-way test, words are usually used instead of symbols in the null hypothesis, and as in the one-way test, the research hypothesis simply states that the null hypothesis is false.

In the two-way test, expected frequencies are calculated by multiplying the column total times the row total and dividing by the overall total. The chi-square critical value may be found in Table D of Appendix D only if degrees of freedom are known. For the two-way test, degrees of freedom equals number of categories for the column variable minus one, times the number of categories for the row variable minus one [df = (C – 1) (R-1)].

Some precautions are necessary in using the chi-square tests. One restriction is that the chi-square test requires that observations be (9)________________________. In this case, independence means that one observation should have no influence on another. One obvious violation of independence occurs when a single subject contributes more than one pair of observations. One way to check that this requirement is not being violated is to remember that the total for all observed frequencies must never exceed the total number of subjects. Using chi-square appropriately also requires that expected frequencies not be too small. Generally, any expected frequency of less than (10)______________ is too small. Small sample sizes should also be avoided, as should unduly large sample size. A sample size that is too large produces a test that detects differences of no practical importance.

When the null hypothesis has been rejected, the researcher should consider using squared (11)______ phi coefficient to determine whether the strength of the relationship is small, medium, or large.

Answer 1

You may recall from Chapter 1, that when observations are classified into categories, the data are (1) qualitative. The hypothesis test for qualitative data is known as chi-square. When the variables are classified along a single variable, the test is a one-way chi-square. The one-way chi-square test makes a statement about two or more population (2) observed frequency that are reflected by expected frequencies.

If the null hypothesis is true, then except for the effects of chance, the hypothesized proportions should be reflected in the sample. The number of observations hypothesized is referred to as (3) expected frequency and is calculated by multiplying the expected proportion by the total sample size. If the discrepancies between the observed and expected frequencies are small enough to be attributed to chance, then the null hypothesis would be retained. But if the discrepancies between the observed and expected frequencies are large enough to qualify as a rare outcome, the null hypothesis would be (4) rejected.

The value of chi-square can never be (5) negative because of the squaring of each difference between observed and expected frequencies.

For the one-way chi-square test, the degrees of freedom always equal the number of (6) mutually exclusive groups minus one.

The chi-square test is non-directional because the squaring of the discrepancies always produces a (7) positive value. However, for the same reason, only the upper tail of the sampling distribution contains the rejection region.

It is possible to cross-classify observation along two qualitative variables. This is referred to as a (8) two-way chi-square test. For the two-way test, the null hypothesis makes a statement about the lack of relationship between the two qualitative variables. In the two-way test, words are usually used instead of symbols in the null hypothesis, and as in the one-way test, the research hypothesis simply states that the null hypothesis is false.

In the two-way test, expected frequencies are calculated by multiplying the column total times the row total and dividing by the overall total. The chi-square critical value may be found in Table D of Appendix D only if degrees of freedom are known. For the two-way test, degrees of freedom equals number of categories for the column variable minus one, times the number of categories for the row variable minus one [df = (C – 1) (R-1)].

Some precautions are necessary in using the chi-square tests. One restriction is that the chi-square test requires that observations be (9) independent. In this case, independence means that one observation should have no influence on another. One obvious violation of independence occurs when a single subject contributes more than one pair of observations. One way to check that this requirement is not being violated is to remember that the total for all observed frequencies must never exceed the total number of subjects. Using chi-square appropriately also requires that expected frequencies not be too small. Generally, any expected frequency of less than (10) 5 is too small. Small sample sizes should also be avoided, as should unduly large sample size. A sample size that is too large produces a test that detects differences of no practical importance.

When the null hypothesis has been rejected, the researcher should consider using squared (11) Cramer's phi coefficient to determine whether the strength of the relationship is small, medium, or large.