Problem 2 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

On average, the number of babies born in Cleveland, Ohio, in the month of September is 1472 . On January 26,1977 , the city was immobilized by a blizzard. Nine months later, in September 1977 , the recorded number of births was 1718 . Can the increase of 246 be attributed to chance? To investigate this, the number of births in the month of September is modeled by a Poisson random variable with parameter \(\mu\), and we test \(H_{0}: \mu=1472\). What would you choose as the alternative hypothesis?

Step-by-Step Solution

Answer

The alternative hypothesis is \( H_1: \mu > 1472 \).

Step 1: Understand the Problem

The problem provides data about the average number of babies born in Cleveland in September and compares two specific instances: the average (1472) and the recorded number (1718) in September 1977, which occurred 9 months after a blizzard. The task is to determine if the increase can be attributed to chance, assuming births follow a Poisson distribution.

Step 2: Identify the Hypotheses

In hypothesis testing, the null hypothesis (\( H_0 \)) is a statement of no effect or no difference. Here, \( H_0 \) is \( \mu = 1472 \), indicating that the number of births is as expected. The alternative hypothesis (\( H_1 \)) needs to reflect an 'increase' due to the blizzard, which would be \( \mu > 1472 \).

Step 3: Formulate the Alternative Hypothesis

Choose the hypothesis that captures the expected effect if the null is not true. Given that the problem suggests testing if there is an increase, the alternative hypothesis should be \( H_1: \mu > 1472 \).

Key Concepts

Poisson Distribution

The Poisson distribution is a probability distribution used to model the number of events occurring within a fixed interval of time or space. It's named after the French mathematician, Siméon Denis Poisson. This distribution is often used when events happen independently, and you know the average number of times the event occurs within the given interval.

To use the Poisson distribution, you need to know the average rate of occurrence, which in this problem is the average number of babies born in Cleveland during the month of September, estimated to be 1472. This average rate, denoted by \( \mu \), serves as the key parameter of the Poisson distribution.

Why the Poisson distribution? Well, it’s perfect for instances where the probabilities of each occurrence are small but the number of occurrences is large, like the number of births happening in a designated time period.

• The Poisson distribution is special because it's defined by only one parameter \( \mu \).
• The events must be independent, and it should be possible for none, one, or more events to occur in the given timeframe or region.

It's these characteristics that make the Poisson distribution an ideal choice for modeling birth occurrences in Cleveland in this instance.

Null Hypothesis

The null hypothesis, denoted as \( H_0 \), is a cornerstone of hypothesis testing. It is essentially the default assumption that there is no effect or no difference present. In the context of our Cleveland birth data, the null hypothesis represents no significant increase in the number of monthly births compared to the usual number.

For this exercise, the null hypothesis \( H_0: \mu = 1472 \) suggests that the actual number of births in September 1977 should align with the average birth rate of any typical September, which means the number there reflects normal variance.

Key traits of the null hypothesis include:

• It's the hypothesis that researchers typically try to disprove or reject.
• Under \( H_0 \), any observed change is attributed to random sampling variability or chance.
• Statistical tests often assume the null hypothesis to start, determining the likelihood of observing the data if \( H_0 \) is true.

The aim is to collect enough evidence to reject \( H_0 \), thereby suggesting that the observed phenomenon is unlikely to be due solely to random fluctuations.

Alternative Hypothesis

The alternative hypothesis, denoted as \( H_1 \) or \( H_a \), provides a statement opposing the null hypothesis. It posits that there is a meaningful effect or difference. In our scenario, the alternative hypothesis suggests that there is indeed an increase in births that September, suggesting the blizzard might be a contributing factor.

The hypothesis \( H_1: \mu > 1472 \) formulates the expectation that, counter to the null, the birth rate exceeded the average due to some influence like the blizzard's potential impact on birth patterns.

Characteristics of an alternative hypothesis include:

• It's mutually exclusive with the null hypothesis, meaning both cannot be true simultaneously.
• The demonstration of its validity is often the research's goal if evidence suggests rejecting \( H_0 \).
• For a one-sided test, as in this Poisson setup, \( H_1 \) specifically tests if the parameter is greater (or less) than what's stated in \( H_0 \).

Thus, by rejecting the null hypothesis in favor of the alternative, one might conclude that the increased birth rate could potentially be non-random.