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 ì = 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 38 waves showed an average wave height of x = 17.2 feet. Previous studies of severe storms indicate that ó = 3.5 feet. Does this information suggest that the storm is (perhaps temporarily) increasing above the severe rating? Use á = 0.01.
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√n
If only one SD is provided, you can use just that to determine SEdiff.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability
To determine whether the storm is increasing above the severe rating, we can conduct a hypothesis test.
Step 1: State the null and alternative hypotheses:
Null hypothesis (H0): The average peak wave height of severe storms remains at µ = 16.4 feet.
Alternative hypothesis (H1): The average peak wave height of severe storms is greater than µ = 16.4 feet.
Step 2: Set the significance level (α):
The significance level (α) is given as α = 0.01.
Step 3: Formulate the test statistic:
We'll use the z-test since we have the population standard deviation (σ) and the sample size (n) is large enough (n > 30).
The formula for the z-test is: z = (x - µ) / (σ / √n)
Where:
x is the sample mean (17.2 feet)
µ is the population mean (16.4 feet)
σ is the population standard deviation (3.5 feet)
n is the sample size (38 waves)
Plugging in the values, we have z = (17.2 - 16.4) / (3.5 / √38)
Step 4: Calculate the test statistic:
z = (0.8) / (3.5 / √38) ≈ 0.140
Step 5: Determine the critical value:
Since our alternative hypothesis is that the average peak wave height is greater, we need to find the critical z-value for a one-tailed test with α = 0.01.
Using a standard normal distribution table or a calculator, the critical z-value at α = 0.01 is approximately 2.33.
Step 6: Compare the test statistic with the critical value:
Since the test statistic (0.140) is less than the critical value (2.33), we fail to reject the null hypothesis.
Step 7: Make a conclusion:
Based on the data and the test results, we do not have enough evidence to suggest that the storm is increasing above the severe rating at a significance level of 0.01.
To determine whether the storm is temporarily increasing above the severe rating, we can conduct a hypothesis test using the given information.
First, let's define the null and alternative hypotheses:
Null Hypothesis (H0): The storm's average peak wave height is not increasing above the severe rating.
Alternative Hypothesis (Ha): The storm's average peak wave height is increasing above the severe rating.
The null hypothesis assumes that there is no significant increase in the storm's average peak wave height, while the alternative hypothesis suggests that there is a significant increase.
Next, we can calculate the test statistic (Z-score) using the formula:
Z = (x - ì) / (ó / sqrt(n))
where x is the sample mean (17.2 feet), ì is the mean for severe storms (16.4 feet), ó is the standard deviation (3.5 feet), and n is the sample size (38 waves).
Calculating the Z-score:
Z = (17.2 - 16.4) / (3.5 / sqrt(38))
= 0.8 / (3.5 / 6.16)
= 0.8 / 0.897
= 0.891
Now, we need to determine the critical value for the given significance level (á = 0.01). Since the test is one-tailed (we are only interested in whether the wave height is increasing), we can use a Z-table or calculator to find the critical Z-score.
The critical Z-score for á = 0.01 (one-tailed test) is approximately 2.33.
Comparing the calculated Z-score (0.891) with the critical Z-score (2.33), we observe that the calculated Z-score is smaller than the critical Z-score. This means that the test statistic does not fall into the rejection region.
Therefore, we do not have enough evidence to reject the null hypothesis. The information provided does not suggest that the storm is temporarily increasing above the severe rating at a significance level of 0.01.