Chapter 9 Three Tests of Significance Winston Jackson and Norine Verberg Methods: Doing Social...

Chapter 9Three Tests of Significance

Winston Jackson and Norine Verberg

Methods: Doing Social Research, 4e

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Inferential Statistics

Inferential statistics do two things:1. Allow us to judge the accuracy of

generalizing from a limited sample to the larger population We can be 95% certain that the sample

mean will be 30%, plus or minus 4%

2. Conduct hypothesis testing Indicate whether study outcome is a fluke or

reflects a true difference in the population i.e., say if the findings are statistically significant

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What Does Statistically Significant Mean?

A test of significance reports the probability that an observed difference or association is a result of sampling fluctuations and not reflective of a “true” difference in the population from which the sample was selected

Three tests of statistical significance introduced in Chapter 9 Chi-square test, t-test, and F-test

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Preliminary Considerations

1. Research and null hypothesis

2. The sampling distribution Standard error of the means

3. One- and two-tailed tests of significance

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1. Research and Null Hypothesis

Tests of significance are used to test hypotheses Set up in the form of a “research hypothesis” and

“null hypothesis” Research Hypothesis (or Alternative Hypothesis)

States one’s prediction of the relationship between the variables

Null Hypothesis States the prediction that there is no relation

between the variables

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Research and Null Hypothesis (cont’d)

Research hypothesis 1: The greater the participation, the higher the self-esteem Null hypothesis 1: There is no relation between

levels of participation and levels of self-esteem Research hypothesis 2: Male university faculty

members earn more money than their female counterparts after controlling for qualifications, achievements, and experience Null hypothesis 2: There is no relation between

gender and earnings of faculty members, after controls

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Research and Null Hypothesis (cont’d)

It is the null hypothesis that is tested Leads us to accept or reject the null hypothesis

If the null hypothesis is accepted: Conclude that the association or difference may

simply be the result of sampling fluctuations and may not reflect an association or difference in the population being studied

Research hypothesis deemed to therefore be false

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Research and Null Hypothesis (cont’d)

If the null hypothesis is rejected: Argue that there is an association between the

variables in the population, and that this association is of a magnitude that probably has not occurred because of chance fluctuations in sampling

Would then examine the data to see if the association is in the predicted direction i.e., consistent with prediction (it could be

different than predicted)

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Findings and Probability

When the results of a study lead to the rejection of the null hypothesis, this only means that there is probably a relationship between the variables under examination It is one piece of evidence that the

relationships exists Other researchers will test it again, and either

confirm or disconfirm the past findings Research conclusions are therefore treated as

tentative, open to disconfirmation

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Did they fail if they accept the null?

Some researchers believe they failed if they accept the null hypothesis (i.e., find no relation between the variables rather than support for the predicted relationship) Not so: it is just as important to show that two

variables are not associated as it is to find out they are associated

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2. The Sample Distribution

Tests of significance report whether an observed relationship could be the result of sample fluctuations or reflect a “real” difference in the population from which the sample has been taken

Sample fluctuation is the idea that each time we select a sample we will get somewhat different results If we draw 1,000 samples of 50 cases, each

will be slightly different from the first sample

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The Sample Distribution (cont’d)

If the means of the same variable for each of the samples were plotted, a normal curve would results, but it would be peaked (or leptokurtic)

Example: The means of weights of respondents are plotted The weights range from 70 to 80 kg, but the

majority of samples would cluster around the true mean weight of 75 kg

Note: We are plotting the mean weights of the respondents in each of the 1000 samples drawn

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The Sample Distribution (cont’d)

The distribution is quite peaked because we are plotting the mean weights for each sample

To measure the dispersion of the means of the samples, we use a statistic called the standard error of the means

Npopulation Sd

means of error Standard

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The Sample Distribution (cont’d)

Relevance to hypothesis testing? In doing tests of significance, we are

assessing whether the results of one sample fall within the null hypothesis acceptance zone (usually 95% of the distribution) or outside the zone, in which we reject the null hypothesis

Four key points that can be made about probability sampling procedures where repeated measures are taken

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Four Key Points: Repeated Samples

1. Plotting the means of repeated samples will produce a normal distribution: it will be more peaked than when raw data are plotted (as shown in Figure 9.1)

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Four Key Points (cont’d)

2. The larger the sample sizes, the more peaked the distribution and the closer the means of the samples to the population mean (shown in Figure 9.2)

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Four Key Points (cont’d)

3. The greater the variability in the population, the greater the variations in the samples

4. When sample sizes are above 100, even if a variable in the population is not normally distributed, the means will be normally distributed when repeated samples are plotted E.g., weight of population of males and

females will be bimodal, but if we did repeated samples, the weights would be normally distributed

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3. One- and Two-Tailed Tests

If the direction of a relationship is predicted, the appropriate test will be one-tailed If the direction of the relationship is not

predicted, conduct a two-tailed test Example:

One tailed: Females are less approving of violence than are males

Two-tailed: There is a gender difference in the acceptance of violence [Note: No prediction about which gender is more approving]

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One- and Two-Tailed Tests (cont’d)

Figure 9.3 (next slide) shows two normal distribution curves

The first one has the 5% rejection area split between the two tails—this would be a two-tailed test

The second one has the 5% rejection area all in one tail, indicating a one tailed test

Same principle applies to 1% level

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Figure 9.3 Five Percent Probability Rejection Area: One- and Two-Tailed Tests

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Chi-Square: Red and White Balls

The Chi-Square test (X2) is used primarily in contingency table analysis, where the dependent variable is nominal level

The formula is:

e

2eo2

f)ff(

X

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One Sample Chi-Square Test

Suppose the following incomes:

INCOME STUDENT SAMPLE

% OF SAMPLE GENERAL POPULATION

Over $100,000 30 15.0 7.8

$40,000 to $99,999 160 80.0 68.9

Under $40,000 10 5.0 23.3

TOTAL 200 100.0 100.0

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The Computation

Chi-squares compare expected frequencies (assuming the null hypothesis is correct) to the observed frequencies.

To calculate the expected frequencies, simply multiply the proportion in each category of the general population times the total number of cases (e.g., 200 students)

Why do you do this?

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Why?

If the student sample is drawn equally from all segments of society, then they should have the same income distribution (this is assuming the null hypothesis is correct)

So what are the expected frequencies in this case?

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Expected Frequencies fe

Frequency Frequency

Observed Expected

30 15.6 (200 x .078)

160 137.8 (200 x .689)

10 46.6 (200 x .233)

Degrees of Freedom = 2

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Decision

Look up critical value: Table 9.2, p. 260 Need to know:

2 degrees of freedom .05 level of significance 1 tailed test (i.e., column one)

Find the critical value = 4.61 Compare to the Chi-Square calculated = 45.61 Decision: Calculated value exceeds critical value so

reject null hypothesis Inspect the data, conclude university students from

higher SES background

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Standard Chi-Square Test

Drug use by Gender (Box 9.4, p. 261) 3 categories of drug use (no experience, once or

twice, three or more times) row marginal x column marginal ÷ total N of

cases = expected frequencies Degrees of freedom = (row – 1)(columns – 1) =

2

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Decision

With 2 degrees of freedom, 2-tailed test, .05 level of significance, the critical value is 5.99

Calculated Chi-Square is 5.69 Does not equal or exceed the critical value So, your decision is what?

Accept the null hypothesis

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The t Distribution: t-Test Groups and Pairs

Used often for experimental data

t-test used when: Sample size is small (e.g.,< 30) Dependent variable measured at ratio level Random assignment to treatment/control

groups Treatment has two levels only Population normally distributed

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The t Distribution: t-Test Groups and PairsThe t-test represents the ratio between the

difference in means between two groups and the standard error of the difference. Thus:

t = difference between the means

standard error of the difference

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Two t-Tests: Between- and Within-Subject Design Between-subjects: used in an experimental

design, with an experimental and a control group, where the groups have been independently established

Within-subjects: In these designs the same person is subjected to different treatments and a comparison is made between the two treatments.

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The F Distribution: Means, ANOVA

Box 9.7 provides an illustration of one-way analysis of variance (“Egalitarianism by Country,” p. 269)

Concern is with how much variation there is within columns compared to variation between columns

The F represents the ratio of between variation divided by within variation

Probabilities looked up on Table 9.4, p. 272

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When Are Tests of Significance Not Appropriate? Total populations studied Non-probability sampling procedures used High nonparticipation rates Nonexperimental research tests for

intervening variables Research is not guided by formal hypotheses