You will need to test if the following sample comes from a Poisson distribution

Question

You will need to test if the following sample comes from a Poisson distribution at alpha = 0.07. Form the null and alternative hypotheses H_0 and H_1. Make a sketch of the test. Specify appropriate equations for the X^2 statistic and degrees of freedom df. Estimate A as the mean of a Poisson distribution if H_0 holds true. Thus lambda^^= x^-. Given lambda^^, use Excel to calculate expected probabilities for x values. Based on the expected probabilities, calculate expected frequencies. Based on expected and observed frequencies, calculate the X^2 statistic and its df. Given alpha = 0.07 and df, use CHlSQ. lNV(*, *) to look up for the critical value X^2*. Compare X^2 statistic with X^2* critical to reject or accept H_0. Interpret your result. Sketch and find the p-value of the test. Would you reject the null if alpha = 0.10?

Explanation / Answer

Use Goodness-of-Fit Test for Poisson to test the hypotheses:

H0: Data follow a Poisson distribution

H1: Data do not follow a Poisson distribution

Welcome to Minitab, press F1 for help.
MTB > PGoodness 'x';
SUBC>   Frequencies 'f';
SUBC>   GBar;
SUBC>   GChiSQ;
SUBC>     Pareto;
SUBC>   RTable.

Goodness-of-Fit Test for Poisson Distribution

Data column: x
Frequency column: f

Poisson mean for x = 3.07

                   Poisson            Contribution
x    Observed Probability Expected     to Chi-Sq
0          14     0.046421   13.9263       0.00039
1          32     0.142513   42.7539       2.70492
2          75     0.218757   65.6272       1.33861
3          58     0.223862   67.1585       1.24896
4          62     0.171814   51.5442       2.12099
5          37     0.105494   31.6481       0.90504
>=6        22     0.091139   27.3418       1.04362

N N* DF   Chi-Sq P-Value
300   0   5 9.36253    0.095

Chart of Observed and Expected Values

Chart of Contribution to the Chi-Square Value by Category

Interpreting the results

Minitab calculates each category's contribution to the chi-square value as the square of the difference in the observed and expected values for a category divided by the expected value for that category. The largest difference between the observed and expected value is for the category with 1 accident and is the highest contributor to the chi-square statistic . However, the contribution is not enough to reject the null hypothesis . If you choose an a-level of 0.07, the p-value for this test is 0.095, which is greater than 0.07. Therefore, you can conclude that you do not have enough evidence to reject that the number of accidents at a particular intersection follow a Poisson distribution.

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