# Math Help - Poisson Goodness-of-Fit Question

1. ## Poisson Goodness-of-Fit Question

I'm trying to review and I have a question:

If 23 random samples of asbestos fibers found the following samples
31, 29, 19, 18, 31, 28, 34, 27, 34, 30, 16, 18, 26, 27, 27, 18, 24, 22, 28, 24, 21, 17, 24

How should I go about using a Goodness-of-Fit test to show that the Poisson distribution is a plausible model in describing the variability of asbestos fiber counts at the 5% level of significance? Should I set up a table and compute the expected counts versus the observed and proceed from there?

Also, how should I go about giving a point estimate and 95% Confidence Interval estimate of the mean asbestos fiber counts? I am totally lost on this one.

Thank you for any help!

2. $(1)$ Find the likelihood function of $Poi(\lambda);$ Take the log and maximize to determine $\hat\lambda_{MLE}.$

$(2)$ $E_i = E(Poi) = \hat\lambda_{MLE}.$ $O_i$ is your data.

$(3)$ Compute $X^2 = \sum \frac{(O_i - E_i)^2}{E_i}.$

$(4)$ Check out the table value of $\chi_{23-1,\color{red}{p}} (X^2)$

If p-value $= 1-\color{red}{p}$ $> \alpha = .05,$ reject the ho.

3. Thank you so much! I have one more question. How should I go about creating the point estimate and 95% confidence interval estimate of the mean fiber counts?