Can you give a real world example of when you would use a chi-square test of independence and a chi-square goodness of fit?
Sure! Let's start with the chi-square test of independence. This test is used to determine if there is a relationship between two categorical variables.
For example, let's say you want to investigate whether there is a relationship between gender and preference for different types of music. You collect data from a sample of individuals, recording their gender (male or female) and their music preference (rock, pop, hip-hop, country).
To conduct the chi-square test of independence, you would first organize your data into a contingency table. This table would show the frequencies of each combination of gender and music preference.
Once you have your contingency table, you can use statistical software or a chi-square calculator to calculate the chi-square test statistic. This statistic measures the difference between the observed frequencies and the expected frequencies assuming independence.
If the chi-square test statistic is statistically significant, it suggests that there is a relationship between gender and music preference. In this case, you could conclude that gender is associated with music preference.
Now, let's move on to the chi-square goodness of fit test. This test is used to compare observed frequencies to expected frequencies in a single categorical variable.
For example, imagine you are a quality control manager in a factory producing copper wires. You want to check if the diameters of the wires conform to the specification. The specification requires that 40% of the wires have a diameter of 0.5 mm, 30% have a diameter of 0.4 mm, and 30% have a diameter of 0.3 mm.
To conduct the chi-square goodness of fit test, you would first collect a random sample of wires and measure their diameters. Then, you would calculate the observed frequencies for each diameter category.
Next, you would calculate the expected frequencies based on the specified proportions (40%, 30%, and 30%).
Using the chi-square test formula, you would determine the chi-square test statistic. If the calculated chi-square value is not statistically significant, it suggests that the observed frequencies match the expected frequencies, and you can conclude that the wires conform to the specification.
In summary, the chi-square test of independence is used to determine if there is a relationship between two categorical variables, whereas the chi-square goodness of fit test is used to compare observed frequencies to expected frequencies in a single categorical variable.