The State Of California Has A Mean Annual Rainfall Of 276 Inches Whe
The state of California has a mean annual rainfall of 27.6 inches, whereas the state of New York has a mean annual rainfall of 48.7 inches. Assume the standard deviation for California is 4.9 inches and for New York is 3.1 inches. Find the probability that, for a sample of 40 years of rainfall for New York, the mean annual rainfall is at least 49 inches.
Understanding the probabilities associated with sample means is fundamental in statistics, especially when analyzing climatic data such as annual rainfall. In this paper, we examine the probability that, in a sample of 40 years, the mean rainfall in New York exceeds a specific threshold, namely 49 inches. Given the population mean and standard deviation, we apply principles of the sampling distribution of the sample mean to compute this probability.
Introduction
Annual rainfall data is crucial for agricultural planning, infrastructure design, and environmental management. Analyzing historical rainfall patterns helps in understanding climate variability and predicting future trends. When dealing with sample data, statisticians rely on probability distributions to estimate the likelihood of specific outcomes, such as the average rainfall exceeding a certain value in a given period. This case involves calculating the probability that the average rainfall over a 40-year span exceeds 49 inches in New York, given the population mean and standard deviation.
Background and Data
From the data provided, the population mean annual rainfall in New York is 48.7 inches, with a standard deviation of 3.1 inches. The sample size, n, is 40 years. The goal is to find the probability that the sample mean, denoted as µ■, is at least 49 inches. Understanding the sampling distribution of the mean is essential for this calculation.
Theoretical Framework
The sampling distribution of the mean, for a sufficiently large sample size, follows approximately a normal distribution due to the Central Limit Theorem. The mean of this sampling distribution is equal to the population mean, µ = 48.7 inches, and its standard error (SE) is given by:
where σ is the population standard deviation, and n is the sample size.
Calculations
Calculate the standard error:
SE = 3.1 / √40 ≈ 3.1 / 6.3246 ≈ 0.4901 inches
Next, calculate the z-score for the observed mean of 49 inches:
z = (X■ - µ) / SE = (49 - 48.7) / 0.4901 ≈ 0.3 / 0.4901 ≈ 0.612
Using standard normal distribution tables or a calculator, find the probability that Z is at least 0.612:
P(Z ≥ 0.612) = 1 - P(Z ≤ 0.612)
From the Z-table, P(Z ≤ 0.612) ≈ 0.7291
Therefore, P(Z ≥ 0.612) ≈ 1 - 0.7291 = 0.2709
Conclusion
There is approximately a 27.09% chance that, in a 40-year sample, the average annual rainfall in New York will be at least 49 inches. This probability indicates a moderate likelihood and underscores the variability inherent in climate data. It also demonstrates the application of statistical principles in environmental and meteorological assessments.
Implications and Significance
This analysis aids in understanding the probability of observing higher-than-average rainfall over extended periods. For policymakers and environmental agencies, such insights can inform water resource management, infrastructure resilience planning, and climate change adaptation strategies. Moreover, the methodology used here exemplifies how statistical tools can quantify uncertainty and variability in real-world data.
Limitations and Further Research
While the normal approximation provides a good estimate here, real-world rainfall data may exhibit skewness or kurtosis that deviates from normality. Further research could involve using non-parametric
methods or examining other periods or regions to validate findings. Additional variables, such as seasonal variations or climate change impacts, could also refine the analysis.
References
Sheskin, D. J. (2011). Handbook of Parametric and Nonparametric Statistical Tests. CRC Press.
Moore, D. S., McCabe, G. P., & Craig, B. A. (2014). Introduction to the Practice of Statistics. W. H. Freeman and Company.
Zar, J. H. (2010). Biostatistical Analysis. Pearson Education.
Johnson, R. A., & Wichern, D. W. (2007). Applied Multivariate Statistical Analysis. Pearson.
Lim, C. (2018). Environmental Statistics: Methods and Applications. Springer.
Wilks, D. S. (2011). Statistical Methods in the Atmospheric Sciences. Academic Press.
Cleveland, W. S., & Devlin, S. J. (1988). Locally weighted regression: An approach to regression analysis by local fitting. Journal of the American Statistical Association, 83(403), 596–610.
Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
Rice, J. (2007). Mathematical Statistics and Data Analysis. Cengage Learning.
Ross, S. M. (2014). Introduction to Probability and Statistics for Engineers and Scientists. Academic Press.