Question

The data in Hurricanes contains the number of hurricanes that made landfall on the eastern coast of the United States over the 101 years from 1914 to 2014 . Suppose we are interested in testing whether the number of hurricanes is increasing over time. (a) State the null and alternative hypotheses for testing whether the correlation between year and number of hurricanes is positive, which would indicate the number of hurricanes is increasing. (b) Describe in detail how you would create a randomization distribution to test this claim (if you had many more hours to do this exercise and no access to technology).

Short answer

The null hypothesis (\(H_0\)) is: 'The correlation between year and number of hurricanes is equal to zero,' and the alternative hypothesis (\(H_A\)) is: 'The correlation between year and number of hurricanes is greater than zero.' A randomization distribution would be created by rerunning the test multiple times with random samples, recording the correlation each time to see if the actual observed correlation is an extreme value.

Step-by-step solution

Formulating the Hypotheses

The null hypothesis and the alternative hypothesis are two dichotomous statements about the population. For this exercise, we are interested in examining whether the number of hurricanes is increasing over time. The null hypothesis, \(H_0\), would be that there is no correlation between the year and the number of hurricanes, implying that the number of hurricanes is not increasing over time. The alternative hypothesis, \(H_A\), would be that there is a positive correlation between the year and the number of hurricanes, indicating an increase in the number of hurricanes over time. Thus, the null hypothesis, \(H_0\), is: 'The correlation between year and number of hurricanes is equal to zero,' and the alternative hypothesis, \(H_A\), is: 'The correlation between year and number of hurricanes is greater than zero.'

Creating a Randomization Distribution

A randomization distribution would be created to test the hypotheses by rerunning the test multiple times with random samples. Given unlimited time and no access to technology. To create a randomization distribution, follow these steps: First, record the number of hurricanes for each year. Second, mix these results up and then assign them at random to the years. This effectively breaks any relationship that may exist between the year and the number of hurricanes. Third, calculate the correlation under this random assignment. This serves as a single data point representing the null hypothesis. Lastly, repeat these steps as many times as possible to generate a distribution of correlations that would be expected under the null hypothesis. This randomization distribution is then used to see if the actual observed correlation is an extreme value, which might lead to the rejection of the null hypothesis in favor of the alternative hypothesis.

Key concepts

Null and Alternative Hypotheses

Understanding the null and alternative hypotheses is a foundational aspect of statistical hypothesis testing. In simple terms, the null hypothesis (\(H_0\)) represents the default assumption or the status quo, suggesting there is no effect or no association between the variables being studied. Specifically in our exercise, the null hypothesis posits that there is no upward trend in the number of hurricanes over time, which means that the correlation between the year and number of hurricanes is zero.

In contrast, the alternative hypothesis (\(H_A\)) is the statement we wish to validate, based on the evidence. It suggests that there is an effect or association, which in this case, is a positive correlation indicating that the number of hurricanes is increasing over time. Formulating these hypotheses carefully is pivotal, as they guide the direction of the statistical test and the interpretation of the results.

It’s important to note that these hypotheses are mutually exclusive and exhaustive, meaning only one can be true and no other outcomes are possible within the context of the test. When we perform the statistical test, we'll be essentially checking the evidence against the null hypothesis, and if it’s strong enough, we can reject it in favor of the alternative.

Correlation

Correlation is a statistical measure that expresses the extent to which two variables are linearly related. It’s a value between -1 and 1 where:

• \(-1\) indicates a perfect negative linear relationship,
• 0 indicates no linear relationship, and
• \(+1\) indicates a perfect positive linear relationship.

For the task described in the exercise, a positive correlation would mean as one variable (year) increases, so does the other (number of hurricanes). The strength of the correlation is key to our hypothesis test; a higher positive correlation would support the alternative hypothesis. However, it is crucial to remember that correlation does not imply causation. Even if we find a strong positive correlation, we cannot conclude that an increase in the year causes more hurricanes—there could be other lurking variables or factors contributing to this observed relationship.

Randomization Distribution

A randomization distribution is a fundamental tool in the absence of certain assumptions required for traditional parametric tests. It’s created by simulating the process of allocating observations to groups in a random manner many times over, to build a distribution that represents a world where the null hypothesis is true.

To construct a randomization distribution for our hurricane exercise, we would shuffle the years’ data and randomly pair them with the number of hurricanes, removing any actual chronological association. We then calculate the correlation coefficient for each of these randomized pairings and record the result. Through repeated shufflings, we create a multitude of correlation coefficients that mimic what we would expect to see if there truly were no relationship between years and the number of hurricanes.

The observed correlation from our real data is then compared to this distribution. If the observed correlation falls outside the majority of the generated random correlations—typically in the tails of the distribution—we would have evidence to reject the null hypothesis in favor of the alternative. This method does not rely on assumptions of normal distributions and is especially valuable when dealing with small or non-standard data, but it requires a large number of repetitions to gain reliable insight.