Question

It is known that a random variable \(X\) has a Poisson distribution with parameter \(\mu .\) A sample of 200 observations from this distribution has a mean equal to 3.4. Construct an approximate 90 percent confidence interval for \(\mu\).

Short answer

The calculated approximate 90% confidence interval for \(\mu\) of the Poisson distribution given the sample is (\(3.4 - 1.645 * \frac{\sqrt{3.4}}{\sqrt{200}}\), \(3.4 + 1.645 * \frac{\sqrt{3.4}}{\sqrt{200}}\)).

Step-by-step solution

Calculate sample mean and standard deviation

We know that in a Poisson distribution, both the mean \(\mu\) and standard deviation \(\sqrt{\mu}\) are equal to the parameter of the distribution. Here, the sample mean is 3.4. So, the standard deviation of the population is also \(\sqrt{3.4}\).

Decide on the level of confidence

We want the confidence level to be 90%. For a two-sided confidence interval and a normal distribution, Z-scores for common confidence levels are: 1.645 for 90% confidence. So, we'll take Z = 1.645.

Use the central limit theorem to construct the confidence interval

Because the sample size is large enough, by the central limit theorem, we can say that the sampling distribution of the sample mean will approximate a normal distribution. We can construct a confidence interval for the population mean (\(\mu\)) using the formula: \((\bar{x} - Z * \frac{\sigma}{\sqrt{n}}, \bar{x} + Z * \frac{\sigma}{\sqrt{n}})\) where \(\bar{x}\) is the sample mean, Z is the Z-score from Step 2, \(\sigma\) is the sample standard deviation, and n is the sample size. Substituting in the values we have: \((3.4 - 1.645 * \frac{\sqrt{3.4}}{\sqrt{200}}, 3.4 + 1.645 * \frac{\sqrt{3.4}}{\sqrt{200}})\)

Compute the confidence interval for \(\mu\).

Calculate the two bounds by performing the operations described in the formula. This will provide the lower and upper bounds for the 90% confidence interval.

Key concepts

Poisson Distribution

The Poisson distribution is a probability model often used for counting events that occur randomly and independently within a fixed interval or space. For instance, it could be applied to counting the number of phone calls received by a call center per hour or the number of decay events from a radioactive source in a given time frame.

What makes this distribution unique is that it is defined by a single parameter, \(\mu\), which represents both the mean and the variance of the events. When using the Poisson distribution, we operate under the assumption that the events occur with a known constant mean rate and independently of the time since the last event.

When given a problem involving a Poisson distribution, calculating the sample mean can provide an estimate of the parameter \(\mu\), which allows for further statistical analysis such as constructing confidence intervals, as seen in our exercise.

Sample Mean

In statistics, the sample mean is one of the most fundamental concepts. It represents the average of a set of observations and is calculated by summing all observations and then dividing by the number of observations. In our exercise, the sample mean is given as 3.4, calculated from 200 observations of a Poisson-distributed variable.

The sample mean is important when trying to estimate population parameters. Our exercise demonstrates how the sample mean of a sufficiently large sample can approximate the population mean, which is crucial for constructing confidence intervals to estimate the true population mean within a specific level of certainty.

Central Limit Theorem

The Central Limit Theorem (CLT) is a statistical theory that states that, given a sufficiently large sample size, the sampling distribution of the sample mean will approximate a normal distribution, regardless of the original distribution of the population. This theorem is foundational as it validates the use of normal probability models for inference in many practical situations.

In the context of our exercise, the CLT allows us to construct a confidence interval for the Poisson-distributed population mean, even though the Poisson distribution is not normal. Because we have a large sample size of 200, we can assume the distribution of the sample mean is approximately normal, which then enables us to use Z-scores associated with the normal distribution to construct our confidence interval.

Z-score

A Z-score, also known as a standard score, measures the number of standard deviations an observation or datum is from the mean. In standard normal distribution, which is symmetrical and centered around a mean of 0, the Z-score indicates how far, in standard deviation units, a data point is from this mean.

In a confidence interval calculation, the Z-score corresponds to the desired confidence level. For example, a 90% confidence level warrants a Z-score of approximately 1.645; this signifies that the interval extends 1.645 standard deviations from the sample mean on both sides.

In our exercise, the Z-score is applied to the formula for constructing the confidence interval, and since the sample size is large, the use of Z-scores from the normal distribution is justified by the Central Limit Theorem. By doing so, it provides a way to create an interval that estimates the possible values of the true mean of the population with a specific level of confidence.