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The test procedure, called the two-proportion z-test, is appropriate when the following conditions are met: The table below shows three sets of hypotheses. Transforming raw scores to z scores. Transforming z scores to raw scores. . Figure 4: Normal curve showing proportion of scores within 1 standard .. ii What is the relation between the mean, μX, of the distribution of sampling. Do the data in the table represent a proportional relationship? Duplicating any part of For each ordered pair, find the value of the ratio ž. For (3, ), the value .
The other two sets of hypotheses Sets 2 and 3 are one-tailed testssince an extreme value on only one side of the sampling distribution would cause a researcher to reject the null hypothesis. When the null hypothesis states that there is no difference between the two population proportions i. It should specify the following elements.
Often, researchers choose significance levels equal to 0. Use the two-proportion z-test described in the next section to determine whether the hypothesized difference between population proportions differs significantly from the observed sample difference. Analyze Sample Data Using sample data, complete the following computations to find the test statistic and its associated P-Value. Compute the standard error SE of the sampling distribution difference between two proportions. The test statistic is a z-score z defined by the following equation.
The P-value is the probability of observing a sample statistic as extreme as the test statistic.
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Since the test statistic is a z-score, use the Normal Distribution Calculator to assess the probability associated with the z-score. See sample problems at the end of this lesson for examples of how this is done. The analysis described above is a two-proportion z-test. Interpret Results If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis.
Typically, this involves comparing the P-value to the significance leveland rejecting the null hypothesis when the P-value is less than the significance level. Test Your Understanding In this section, two sample problems illustrate how to conduct a hypothesis test for the difference between two proportions.
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The first problem involves a a two-tailed test; the second problem, a one-tailed test. The company states that the drug is equally effective for men and women. To test this claim, they choose a a simple random sample of women and men from a population ofvolunteers. Based on these findings, can we reject the company's claim that the drug is equally effective for men and women?
Hypothesis Test: Difference Between Proportions
Labor force status by Marital status This table is a crosstabulation of Labor force status by Marital status, with column proportions shown as the summary statistic. Comparisons of column proportions The column proportions test table assigns a letter key to each category of the column variables.
For Marital status, the category Married is assigned the letter A, Widowed is assigned the letter B, and so on, through the category Never married, which is assigned the letter E. For each pair of columns, the column proportions are compared using a z test. Seven sets of column proportions tests are performed, one for each level of Labor force status.
For each significant pair, the key of the smaller category is placed under the category with the larger proportion. For the set of tests associated with Working full time, the B key appears in each of the other columns.
- Comparing Column Proportions
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Also, the A key appears in the C column. No other keys are reported in other columns. Thus, you can conclude that the proportion of divorced persons who are working full time is greater than the proportion of married persons working full time, which in turn is greater than the proportion of widowers working full time. The proportions of people who are separated or never married and working full time cannot be differentiated from people who are divorced or married and working full time, but these proportions are greater than the proportion of widowers working full time.
Thus, the proportions of people who have never been married and are in school or are working part time are greater than the proportions of married, widowed, or divorced people who are in school or working part time. For the tests associated with Temporarily not working or with Other labor status, no other keys are reported in any columns.
Thus, there is no discernible difference in the proportions of married, widowed, divorced, separated, or never-married people who are temporarily not working or are in an otherwise uncategorized employment situation. The tests associated with Retired show that the proportion of widowers who are retired is greater than the proportions of all other marital categories who are retired.
Moreover, the proportions of married or divorced people who are retired is greater than the proportion of never-married persons who are retired.
There are greater proportions of people married, widowed, or separated and keeping house than proportions of people divorced or never married and keeping house. The proportion of people who have never been married and are Unemployed, laid off is higher than the proportions of people who are married or widowed and unemployed. Also, note that the Separated column is marked with a ".
Significance Results in APA-style Notation If you do not want the significance results in a separate table, you can choose to display them in the main table.P Values, z Scores, Alpha, Critical Values