In a particular chi-square goodness-of-fit test there are four categories and 200 observations. The significiance level is 0.05. How many degrees of freedom are there? What is the critical value of chi-square? compute the value of chi-square?
Use a chi-square table with k-1 degrees of freedom (4 - 1 = 3 degrees of freedom) at 0.05 level of significance to determine your critical value.
To determine the degrees of freedom in a chi-square goodness-of-fit test, we need to subtract 1 from the number of categories. In this case, there are four categories, so the degrees of freedom will be (4 - 1) = 3.
To find the critical value of chi-square, we need to refer to the chi-square distribution table with the given significance level of 0.05 and the degrees of freedom of 3. The critical value can be found at the intersection of the row corresponding to 3 degrees of freedom and the column for the cumulative probability of 0.05.
Please note that the actual critical value may vary depending on the specific chi-square distribution table you are using.
To compute the chi-square value, we need to perform the following steps:
1. State the null hypothesis (H0) and the alternate hypothesis (H1). For a goodness-of-fit test, the null hypothesis states that the observed frequencies in each category are equal to the expected frequencies. The alternate hypothesis suggests that there is a difference between the observed and expected frequencies.
2. Determine the expected frequencies for each category. In this case, if there are 200 observations equally distributed among the four categories, the expected frequency for each category would be 200/4 = 50.
3. Calculate the chi-square test statistic using the formula:
χ^2 = Σ((O - E)^2 / E)
where:
χ^2 is the chi-square test statistic,
O is the observed frequency,
E is the expected frequency, and
Σ represents the summation over all categories.
Calculate this value for each category and sum them up to obtain the chi-square test statistic.
4. Compare the computed chi-square value to the critical value obtained from the chi-square distribution table. If the computed chi-square value is greater than the critical value, reject the null hypothesis; otherwise, fail to reject the null hypothesis.
Remember, without access to the actual observations and their distribution across the categories, it is not possible to compute the specific values in your case. Be sure to substitute the appropriate observed values in the above formula to get the actual chi-square value.