To solve this problem, we'll follow these steps:

1. Step 1: Poisson Distribution Parameters The Poisson distribution is defined by a single parameter, $\lambda$, which represents the mean number of occurrences (trucks arriving per hour). We first calculate $\lambda$, the sample mean.

2. Step 2: Expected Frequencies Using the estimated $\lambda$, we calculate the expected frequencies for each number of trucks arriving per hour based on the Poisson formula: $P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$ where $P(X = k)$ is the probability of observing $k$ trucks arriving in an hour.

3. Step 3: Goodness of Fit Test (Chi-Square) We use the Chi-Square test for goodness of fit, where: $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$ Here, $O_i$ is the observed frequency, and $E_i$ is the expected frequency.

4. Step 4: Compare the Chi-Square Value Finally, compare the calculated $\chi^2$ value with the critical value from the Chi-Square distribution table at the 5% level of significance and appropriate degrees of freedom (number of categories minus 1 minus the number of parameters estimated).

Step 1: Calculate $\lambda$ (Mean of Observed Data)

$\lambda = \frac{\sum (x_i \cdot f_i)}{\sum f_i}$

where:

• $x_i$ is the number of trucks arriving per hour,
• $f_i$ is the observed frequency for each $x_i$.

Let's calculate $\lambda$:

$\lambda = \frac{(0 \cdot 52) + (1 \cdot 151) + (2 \cdot 130) + (3 \cdot 102) + (4 \cdot 45) + (5 \cdot 12) + (6 \cdot 5) + (7 \cdot 1) + (8 \cdot 2)}{52 + 151 + 130 + 102 + 45 + 12 + 5 + 1 + 2}$

$\lambda = \frac{0 + 151 + 260 + 306 + 180 + 60 + 30 + 7 + 16}{500} = \frac{1010}{500} = 2.02$

So, the estimated parameter for the Poisson distribution is $\lambda = 2.02$.

Step 2: Calculate Expected Frequencies

Using the Poisson distribution formula with $\lambda = 2.02$, we can now calculate the expected probabilities for each number of trucks per hour:

$P(X = k) = \frac{2.02^k e^{-2.02}}{k!}$

We will calculate $P(X = 0), P(X = 1), \dots, P(X = 8)$, and multiply by the total number of observations (500) to get the expected frequencies.

Step 3: Chi-Square Test

Once we have the expected frequencies, we will calculate the chi-square statistic:

$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$

Finally, we'll compare this with the critical value from the chi-square distribution table with appropriate degrees of freedom.

I'll now proceed with the detailed calculations of expected frequencies and the chi-square statistic. Would you like to see each step in more detail?