Making sure whether the test of goodness of fit was done correctly.

Question

The Elves Toy Co. manufactures toy trains. For quality control purpose it checks 5 random trains from production line . The probability that a train is defective and is thrown away=$0.25$. The number of trains thrown away vs the number of weeks in which that amount of trains were thrown is given in the picture.. Propose an appropriate probability distribution and check for its correctness. Sorry I dont know how to make table :(. $$\text {Attempt} $$. This distribution has to be discrete as number of discarded trains have to integers. Now I think this has to be binomial as poisson is for large number of data set. Thus I got $ E_{i }$ for each number of trains as $12,21,14,5,1,0$ . I then clube last three columns as $\text E_{i} $ has to $\geq 5$. Then $\chi $ came out to be $0.5$ . Now $\chi_{0.05}$ for $3$ df is $7.8$ as this is greater than calculated one thus our test holds good and the distribution is binomial. I am still not able to convince myself thoroughly that its true. Am I correct or am I missing something? Thanks!

Answer 1

Of course, the 'observed counts' $10, 21, 14, \dots$ for 'categories' $0, 1, 2, \dots$ are integers. But it is incorrect to round 'expected counts' to integers; perhaps they can be rounded to a couple of decimal places.

My four expected counts for categories $0, 1, 2, 3^+$ are $12.34, 20.57, 13.71, 5.38, $ respectively. And my chi-squared statistic is $ 0.945 < 7.815,$ so I do not reject the null hypothesis that the number of defective trains out of 5 inspected each week is distributed as $\mathsf{Binom}(5, .25).$

Data are "consistent with" this binomial model. However, the same data might also be consistent with other similar models. (So one should not declare that $\mathsf{Binom}(5, .25)$ is the correct model.)