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The Chi-Squared Test is a statistical method used to determine if there is a significant difference between the observed frequencies and the expected frequencies under a given hypothesis. In this context, we are using it to decide if the observed call frequencies fit a Poisson distribution model.

To perform a Chi-Squared test, we first calculate the expected frequencies based on our hypothesis. These expected frequencies come from a theoretical distribution (in this case, Poisson) defined by a calculated mean. With these expected values, the Chi-Squared statistic is computed as follows:

- Calculate the observed frequencies from the data.
- Calculate the expected frequencies using the Poisson distribution formula for each call count value.
- Use the formula: \[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \] where \( O_i \) are the observed frequencies and \( E_i \) are the expected frequencies.

The result, known as the Chi-Squared statistic, is then compared to a threshold (critical value) from Chi-Squared distribution tables with the corresponding degrees of freedom. This comparison helps determine whether to support or reject the proposed distribution model.

Hypothesis testing is a fundamental aspect of statistical analysis that helps make inferences about a population based on sample data. It involves setting up two contrasting hypotheses: the null hypothesis \( H_0 \) and the alternative hypothesis \( H_1 \).

In this exercise, the null hypothesis \( H_0 \) posits that the number of calls follows a Poisson distribution with parameter \( \lambda = \bar{x} = 10.23 \). This is the hypothesis we assume to be true unless evidence suggests otherwise. The alternative hypothesis \( H_1 \) suggests that the call distribution does not follow a Poisson distribution.

The hypothesis test uses a Chi-Squared Test at a significance level \( \alpha = 0.05 \). This significance level represents a 5% risk of rejecting the null hypothesis if it is indeed true. The test involves calculating a test statistic and comparing it with the critical value from the Chi-Squared distribution. A decision is made based on whether this statistic exceeds the critical value. If it does, \( H_0 \) is rejected in favor of \( H_1 \). Otherwise, we lack sufficient evidence to reject \( H_0 \). This systematic approach ensures consistent decision-making in statistical inference.

Expected frequencies are crucial in determining if our sample data follows a specific statistical model, like the Poisson distribution. They represent the counts we would expect to observe, assuming that the underlying distribution model accurately reflects reality.

To calculate expected frequencies for a Poisson distribution, we use the probability mass function (PMF) of the Poisson distribution: \[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \] where \( \lambda \) is the mean number of occurrences. For each observed value \( k \), this function gives the probability of observing exactly \( k \) events. Multiply this by the total number of observations to get the expected frequency for each \( k \).

For example, to find the expected frequency for \( k = 5 \), plug \( \lambda = 10.23 \) into the PMF and compute: \[ P(X = 5) = \frac{e^{-10.23} \times 10.23^5}{5!} \approx 0.036 \] Then multiply by the total observations (30 days) to yield an expected frequency of approximately 1.08 for \( X = 5 \). This process is repeated for each distinct call count value. Expected frequencies provide a basis for the Chi-Squared Test, allowing comparisons between what we theoretically expect and what we actually observe.