91Ó°ÊÓ

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 Problem

The problem requires setting up hypotheses to analyze whether Nor'easter storms are worsening (higher wave height) or improving (lower wave height), or if the wave height is simply different from the severe storm average of 16.4 feet.

02

Define the Null Hypothesis

(a) The null hypothesis (\(H_0\)) is a statement that there is no change or effect. So, for this scenario, the null hypothesis would be that the average wave height is equal to 16.4 feet: \(H_0: \mu = 16.4\) feet.

03

Worse Storm Hypothesis

(b) An alternate hypothesis (\(H_a\)) is used to test whether a condition is true. If we hypothesize that the storm is getting worse (higher wave height), the alternate hypothesis would be: \(H_a: \mu > 16.4\) feet.

04

Dying Down Storm Hypothesis

(c) If we hypothesize that the waves are dying down (lower wave height), the alternate hypothesis would be: \(H_a: \mu < 16.4\) feet.

05

Different Wave Height Hypothesis

(d) If we are testing the hypothesis that the average wave height is simply different from 16.4 feet (could be higher or lower), the alternate hypothesis would be: \(H_a: \mu eq 16.4\) feet.

06

Determine the Position of P-value

(e) - For part (b), since we are testing if the storm is getting worse, the \(P\)-value will be on the right since it's a 'greater than' test.- For part (c), testing if the storm is dying down means the \(P\)-value will be on the left as it's a 'less than' test.- For part (d), the \(P\)-value will be on both sides since it's a 'not equal to' test, which is two-tailed.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

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 statistical hypothesis testing. It represents a statement of no effect or no difference, and it is the hypothesis that we initially assume to be true.
In our specific context of Nor'easter storms, the null hypothesis is constructed to reflect the idea that the average wave height remains stable at a known severe storm peak height of 16.4 feet. This presumption implies that there is no increase or decrease in wave activity; thus, the storm's intensity hasn't changed.
In symbolic form, the null hypothesis is expressed as:

• \( H_0: \mu = 16.4 \) feet

By stating the null hypothesis, we are setting a baseline against which to measure any observed data. If the data suggests a significant deviation from this hypothesis, we might then consider alternative explanations. However, the null hypothesis serves as our starting point, embodying the "status quo" or the "no change" scenario.

Alternate Hypothesis

The alternate hypothesis is the complement to the null hypothesis and is what we test against. It denotes that there is a statistically significant effect or difference.
There are a few scenarios where alternate hypotheses differ based on what we are looking to investigate about the Nor'easter's wave activities.
When testing if the storm is worsening, the hypothesis would be:

• \( H_a: \mu > 16.4 \) feet

This implies that the average wave height is greater than 16.4 feet, suggesting the storm is intensifying.
When considering if the storm is dying down, the alternate hypothesis is:

• \( H_a: \mu < 16.4 \) feet

This implies that the average wave height is less than 16.4 feet, suggesting a decrease in storm activity.
For cases where we are testing for any difference, whether the wave height is higher or lower, the hypothesis becomes two-tailed:

• \( H_a: \mu eq 16.4 \) feet

Here, we are open to any deviation from the known measure, indicating simply that change has occurred.

P-value

The P-value is a crucial component in hypothesis testing, helping us decide whether to reject the null hypothesis. It is the probability of obtaining test results at least as extreme as the observed data, assuming that the null hypothesis is true.
Depending on the nature of our alternate hypothesis, the P-value can fall on different parts of the distribution.

• Right-tailed test: If we are testing whether the storm is worsening, the P-value is on the right, as we are checking for values greater than 16.4 feet.
• Left-tailed test: If testing whether the storm is calming down, the P-value is on the left, focusing on values less than 16.4 feet.
• Two-tailed test: If exploring any change from 16.4 feet, the P-value spans both tails—calculating the probability of observing deviations on either side.

A small P-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis, suggesting that the observed data is unlikely under the assumption that the null hypothesis is true. Conversely, a larger P-value indicates insufficient evidence to reject the null hypothesis.