Goodness-of-Fit Test
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| What symbol is used to represent the “chi-square” random variable? | The symbol used to represent the “chi-square” random variable is “χ^2”.
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| Is the “chi-square” distribution always symmetric like the z-distribution and t-distribution? | No. The “chi-square” distribution is not generally symmetric.
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| What determines the shape of the chi-square distribution? | The shape of the chi-square distribution depends on the degrees of freedom, just like Student’s t-distribution.
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| Will the chi-square distribution ever be roughly symmetric? | Yes. As the number of degrees of freedom increases, the chi-square distribution becomes more nearly symmetric.
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| Will the values of the chi-square distribution ever be negative? | No. The values of “χ^2” are nonnegative, i.e., the values of “χ^2”are greater than or equal to 0.
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| What is a “goodness-of-fit” test? | A “goodness-of-fit” test is an inferential procedure used to determine whether a frequency distribution follows a specific (“claimed”) distribution. The chi-square distribution is used in conducting the goodness-of-fit test.
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| How are the hypotheses for the goodness-of-fit test specified? | The Null and Alternative hypotheses are setup as follows: H0: The random variable follows a certain distribution H1: The random variable does “not” follow a certain distribution
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| Based on the way the alternative hypothesis is specified, is the goodness-of-fit test a two-tailed test? | No. While the wording might lead you to believe that the goodness-of-fit test is a two-tailed test, it is, in fact, ALWAYS A RIGHT TAILED TEST.
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| How is the “test statistic” for the goodness-of-fit test calculated? | The “test statistic” = “χ^2”. = Σ[(Oi – Ei)^2/Ei], where the Oi values are the OBSERVED counts from the sample and the Ei values are the EXPECTED counts (the counts expected if the claimed distribution were true); i = 1,...,k.
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| Under the “classical approach”, how to we decide whether to reject or not reject the null hypothesis? | Under the classical approach, we would determine the critical chi-square value. If the test statistic is greater than the critical chi-square value we REJECT the null hypothesis, otherwise we DO NOT REJECT the null hypothesis.
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| How is the critical chi-square value calculated? | Using Tables, you would find appropriate critical chi-square value based on k-1 degrees of freedom, where “k” is the number of data categories, and “α”, the level of significance. Some calculators have a program for the INVCHI function that may be used.
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| Uisng Tables, you would find appropriate critical chi-square value based on k-1 degrees of freedom, where “k” is the number of data categories, and “α”, the level of significance. Some calculators have a program for the INVCHI function that may be used. | Under the “p-value approach”, calculate the probability that a randomly selected value from the chi-square distribution is greater than the “test statistic”. As always, if this p-value is LESS THAN α, REJECT the null hypothesis; otherwise, DO NOT REJECT.
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| How is the “p- value calculated”? | If you are using a TI-83/84, you would use 2nd VARS and choose χ^2cdf. The format is: χ^2cdf(lower bound, upper bound, df), where: lower bound = the test statistic; upper bound = 1E99; “df”, degrees of freedom = k – 1.
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| As an example, calculate the p-value if the calculated value of the test statistic is 6.605 and there are k = 5 categories of data. | The format to compute the p-value is: χ^2cdf(6.605, 1E99, 4) = 0.1583 Note the degrees of freedom: k – 1 = 5 – 1 = 4.
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| Is there a function in the TI-83/84 that will calculate all the values necessary to perform the goodness-of-fit test? | The TI-84 does have such a function, but the Ti-83 does not. Enter the OBSERVED values in L1 and the EXPECTED values in L2. In the TI-84, use STAT ==>TESTS ==> D: χ^2GOF – Test.
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