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Chapter 11: Problem 81

You are performing a goodness-of-fit test with four categories, all of which are supposed to be equally likely. You have a total of 100 observations. The observed frequencies are \(21,26,31\), and 22, respectively, for the four categories. a. Show that you would fail to reject the null hypothesis for these data for any reasonable significance level. b. The sum of the absolute differences (between the expected and the observed frequencies) for these data is 14 (i.e., \(4+1+6+3=14\) ). Is it possible to have different observed frequencies keeping the sum at 14 so that you get a \(p\) -value of \(.10\) or less?

Short Answer

Expert verified
a. The null hypothesis that the observed frequencies follow the specified distribution will be rejected or accepted based on the p value obtained by computing the Chi-square statistic. Here, it won't be rejected for any reasonable significance level. b. Yes, it is possible to have different observed frequencies keeping the sum of absolute differences constant and achieving a p-value of .10 or less. This involves an iterative process of adjusting frequencies and checking the resulting p-value.

Step by step solution

01

Calculation of Chi-square statistic for current data

The Chi-square statistic can be computed using the formula: \[X^2 = \sum \frac{(O_i - E_i)^2}{E_i}\] Where \(O_i\) and \(E_i\) are the observed and expected frequencies, respectively. Plug the observed and expected frequencies for the provided data and compute the Chi-square statistic.
02

Determination of the p-value

The degrees of freedom in this scenario are \(k-1\), where \(k\) is the number of categories. Here, \(k=4\), hence the degrees of freedom=3. The p-value for the calculated Chi-square value can be obtained from the Chi-square distribution table or using statistical software. This p-value is then compared with the significance level. If the p-value is less than the significance level, the null hypothesis is rejected, otherwise it is accepted.
03

Change in observed frequencies

In part b, you are asked to adjust the observed frequencies while keeping the total sum of the absolute differences from the mean constant in such a way that the p-value would decrease to 0.1 or less. Adjust the observed frequencies and repeat Steps 1 and 2 until a p-value less than or equal to 0.1 is obtained.

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Most popular questions from this chapter

Describe the four characteristics of a multinomial experiment.

Of all students enrolled at a large undergraduate university, \(19 \%\) are seniors, \(23 \%\) are juniors, \(27 \%\) are sophomores, and \(31 \%\) are freshmen. A sample of 200 students taken from this university by the student senate to conduct a survey includes 50 seniors, 46 juniors, 55 sophomores, and 49 freshmen. Using the \(10 \%\) significance level, test the null hypothesis that this sample is a random sample. (Hint: This sample will be a random sample if it includes approximately \(19 \%\) seniors, \(23 \%\) juniors, \(27 \%\) sophomores, and \(31 \%\) freshmen.)

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