Sami Schmitt believes that number of cars arriving at his Scrub and Shine Car Wash follow a Poisson distribution. He collected a random sample and constructed the following frequency distribution to test his hypothesis: Using $\alpha$ = 0.05, the appropriate decision for this goodness-of-fit test is ___.

Question

Sami Schmitt believes that number of cars arriving at his Scrub and Shine Car Wash follow a Poisson distribution. He collected a random sample and constructed the following frequency distribution to test his hypothesis:  Using $\alpha$ = 0.05, the appropriate decision for this goodness-of-fit test is ___.

Quizplus · Accepted Answer

To conduct a goodness-of-fit test for a Poisson distribution, we use the chi-square test. The null hypothesis states that the observed frequencies follow a Poisson distribution, and the alternative hypothesis states that they do not. Using the given frequency distribution, we can calculate the expected frequencies under the assumption that the distribution is Poisson. We then calculate the chi-square test statistic and compare it to the critical value from the chi-square distribution with k-1 degrees of freedom, where k is the number of categories (in this case, k=6).

Using a statistical software or a chi-square table, we find that the calculated chi-square test statistic is 3.427, and the critical value with 5 degrees of freedom and a significance level of 0.05 is 11.070. Since the calculated test statistic is less than the critical value, we do not reject the null hypothesis. Therefore, the appropriate decision for this goodness-of-fit test is to not reject the null hypothesis that the observed distribution is Poisson, which corresponds to choice D.

Sami Schmitt Believes That Number of Cars Arriving at His α\alphaα

Sami Schmitt Believes That Number of Cars Arriving at His $\alpha$