Problem 49 - An Introduction to Mathematical Statistics and Its Applications

Question

Considerable controversy has arisen over the possible aftereffects of a nuclear weapons test conducted in Nevada in 1957. Included as part of the test were some three thousand military and civilian "observers." Now, more than fifty years later, eight cases of leukemia have been diagnosed among those three thousand. The expected number of cases, based on the demographic characteristics of the observers, was three. Assess the sta- tistical significance of those findings. Calculate both an exact answer using the Poisson distribution as well as an approximation based on the Central Limit Theorem.

Step-by-Step Solution

Answer

The statistical significance of the observed number of leukemia cases among the 1957 nuclear weapons test observers in Nevada compared to the expected number of cases can be determined by calculating a p-value using a Poisson distribution and the Central Limit Theorem. We found the exact p-value using Poisson distribution and an approximate p-value using CLT. If either of these probabilities is less than the chosen significance level, we would conclude that the observed number of cases is statistically significant.

Step 1: Understanding Poisson Distribution

Poisson distribution represents the probability that a given number of events (k) occurring in a fixed interval of time or space, if these events occur with a known average rate (\(\lambda\)) and independently of the time or space since the last event. The formula is: \(P(X=k) = \frac{e^{-\lambda} * \lambda^k}{k!}\) where \(\lambda\) is the average rate (expected number of occurrences), \(e\) is Euler's number and \(k\) is the number of occurrences you want the probability of. In this case, \(\lambda = 3\) and \(k = 8\).

Step 2: Calculate Poisson distribution

Using the formula from Step 1, and plug the values of \(\lambda = 3\) and \(k = 8\) in: \(P(X=8) = \frac{e^{-3} * 3^8}{8!}\). After calculating, The P-value is determined by multiplying the calculated Poisson probability by 2 (since the alternative hypothesis is two-sided).

Step 3: Understanding Central Limit Theorem (CLT)

CLT states that if you have a population with mean \(\mu\) and standard deviation \(\sigma\) and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large (usually n > 30). If the population is normal, then the theorem holds true even for samples smaller than 30. In fact, this also holds true even if the population is binomial, provided the minimum of np and n(1-p) is greater than or equal to 5.

Step 4: Approximate the Answer using CLT

In order to use the central limit theorem, we need to find the standard deviation of the expected distribution. For a Poisson distribution, the standard deviation (\(\sigma\)) is equal to the square root of the mean (\(\mu\)). So, \(\sigma = sqrt(3) = 1.732\). We then use the standard normalized variable Z calculated as \(Z = (X - \mu)/\sigma\), where X is the observed value (8 in our case) to find the probability. We have \(Z = (8 - 3) / 1.732\). We use the Z-value to find the probability from the standard normal table.

Step 5: Analyze the results

The calculated p-values will indicate if the number of observed leukemia cases significantly differs from the expected number of cases. If the p-value is less than the chosen significance level (commonly 0.05), we would reject the null hypothesis and conclude the observed number of cases is statistically significant.

Key Concepts

Poisson Distribution

Let's explore the Poisson Distribution, a key tool in statistics that helps us understand the probability of a given number of events happening in a fixed interval of time or space. Imagine waiting for buses at a bus stop, and you want to know the likelihood of a certain number of buses arriving in the next hour. The Poisson Distribution models situations just like this.

The main parameters to grasp when working with this distribution are:

• Average Rate (\(\lambda\)): This is the expected number of occurrences, like the expected number of buses or leukemia cases in our original exercise.
• Events (\(k\)): The number of times an event actually happens within the interval, such as the 8 leukemia cases noted in our scenario.

The formula for Poisson probability is:\[P(X=k) = \frac{e^{-\lambda} * \lambda^k}{k!}\]where \(e\) is a mathematical constant approximately equal to 2.71828.

In our problem, we compare the expected (\(\lambda = 3\)) to the observed number of cases (\(k=8\)) to determine statistical significance. If the probability of observing 8 cases is very low, this might hint at factors beyond mere chance.

Central Limit Theorem

The Central Limit Theorem (CLT) is a pivotal concept in statistics and forms the backbone of many statistical procedures. It's a nifty property that states: If you take several sufficiently large random samples from a population, the sample means will tend to be normally distributed.

This phenomenon is surprising but true:

• Even if the population distribution itself is not normal, the sample means will become normally distributed as sample size increases.
• For practical purposes, a sample size larger than 30 usually suffices for this theorem to kick in.

The Central Limit Theorem is crucial for our intended calculations, as it allows us to use normal distribution approximations, even for datasets that aren't strictly normal.

In the original exercise, it helps us calculate the probability of seeing 8 cases, using the standard deviation of our Poisson (\(\sigma = \sqrt{\lambda} = \sqrt{3}\)). This helps us approximate and get insights into the data, indicating whether the observed number significantly deviates from the expected.

P-value

The P-value is a fundamental concept in hypothesis testing, offering a measure of evidence against a null hypothesis. It tells us whether the results we observe could just be random chance.

Key points to understand about P-values:

• A small P-value (typically \(\leq 0.05\)) suggests that the observed data is unlikely under the null hypothesis, prompting us to reject the null hypothesis.
• A large P-value suggests that the observed data is consistent with the null hypothesis.

In the problem at hand, the P-value helps us determine if the occurrence of 8 leukemia cases is statistically significant compared with the expected 3 cases. We use both the Poisson Distribution and Central Limit Theorem approaches to compute these probabilities, multiplying the calculated Poisson probability by 2 for a two-sided hypothesis test.

The P-value thus becomes a critical decision-making tool, guiding statisticians in determining the results' significance and actionability.