Weatherwise Is A Magazine Published By The American Meteorol
Weatherwise Is A Magazine Published By The American Meteorological Soc
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 31 waves showed an average wave height of x = 17.2 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? (b) 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.)
Let x be a random variable that represents the pH of arterial plasma (i.e., acidity of the blood). For healthy adults, the mean of the x distribution is μ = 7.4. A new drug for arthritis has been developed. However, it is thought that this drug may change blood pH. A random sample of 41 patients with arthritis took the drug for 3 months. Blood tests showed that x = 8.4 with sample standard deviation s = 2.7. Use a 5% level of significance to test the claim that the drug has changed (either way) the mean pH level of the blood. (a) What is the level of significance? What is the value of the sample test statistic? (Round your answer to three decimal places.) (c) Find the P-value. (Round your answer to four decimal places.)
Determine the probabilities for the following normal distribution problems. Round the values of z to 2 decimal places. Round your answers to 4 decimal places:
- (a) μ = 604, σ = 56.8, x ≤ 635
- (c) μ = 111, σ = 33.8, 100 ≤ x
- (d) μ = 264, σ = 10.9, 250
In a recent year, the average price of a Microsoft Windows Upgrade was $90.28 according to PC Data. Assume that prices of the Microsoft Windows Upgrade that year were normally distributed, with a standard deviation of $8.53. If a retailer of computer software was randomly selected that year:
- (a) What is the probability that the price of a Microsoft Windows Upgrade was below $80?
- (b) What is the probability that the price was above $95?
- (c) What is the probability that the price was between $83 and $87?
Suppose you are working with a data set that is normally distributed, with a mean of 300 and a standard deviation of 44. Determine the value of x from the following information. Round your answers and z values to 2 decimal places:
- (a) 70% of the values are greater than x.
- (b) x is less than 13% of the values.
- (c) 25% of the values are less than x.
Paper For Above instruction
Introduction
The application of statistical analysis in meteorology is crucial for understanding and predicting severe weather phenomena such as Nor'easter storms. These storms, prevalent along the northeastern coast of the United States, can lead to significant damage due to high wave heights and atmospheric changes. Accurate measurement, hypothesis testing, and understanding probability distributions are integral for meteorologists and researchers aiming to assess storm severity and predict their impacts effectively.
Statistical Analysis of Nor’easter Storm Wave Heights
The first scenario explores whether a recent measurement of wave heights indicates an escalation beyond the severe storm classification. The previous studies suggest that the average peak wave height (μ) for severe storms is 16.4 feet, with a standard deviation (σ) of 3.5 feet. A sample of 31 waves yielded an average of 17.2 feet. To evaluate whether this indicates an increase, a hypothesis test was conducted using a significance level (α) of 0.01.
The null hypothesis (H₀) assumes that the true mean wave height remains at 16.4 feet, whereas the alternative hypothesis (H₁) suggests it exceeds this value. The test statistic employed is the z-test for the mean:
Z = (x̄ - μ) / (σ / √n) = (17.2 - 16.4) / (3.5 / √31) ≈ 1.58
Calculating the P-value associated with this z-score, we find P ≈ 0.057, which exceeds the significance level of 0.01. Hence, there is insufficient evidence at the 1% level to conclude that the storm's wave height has increased beyond the severe classification at this point, although the data suggest a trend towards increase.
Blood pH and Effects of New Arthritis Drug
The second scenario involves testing whether the new arthritis drug impacts blood pH levels. The known mean (μ) for healthy adults is 7.4, but a sample of 41 patients taking the drug showed a mean of 8.4 with a standard deviation of 2.7. The hypotheses test whether the mean blood pH has changed:
H₀: μ = 7.4 (no change); H₁: μ ≠ 7.4 (change occurs). The significance level (α) is 0.05.
The test statistic, employing the t-distribution since the standard deviation is from a sample, is:
t = (x̄ - μ) / (s / √n) = (8.4 - 7.4) / (2.7 / √41) ≈ 2.42
With degrees of freedom df = 40, the two-tailed P-value associated with t = 2.42 is approximately 0.021. Since this P-value is less than 0.05, we reject the null hypothesis, indicating significant evidence that the drug alters blood pH levels.
Normal Distribution Probabilities
Calculating specific probabilities involves standardizing x values into z-scores for normal distributions:
- (a) μ = 604, σ = 56.8, x ≤ 635:
Z = (635 - 604) / 56.8 ≈ 0.58; P(Z ≤ 0.58) ≈ 0.7810
- (c) μ = 111, σ = 33.8, 100 ≤ x
Z₁ = (100 - 111) / 33.8 ≈ -0.34; Z₂ = (150 - 111) / 33.8 ≈ 1.16; P(100 ≤ x
- (d) μ = 264, σ = 10.9, 250
Z₁ = (250 - 264) / 10.9 ≈ -1.28; Z₂ = (255 - 264) / 10.9 ≈ -0.83; P(250
Probability Calculations for Microsoft Windows Upgrade Prices
Given the average price of $90.28 with a standard deviation of $8.53, the probabilities are computed using standardization:
- (a) Probability price
Z = (80 - 90.28) / 8.53 ≈ -1.22; P(Z
- (b) Probability price > $95:
Z = (95 - 90.28) / 8.53 ≈ 0.55; P(Z > 0.55) ≈ 0.2912
- (c) Probability between $83 and $87:
Z₁ = (83 - 90.28) / 8.53 ≈ -0.78; Z₂ = (87 - 90.28) / 8.53 ≈ -0.39; P(83
Computing X-Values From Normal Distribution Percentiles
With mean = 300 and SD = 44, the x-values can be derived from z-scores:
- (a) 70% of values > x:
Z for 30% (since greater than x): approximately -0.52. X = μ + Zσ = 300 + (-0.52)(44) ≈ 300 - 22.88 ≈ 277.12
- (b) x less than 13% of values:
Z for 13% (lower percentile) is approximately -1.13. X = 300 + (-1.13)(44) ≈ 300 - 49.72 ≈ 250.28
- (c) x less than 25% of values:
Z at 25% percentile is approximately -0.67. X = 300 + (-0.67)(44) ≈ 300 - 29.48 ≈ 270.52
Conclusion
The statistical analyses detailed above demonstrate the critical role of hypothesis testing and probability calculations in meteorology and biomedical contexts. Monitoring storm wave heights ensures timely alerts for severe conditions, while tests on blood pH and distribution probabilities inform medical research and commercial decisions. Mastery of these statistical tools equips researchers to make informed, data-driven decisions across diverse scientific fields.
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