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Chapter 4: Problem 9

It is proposed to fit the Poisson distribution to the following data: \begin{tabular}{c|ccccc} \(x\) & 0 & 1 & 2 & 3 & \(3

Short Answer

Expert verified
After conducting the analysis through the steps outlined above, a \(\chi^2\) value is obtained along with the degrees of freedom. If the \(\chi^2\) value exceeds the critical value for a Chi-Square distribution with 4 degrees of freedom at a 0.05 significance level, the data does not follow a Poisson distribution. Otherwise, the data is consistent with a Poisson distribution.

Step by step solution

01

Calculate the mean

First, calculate the mean (\(\lambda\)) of the distribution by taking the sum of the products of \(x\) and their respective frequencies, and then dividing by the total frequency. The total frequency is the sum of all frequencies, and here is \(20+40+16+18+6=100\). So, the mean is \((0*20 + 1*40 + 2*16 + 3*18 + 4*6) / 100 = 1.38\).
02

Calculate the expected frequency

To calculate the expected frequency for each \(x\), use the formula \(E(x) = n * P(x)\), where \(E(x)\) is the expected frequency, \(n\) is the total frequency (100 in this case), and \(P(x)\) is the probability of \(x\) under the Poisson distribution, which is \((e^{-\lambda} * \lambda^x) / x!\). Use the values for \(\lambda\) from Step 1, calculate \(P(x)\) for each \(x\), and then use them to calculate \(E(x)\).
03

Calculate Chi-Square Statistic

Next, the Chi-Square statistic is calculated using the formula \(\chi^2=\Sigma((O-E)^2/E)\), where \(O\) is the observed frequency, \(E\) is the expected frequency, and the summation (\(\Sigma\)) is over all \(x\). Using the observed and expected frequencies from Steps 1 and 2, calculate \(\chi^2\). For example, for \(x=0\), \((O-E)^2/E = (20-E_0)^2/E_0\). Do this for all \(x\) and sum up the results.
04

Calculate Degrees of Freedom

The degrees of freedom (\(df\)) for a Chi-Square test is the number of outcomes minus 1, so for this dataset, \(df = 5-1 = 4\).
05

Compare to Chi-Square Distribution

The last step is to compare the calculated Chi-Square statistic from Step 3 to the Chi-Square distribution with \(df = 4\) at the \(\alpha = 0.05\) level. If our calculated \(\chi^2\) value is greater than the critical value from the Chi-Square distribution, we reject the null hypothesis that the data follows a Poisson distribution. If not, we do not reject the null hypothesis.

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