Final answer:
To determine if the observed frequencies of a variable differ from a given distribution, we can use the chi-square goodness-of-fit test. The test compares the expected frequencies to the observed frequencies and if the difference is too large, we reject the null hypothesis. In this case, we calculate the expected frequencies and the chi-square test statistic to determine if the observed frequencies differ from the given distribution.
Step-by-step explanation:
To determine if the observed frequencies of a variable differ from a given distribution, we can use the chi-square goodness-of-fit test. The null hypothesis for this test states that the observed frequencies follow the given distribution. The test compares the expected frequencies (calculated from the given distribution) to the observed frequencies, and if the difference is too large, we reject the null hypothesis.
In this case, the given distribution is 0.1875, 0.1875, 0.3125, 0.3125. The observed frequencies are 20, 21, 22, 33. To perform the chi-square test, we need to calculate the expected frequencies. Multiply each distribution value by the total sample size to get the expected frequencies: 0.1875 * 96 = 18, 0.1875 * 96 = 18, 0.3125 * 96 = 30, 0.3125 * 96 = 30.
Next, we calculate the chi-square test statistic using the formula: X^2 = (O1 - E1)^2 / E1 + (O2 - E2)^2 / E2 + (O3 - E3)^2 / E3 + (O4 - E4)^2 / E4.
Finally, we compare the calculated test statistic with the critical value from the chi-square distribution table at the specified significance level (0.05). If the calculated test statistic is greater than the critical value, we reject the null hypothesis and conclude that the observed frequencies differ from the given distribution.