A psychologist is interested in comparing the demographics of grand jury members to demographics of the population to see if grand jury panels are really representative of the population. The first variable she examines is age. The percentage of people over 65 in the population is 25%, but out of 200 people empaneled for grand jury trials, 76 (38%) were aged 65 or more. She wants to know if the proportion of people over 65 on grand juries is significantly different than that proportion in the population. What test should she use?
t-test for independent samples
A t-test to compare means requires a continuous dependent measure. We do not have means; instead we have frequency data and proportions.
t-test for paired data
A paired t-test is appropriate for continuous data where observations are paired or matched, as with before-after measures. In this example, we have frequency data.
Chi-square goodness of fit test
The chi-square goodness of fit test can be used to test observed frequency data such as the number of individuals over 65 who serve on grand jury trials against an expected value based on the percentage of individuals over 65 in the population.
If the grand jury panel is selected randomly from the population we would expect 25% of the 200 members to be over age 65, which is 50 members. With the chi-square test we can see whether the observed number is significantly different from the expected number of 50 members.
Chi-square test of independence
The chi-square test of independence can be used to test for a difference in proportions based on two samples. In this example we have only one sample to be compared to a population value.
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The Expert says…
The null hypothesis is that the proportion of people over age 65 among empaneled jurors is the same as the proportion of people over age 65 in the population.
We have frequency data from a sample (200 empaneled jurors) that we wish to compare to expectation under the null hypothesis that the sample is drawn randomly from the population. Knowledge of the population proportion allows us to compute the expected number of people who are over age 65 in a random sample of 200 cases. In this case, we would expect about 50 of the 200 jurors to be over age 65 if the sample was drawn randomly from the population. We can use the chi-square goodness of fit test to see whether the observed number is significantly different from this expected number. Do the observed data ‘fit’ the model?
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