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Weatherwise magazine is published in association with the American Meteorological Society. Volume \(46,\) Number 6 has a rating system to classify Nor'easter storms that frequently hit New England states and can cause much damage near the ocean coast. A severe storm has an average peak wave height of 16.4 feet for waves hitting the shore. Suppose that a Nor'easter is in progress at the severe storm class rating. (a) Let us say that we want to set up a statistical test to see if the wave action (i.e., height) is dying down or getting worse. What would be the null hypothesis regarding average wave height? (b) If you wanted to test the hypothesis that the storm is getting worse, what would you use for the alternate hypothesis? (c) If you wanted to test the hypothesis that the waves are dying down, what would you use for the alternate hypothesis? (d) Suppose you do not know whether the storm is getting worse or dying out. You just want to test the hypothesis that the average wave height is different (either higher or lower) from the severe storm class rating. What would you use for the alternate hypothesis? (e) For each of the tests in parts (b), (c), and (d), would the area corresponding to the \(P\) -value be on the left, on the right, or on both sides of the mean? Explain your answer in each case.

Step by step solution

01

Understanding the Null Hypothesis

The null hypothesis is a statement that there is no effect or no difference and is often the hypothesis that researchers aim to test against. In this context, the null hypothesis ( \( H_0 \) ) regarding average wave height would be that the average peak wave height is equal to the severe storm class rating, which is 16.4 feet. Therefore, \( H_0: \mu = 16.4 \) where \( \mu \) is the true average peak wave height.

02

Alternate Hypothesis for Worsening Storm (Part b)

If we want to test the hypothesis that the storm is getting worse, the alternate hypothesis ( \( H_a \) ) would indicate that the average peak wave height is greater than 16.4 feet. Thus, \( H_a: \mu > 16.4 \) .

03

Alternate Hypothesis for Dying Down Storm (Part c)

If we want to test the hypothesis that the waves are dying down, the alternate hypothesis ( \( H_a \) ) would indicate that the average peak wave height is less than 16.4 feet. Thus, \( H_a: \mu < 16.4 \) .

04

Alternate Hypothesis for Any Change in Storm (Part d)

If we want to test the hypothesis that the average wave height is different from 16.4 feet, we use a two-tailed test. The alternate hypothesis ( \( H_a \) ) would express that the average peak wave height is not equal to 16.4 feet. Thus, \( H_a: \mu eq 16.4 \) .

05

Determining the P-value Region (Part e)

For each test:- Part (b): Since the hypothesis is one-sided ( \( H_a: \mu > 16.4 \) ), the \( P \)-value corresponds to the area on the right side of the mean.- Part (c): As this is also one-sided ( \( H_a: \mu < 16.4 \) ), the \( P \)-value corresponds to the area on the left side of the mean.- Part (d): This is a two-tailed test, so the \( P \)-value area is on both sides of the mean.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis

The null hypothesis is a fundamental concept in hypothesis testing. It represents the statement or condition where no effect or no difference exists. It is the assumption that the observed phenomenon has no influence or that the outcome is the same as an established fact. For example, in the case of the Nor'easter storm, the null hypothesis is that the average peak wave height remains at 16.4 feet, the same as the severe storm class rating. This can be mathematically expressed as \( H_0: \mu = 16.4 \), where \( \mu \) is the true average peak wave height. The role of the null hypothesis is to serve as a default position that indicates no change or effect until evidence suggests otherwise. Hypothesis testing involves trying to gather enough evidence to reject this null hypothesis in favor of an alternate hypothesis.

Alternate Hypothesis

The alternate hypothesis offers a statement opposite to the null hypothesis. If the null hypothesis suggests no change, the alternate hypothesis supports a change or effect. In hypothesis testing, researchers aim to provide evidence for the alternate hypothesis, which represents the actual claim tested. In the context of the severe storm class rating scenario, if the hypothesis is that the storm is worsening, the alternate hypothesis is that the peak wave height is greater than 16.4 feet, noted as \( H_a: \mu > 16.4 \). Alternatively, if the hypothesis is that the storm is dying out, \( H_a: \mu < 16.4 \). Finally, if there is no predetermined direction of change, we simply test for any difference, using \( H_a: \mu eq 16.4 \). Each of these reflects a different aspect of change we are testing against the null hypothesis.

P-Value

In hypothesis testing, the P-Value measures the strength of evidence against the null hypothesis. It is a probability that helps us determine the significance of our results. A smaller P-Value suggests that the observed data would be unlikely under the null hypothesis. If the P-Value is less than a pre-determined threshold (often 0.05), we have enough evidence to reject the null hypothesis in favor of the alternate hypothesis. Conversely, a higher P-Value means we retain the null hypothesis as there isn’t enough evidence against it. In our storm example, if the test for worsening conditions (\( H_a: \mu > 16.4 \)) produced a small P-Value, it would suggest the storm was indeed worsening.

One-Tailed Test

A one-tailed test examines the relationship in a single direction of interest. It is used when researchers have a specific direction of change in mind, such as testing if something is greater or less than a particular value. The test only considers this one scenario, making it powerful when there is a clear hypothesis regarding direction. For instance, in detecting if the storm is getting worse (\( H_a: \mu > 16.4 \)), we use a one-tailed test. The P-Value calculated will correspond to the area on the right side of the mean, reflecting the probability of observing values greater than 16.4 feet. Thus, one-tailed tests focus on deviations in only one critical direction.

Two-Tailed Test

A two-tailed test, unlike a one-tailed, accounts for variations in both directions from the mean. It is used when there is no predetermined direction of change and the focus is on detecting a difference, regardless of its direction. For the Nor'easter example, where we test whether the average wave height is simply different from the norm (\( H_a: \mu eq 16.4 \)), a two-tailed test applies. Here, the P-Value region exists on both sides of the mean, capturing deviations that may result in either higher or lower wave heights than 16.4 feet. Two-tailed tests are thorough and offer a moderate approach when direction isn't a prior concern, ensuring that any possible significant change is not overlooked.