Problem 40 - An Introduction to Mathematical Statistics and Its Applications

Question

A random sample of 747 obituaries published recently in Salt Lake City newspapers revealed that 344 (or \(46 \%\) ) of the decedents died in the three- month period following their birthdays (131). Assess the statistical significance of that finding by approximating the probability that \(46 \%\) or more would die in that particular interval if deaths occurred randomly throughout the year. What would you conclude on the basis of your answer?

Step-by-Step Solution

Answer

The final conclusion will be that if the p-value is less than \(0.05\), we reject the null hypothesis and conclude that people are more likely to die in the three-month period following their birthdays. If the p-value is greater than \(0.05\), we do not reject the null hypothesis, and we conclude that deaths occur randomly throughout the year.

Step 1: Define the Null Hypothesis and The Alternative Hypothesis

The null hypothesis (\(H_0\)) is that deaths occur randomly throughout the year, i.e., probability of death in any 3 month period is \(0.25\) or \(25\%\). The Alternative Hypothesis (\(H_1\)) is that the deaths do not occur randomly throughout the year i.e., probability of death in the three-month period following someone's birthday is greater than \(0.25\) or \(25\%\).

Step 2: Compute the Test Statistic

The test statistic for a proportion test is defined as \((\hat{p} - p_0) / \sqrt{p_0*(1 - p_0) / n}\), where \(\hat{p}\) is the sample proportion, \(p_0\) is the proportion under \(H_0\), and \(n\) is the sample size. Here, \(\hat{p} = 0.46\), \(p_0 = 0.25\), and \(n = 747\). Substituting these values, we get a test statistic.

Step 3: Compute the p-Value

Using the normal distribution table or a statistical software, find the p-value associated with the calculated test statistic. The p-value is the probability of obtaining a test statistic as extreme or more extreme than the one calculated, assuming \(H_0\) is true. We want to find \(P(Test Statistic > Z)\), where \(Z\) is the computed test statistic.

Step 4: Make the Decision

If the p-value is less than the chosen significance level (typically \(0.05\)), reject the null hypothesis in favor of the alternative. A small p-value indicates that the chance of observing such a result, assuming the null hypothesis is true, is small. Here, if we find that the p-value is smaller than our significance level, we can conclude that the percentage of people dying in the three-month period following their birthdays is significantly higher than what would be expected if deaths were to occur randomly.

Key Concepts

Null Hypothesis

In statistics, the null hypothesis, symbolized as \(H_0\), is a default assumption indicating that there is no effect or no difference. In the case of the obituary exercise, the null hypothesis is that deaths occur randomly throughout the year. This means each person has an equal 25% chance of dying in any given three-month period. This hypothesis acts as a baseline or reference point against which we compare the observed data. By assuming randomness, we can then test whether the actual observed data significantly deviates from this assumption.

Proportion Test

A proportion test is used to determine if the observed proportion of a particular event differs significantly from a known or assumed proportion. In our exercise, we are investigating whether the proportion of deaths (46%) in the three months after a birthday is significantly different from the expected probability of 25% if deaths were random.
To perform the test, we calculate a test statistic using the formula:

• \(\hat{p}\) = observed proportion (0.46)
• \(p_0\) = expected proportion (0.25)
• \(n\) = sample size (747)

This formula helps us understand how far the observed proportion diverges from the expected proportion under the null hypothesis.

p-value

The p-value is a crucial component in hypothesis testing. It represents the probability of obtaining a result as extreme as, or more extreme than, the observed one, under the assumption that the null hypothesis is correct. In simpler terms, it tells us how likely our observed data would occur by random chance.
If the p-value is low, typically less than the significance level of 0.05, it suggests that such an extreme result is unlikely to occur under the null hypothesis, and thus, we might consider the alternative hypothesis instead. In our case, if the p-value is less than 0.05, we might conclude that deaths do not occur randomly throughout the year.

Significance Level

The significance level, often denoted by \(\alpha\), is a threshold used in hypothesis testing to decide whether to reject the null hypothesis. Commonly set at 0.05, it reflects a 5% risk of concluding that a difference exists when there is none.
When the p-value calculated from our test statistic is less than the significance level, it provides strong evidence against the null hypothesis. This implies that it is unlikely that the observed pattern emerged by chance. The significance level thus helps determine the threshold for making statistically confident decisions.
In this exercise, if our p-value is below 0.05, we conclude that the observed 46% death rate following birthdays is statistically significant compared to the assumed random rate of 25%.