Question 1173255: a. A chi-square goodness of fit test, a chi-square independence test, or a chi-square homogeneity test is always right-tailed. Why?
b. Suppose you are conducting chi-square goodness of fit test. If the sum of the expected frequencies does not equal the sample size, what do you conclude?
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Let's break down these chi-square concepts:
**a. Why Chi-Square Tests Are Always Right-Tailed**
* **Chi-Square Statistic:** The chi-square statistic is calculated by summing the squared differences between observed and expected frequencies, divided by the expected frequencies.
* The formula is: χ² = Σ [(O - E)² / E]
* Where:
* O = Observed frequency
* E = Expected frequency
* **Squared Differences:** Notice that the numerator of the formula involves squaring the differences (O - E)². Squaring any number, whether positive or negative, always results in a non-negative value.
* **Non-Negative Statistic:** Since all the terms in the summation are non-negative, the chi-square statistic itself is always non-negative.
* **Right-Skewed Distribution:** The chi-square distribution is a right-skewed distribution. This means that it has a long tail extending towards the right.
* **Rejection Region:** Large values of the chi-square statistic indicate a significant difference between the observed and expected frequencies. Therefore, the rejection region for chi-square tests is always in the right tail of the distribution.
* **Focus on Discrepancies:** The purpose of the chi-square test is to determine if there are significant discrepancies between observed and expected frequencies. Larger discrepancies lead to larger chi-square values, which fall into the right tail of the distribution.
In essence, because the chi-square statistic is always non-negative and we're interested in detecting large deviations from expected values, the test is always right-tailed.
**b. Chi-Square Goodness of Fit Test and Unequal Sums**
* **Expected Frequencies:** In a chi-square goodness of fit test, the expected frequencies represent the frequencies you would expect to see if the null hypothesis were true. They are calculated based on a theoretical distribution or a hypothesized proportion.
* **Sample Size:** The sample size is the total number of observations in your data.
* **Sum of Expected Frequencies:** The sum of the expected frequencies should always equal the sample size.
* **Discrepancy:** If the sum of the expected frequencies does not equal the sample size, it indicates an error in your calculations or in the setup of your test.
* **Conclusion:**
* You have made a mistake in calculating the expected frequencies.
* There is a mistake in the data.
* The test has been setup incorrectly.
* You cannot proceed with the chi-square goodness of fit test until you identify and correct the error.
It is a basic requirement that the total of the expected frequencies match the total of the observed frequencies (the sample size). If they don't, the test is invalid.
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