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A Confidence Interval (CI) gives us a range within which we expect the true parameter, in this case, the Poisson parameter \(\lambda\), to lie. When we talk about a CI, it gives you an idea of the reliability of the estimate you have made. Here, the CI is constructed around the sample mean, since for a Poisson distribution, the mean is equal to its parameter \(\lambda\). In this exercise, we aim to construct a CI for \(\lambda\) based on the sample mean from the absences data. The formula used is derived by approximating the distribution of the sample mean with a normal distribution. This becomes valid especially when dealing with large sample sizes. For the CI, we use the formula:

• \(\lambda = \bar{x} \pm z_{\alpha/2}\sqrt{\frac{\bar{x}}{n}}\)

Here, \(z_{\alpha/2}\) is the critical value from the standard normal distribution for our chosen level of confidence. Most commonly, a 95% confidence level is chosen, where \(z_{0.025}\approx 1.96\).
The CI tells us that we can be 95% confident that the true \(\lambda\) is within this derived interval.

The Sample Mean is a central concept in statistics and is used as an estimate of the population mean. Here, it serves as an estimate for the Poisson parameter \(\lambda\) since, for a Poisson distribution, the mean has the same value as \(\lambda\). To find the sample mean, you multiply each observed value (e.g., number of absences) by its frequency, sum those products, and then divide by the total number of observations, \(n\). In this problem, the sample mean \(\bar{x}\) is calculated as follows:

• \(\bar{x} = \frac{(0\times 1 + 1\times 4 + 2\times 8)}{13} = \frac{20}{13} \approx 1.54\)

This mean stands for the average number of absences per day based on the data collected from the 50 recorded days. Calculating the sample mean is a key step in constructing the confidence interval for \(\lambda\).
This value not only estimates the central tendency of our data but is also pivotal for further statistical inference, such as constructing confidence intervals or hypothesis testing.

The Standard Normal Distribution is a special case of a normal distribution with a mean of 0 and a standard deviation of 1. It is often denoted as \(Z\). This distribution is fundamental as it allows for standardizing data to compare different datasets or variables. When we standardize data, we convert an original data value \(X\) to a standardized value \(Z\) using the formula:

• \(Z = \frac{X - \mu}{\sigma}\)

where \(\mu\) is the mean and \(\sigma\) is the standard deviation of the original data. In this exercise, we approximate the sampling distribution of our sample mean using a normal distribution because of the Central Limit Theorem, which facilitates using \(Z\) for large sample sizes.The Standard Normal Distribution provides the critical values \(z_{\alpha/2}\) necessary for constructing the confidence interval. For a 95% confidence level, this critical value is approximately 1.96. This process helps transform any normal random variable to a standard form, making it much easier to calculate probabilities and critical values for hypothesis testing and confidence intervals.

Sample Variance measures the spread or variability of sample data points around the mean. In the context of a Poisson distribution, the variance is theoretically equal to \(\lambda\), the parameter of the distribution. However, calculating the sample variance provides a robustness check, confirming if our assumption for the variance's similarity to the mean holds true.To compute the sample variance \(s^2\), use the formula:

• \(s^2 = \frac{\Sigma ((x_i - \bar{x})^2 \cdot f_i)}{n-1}\)

where \(x_i\) represents observed values, \(\bar{x}\) is the sample mean, and \(f_i\) is the frequency. This calculation ensures that the variability of data is as expected.
Since Poisson's mean and variance are equal, you expect the sample variance to be approximately the same as the sample mean. Nonetheless, knowing the spread of data through the sample variance is crucial in determining how much data points differ from the mean, providing a deeper insight into the data distribution's nature.