Question

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 μ = 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 x = 17.4 feet. Previous studies of severe storms indicate that σ = 3.5 feet. Does this information suggest that the storm is (perhaps temporarily) increasing above the severe rating? Use α = 0.01.

(a) What is the level of significance?

State the null and alternate hypotheses.

H0: μ = 16.4 ft; H1: μ < 16.4 ft

H0: μ > 16.4 ft; H1: μ = 16.4 ft

H0: μ = 16.4 ft; H1: μ > 16.4 ft

H0: μ < 16.4 ft; H1: μ = 16.4 ft

H0: μ = 16.4 ft; H1: μ ≠ 16.4 ft

(b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution.

The Student's t, since the sample size is large and σ is unknown.

The standard normal, since the sample size is large and σ is known.

The Student's t, since the sample size is large and σ is known.

The standard normal, since the sample size is large and σ is unknown.

What is the value of the sample test statistic? (Round your answer to two decimal places.)

(c) Find the P-value. (Round your answer to four decimal places.)

Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) Interpret your conclusion in the context of the application.

There is sufficient evidence at the 0.01 level to conclude that the storm is increasing above the severe rating.

There is insufficient evidence at the 0.01 level to conclude that the storm is increasing above the severe rating.

Answer 1

Suppose that a reading of 36 waves showed an average wave height of x = 17.4 feet. Previous studies of severe storms indicate that σ = 3.5 feet.

a) level of significance

The null and the alternative hypotheses are,

H0: μ = 16.4 ft; H1: μ > 16.4 ft

b) Here, we used the standard normal, since the sample size is large and σ is known.

The sample test statistic is,

c) since, it is right-tailed test,

p-value=P(Z > 1.71) = 1 - P(Z < 1.71)=1 - 0.9564 = 0.0436

=> p-value = 0.0436

d) Here, p-value = 0.0436 > 0.01

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant

e) Conclusion : There is insufficient evidence at the 0.01 level to conclude that the storm is increasing above the severe rating.