Final answer:
The expected May rainfall is 3.5 inches if it's uniformly distributed between 3 and 4 inches. The standard deviation is approximately 0.2887 inches. However, since the distribution is uniform, 100% of the data will fall within two standard deviations of the mean.
Step-by-step explanation:
Let x be the amount of May rainfall in the local area, and assume that x is uniformly distributed over the interval 3 to 4 inches.
Expected May Rainfall Calculation
The expected value (mean) of a uniform distribution is the midpoint of the interval, calculated as:
Mean (μ) = (3 + 4) / 2 = 3.5 inches
Standard Deviation Calculation
The formula for the standard deviation (σ) of a uniform distribution is:
Standard Deviation (σ) = (b - a) / √12
where a and b are the limits of the distribution.
Applying the formula we get:
σ = (4 - 3) / √12 = 1 / √12 ≈ 0.2887 inches
Empirical Rule
For a uniform distribution, the probability of observing a value within one standard deviation and two standard deviations of the mean doesn't apply as it would for a normal distribution.
In this case, since all values within the interval are equally likely, the probability that the observed May rainfall will fall within two standard deviations of the mean is 100%, because the interval covered by two standard deviations will certainly include the entire range of the uniform distribution.
Applying the empirical rule for a uniform distribution doesn't make sense as it does for a normal distribution since the chance of x falling between any two values within the range is proportional to the length of the interval between those values.