chi-squared test

(noun)

In probability theory and statistics, refers to a test in which the chi-squared distribution (also chi-square or χ-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables.

Related Terms

Examples of chi-squared test in the following topics:

  • Student Learning Outcomes

  • Structure of the Chi-Squared Test

    • The chi-square test is used to determine if a distribution of observed frequencies differs from the theoretical expected frequencies.
    • If a chi squared test is conducted on a sample with a smaller size, then the chi squared test will yield an inaccurate inference.
    • The researcher, by using chi squared test on small samples, might end up committing a Type II error.
    • This means chi-squared cannot be used to test correlated data (like matched pairs or panel data).
    • First, we calculate a chi-square test statistic.

  • Example: Test for Independence

    • The chi-square test for independence is used to determine the relationship between two variables of a sample.
    • The chi-square test for independence is used to determine the relationship between two variables of a sample.
    • It is important to keep in mind that the chi-square test for independence only tests whether two variables are independent or not.
    • Using the chi-square test for independence, who gets into more trouble between boys and girls cannot be evaluated directly from the hypothesis.
    • In the chi-square test for independence, the degrees of freedom are found as follows:

  • Randomization for two-way tables and chi-square

    • In short, we create a randomized contingency table, then compute a chi-square test statistic.
    • We repeat this many times using a computer, and then we examine the distribution of these simulated test statistics.
    • When the minimum threshold is met, the simulated null distribution will very closely resemble the chi-square distribution.

  • Using simulation for goodness of fit tests

    • In short, we simulate a new sample based on the purported bin probabilities, then compute a chi-square test statistic $X^2_{sim}$.
    • We do this many times (e.g. 10,000 times), and then examine the distribution of these simulated chi-square test statistics.
    • Since the minimum bin count condition was satisfied, the chi-square distribution is an excellent approximation of the null distribution, meaning the results should be very similar.
    • Figure 6.21 shows the simulated null distribution using 100,000 simulated values with an overlaid curve of the chi-square distribution.
    • The precise null distribution for the juror example from Section 6.3 is shown as a histogram of simulated $X^2_{sim}$ statistics, and the theoretical chi-square distribution is also shown.

  • Finding a p-value for a chi-square distribution

    • According to the rule above, the test statistic should then follow a chi-square distribution with k − 1 = 3 degrees of freedom if H 0 is true.
    • If the null hypothesis is true, the test statistic = 5.89 would be closely associated with a chi-square distribution with three degrees of freedom.
    • The p-value for this test statistic is found by looking at the upper tail of this chi- square distribution.
    • There are three conditions that must be checked before performing a chi-square test:
    • The p-value for the juror hypothesis test is shaded in the chi-square distribution with df = 3.

  • Chi Square Distribution

    • Define the Chi Square distribution in terms of squared normal deviates
    • A Chi Square calculator can be used to find that the probability of a Chi Square (with 2 df) being six or higher is 0.050.
    • The Chi Square distribution is very important because many test statistics are approximately distributed as Chi Square.
    • Two of the more common tests using the Chi Square distribution are tests of deviations of differences between theoretically expected and observed frequencies (one-way tables) and the relationship between categorical variables (contingency tables).
    • Numerous other tests beyond the scope of this work are based on the Chi Square distribution.

  • Facts About the Chi-Square Distribution

    • The test statistic for any test is always greater than or equal to zero.
    • When df > 90, the chi-square curve approximates the normal.
    • In the next sections, you will learn about four different applications of the Chi-Square Distribution.
    • These hypothesis tests are almost always right-tailed tests.
    • Think about the implications of right-tailed versus left-tailed hypothesis tests as you learn the applications of the Chi-Square Distribution.

  • Estimating a Population Variance

    • The chi-square distribution is used to construct confidence intervals for a population variance.
    • A chi-square distribution can be used to construct a confidence interval for this variance.
    • The chi-square distribution with a $k$ degree of freedom is the distribution of a sum of the squares of $k$ independent standard normal random variables.
    • The chi-squared distribution is a special case of the gamma distribution and is used in the common chi-squared tests for goodness of fit of an observed distribution to a theoretical one, the independence of two criteria of classification of qualitative data, and in confidence interval estimation for a population standard deviation of a normal distribution from a sample standard deviation.
    • The chi-square distribution is a family of curves, each determined by the degrees of freedom.

  • Chi-Square Probability Table

    • Areas in the chi-square table always refer to the right tail.