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Rainfall statistics are often modeled using the Normal Distribution. This is because many natural phenomena, like rainfall, tend to cluster around an average value with symmetrical deviations. The Normal Distribution is characterized by its bell-shaped curve, where the mean (\( \mu \)) is at the center, and the spread is determined by the standard deviation (\( \sigma \)). For Ithaca, NY, the mean annual rainfall is \( 35.4^{\prime \prime} \), with a standard deviation of \( 4.2^{\prime \prime} \). This suggests that most rainfall measurements will fall within this range:

• Approximately 68% falls within one standard deviation (\( 31.2^{\prime \prime} \) to \( 39.6^{\prime \prime} \)).
• About 95% falls within two standard deviations.

Thus, the Normal Distribution helps us predict the likelihood of certain rainfall amounts.

Calculating a Z-score is essential for understanding how far a specific data point is from the mean in terms of standard deviations. The Z-score formula is \( z = \frac{x - \mu}{\sigma} \), where \( x \) is the value of interest. It allows us to compare different data points by standardizing them. For instance, to find how unusual \(40^{\prime \prime}\) of rainfall is in Ithaca, we calculate:

• Z-score for \( 40^{\prime \prime} = \frac{40 - 35.4}{4.2} \approx 1.095\)

This tells us that \(40^{\prime \prime}\) is \(1.095\) standard deviations above the mean. The Z-score helps us reference standard normal distribution tables or calculators.

A Sampling Distribution is concerned with the distribution of sample means rather than individual data points. If you take multiple samples from a population and calculate their mean, the distribution of these means will be normal if the sample size is large enough, a principle known as the Central Limit Theorem.
In our case, with a sample size of 4 years, we are interested in the average rainfall over these years. Given a population with mean \( \mu \) and standard deviation \( \sigma \), the sampling distribution will have:

• The same mean: \( \mu = 35.4^{\prime \prime} \)
• A standard deviation of \( \sigma_{\bar{y}} = \frac{4.2}{\sqrt{4}} = 2.1^{\prime \prime} \)

This makes the sampling distribution for 4 years’ average rainfall: \( N(35.4, 2.1)\).

Probability Analysis involves calculating the likelihood of a particular event occurring, using statistical methods such as Z-score calculations and normal distribution properties. This helps in making informed predictions based on data.
For instance, determining the probability that the average rainfall over four years is less than \(30^{\prime \prime}\) involves calculating the Z-score for the average:

• Calculate Z-score: \( z = \frac{30 - 35.4}{2.1} \approx -2.571 \)

Using standard normal distribution tables or tools, the Z-score tells us the probability of this event. Here, \( P(Z < -2.571) \approx 0.005 \), which means there's only a 0.5% chance that the four-year average rainfall is less than \(30^{\prime \prime}\). These analyses provide crucial insights for decision-making and understanding weather patterns.