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Weatherwise is a magazine published by the American Meteorological Society. One issue gives a rating system used to classify Nor'easter storms that frequently hit New England and can cause much damage near the ocean. A severe storm has an average peak wave height of \(\mu=16.4\) feet for waves hitting the shore. Suppose that a Nor'easter is in progress at the severe storm class rating. Peak wave heights are usually measured from land (using binoculars) off fixed cement piers. Suppose that a reading of 36 waves showed an average wave height of \(\bar{x}=17.3\) feet. Previous studies of severe storms indicate that \(\sigma=3.5\) feet. Does this information suggest that the storm is (perhaps temporarily) increasing above the severe rating? Use \(\alpha=0.01\).

Step by step solution

01

Understanding the Hypotheses

We need to determine if the storm is increasing above the severe rating. We will form the null hypothesis, which states that the average peak wave height is equal to the severe storm rating mean, \(H_0: \, \mu = 16.4\) feet. The alternative hypothesis, reflecting an increase, is \(H_a: \, \mu > 16.4\) feet.

02

Collecting the Given Data

We have the following data: sample mean \(\bar{x} = 17.3\) feet, population standard deviation \(\sigma = 3.5\) feet, sample size \(n = 36\), and significance level \(\alpha = 0.01\).

03

Calculating the Test Statistic

The test statistic for a one-sample z-test is calculated using the formula: \[ z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \] Substituting the known values: \[ z = \frac{17.3 - 16.4}{\frac{3.5}{\sqrt{36}}} = \frac{0.9}{0.5833} \approx 1.54 \]

04

Determining the Critical Value for z

For a one-tailed test with \(\alpha = 0.01\), we check the z-table to find the critical z-value, which is approximately 2.33.

05

Making the Decision

Compare the calculated z-value (1.54) to the critical z-value (2.33). Since 1.54 is less than 2.33, we do not reject the null hypothesis.

06

Drawing a Conclusion

Since the z-value does not exceed the critical value, there is not enough statistical evidence at the 0.01 significance level to conclude that the storm is increasing above the severe rating.

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Key Concepts

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

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about data, especially when determining the likelihood of a particular hypothesis being true. At its core, hypothesis testing helps you figure out if what you are observing is just due to random chance or if there is actually something significant happening. In this exercise, the peak wave heights are being tested to see if they hit a severe storm level. To do this, we follow these steps:

• Identify your hypotheses: Null hypothesis ( $H_0$ ) and alternative hypothesis ( $H_a$ )
• Collect and analyze data: Here, the average wave height is measured
• Use statistical calculations to form conclusions

The goal is to gather enough evidence to either support or reject $H_0$ .

Z-test

The z-test is a type of hypothesis test that determines if there is a significant difference between sample data and a known population mean. It's handy when we want to know if a sample comes from a population with a specific mean, and we know the population's standard deviation. For this problem, we perform a one-sample z-test since we are comparing one group of wave data to a known severe storm height mean. The formula used is: \[ z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \] Where:

• \(\bar{x}\): Sample mean
• \(\mu\): Population mean
• \(\sigma\): Population standard deviation
• \(n\): Sample size

The calculated z-score tells us how many standard deviations our sample mean is from the actual mean.

Significance Level

The significance level, often denoted as \(\alpha\), is the threshold used in hypothesis testing to decide whether a result is statistically significant. In simpler terms, \(\alpha\) is your risk of incorrectly rejecting the null hypothesis. For this exercise, \(\alpha\) is set at 0.01, suggesting we need very strong evidence to prove an increase in wave heights.

• If the p-value is less than \(\alpha\), reject \(H_0\)
• If the p-value is greater than \(\alpha\), do not reject \(H_0\)

Choosing an \(\alpha\) level is a bit like drawing a line in the sand, dictating how confident we want to be about our results.

Null Hypothesis

The null hypothesis (\(H_0\)) acts like the default assumption in hypothesis testing—it represents no effect or no difference.In this context, our \(H_0\) states that the average wave height is equal to the severe storm rating mean, or \(H_0: \mu = 16.4\) feet. Testing \(H_0\) allows us to explore the alternative hypothesis (\(H_a\)), which expects some change, like an increase in wave height. Essentially, accepting or rejecting \(H_0\) helps guide our conclusions:

• If we fail to reject \(H_0\), the storm might not be any worse than predicted
• If we reject \(H_0\), there’s statistical evidence suggesting the storm could be stronger

Understanding \(H_0\) provides clarity in verifying or challenging existing assumptions with data.