Goodness of fit tests：練習問題

A Goodness of Fit test is being conducted to see if a set of data fits a discrete uniform distribution with 6 categories. If the total frequency is 120, what is the expected frequency for each category?

A Goodness of Fit test is performed to determine if a data set follows a Normal distribution. The data was grouped into 8 classes. The mean and standard deviation were estimated from the sample data. Calculate the degrees of freedom for the \(\chi^2\) test statistic, assuming no pooling was required.

An experimenter is testing the Goodness of Fit of a Poisson distribution to a set of data with 6 possible outcomes (0, 1, 2, 3, 4, \(\ge 5\)). The mean \(\lambda\) was estimated from the data. During calculation, the expected frequencies were: 12.5, 22.0, 15.4, 8.2, 4.1, and 2.8. After pooling necessary classes to satisfy the standard requirement for \(\chi^2\) tests, how many degrees of freedom are available?

A researcher is conducting a \(\chi^2\) test for independence using a contingency table. The table consists of \(4\) rows and \(3\) columns. Calculate the degrees of freedom that should be used for this test.

A die is rolled 60 times to test if it is fair. The observed frequency for the outcome '6' is 15. Calculate the contribution to the \(\chi^2\) statistic for this specific category (outcome '6') under the null hypothesis that the die is fair.

A researcher is investigating whether a specific Poisson distribution with a mean of 2 is an appropriate model for the number of emails received by a small business in 100 random one-hour intervals. The observed data is summarized below:

Number of Emails (x) | Observed Frequency (O)
0 | 11
1 | 29
2 | 26
3 | 18
4 | 9
\(\ge 5\) | 7

(a) Calculate the expected frequencies for each category using a Poisson distribution with \(\lambda = 2\). Round values to two decimal places.

(b) Explain why some categories might need to be pooled before performing a Goodness of Fit test.

(c) Perform the test at a 5% significance level to determine if the Poisson model is a good fit. (Critical value for \(\chi^2\) with 4 degrees of freedom at 5% is 9.488).