The state of California has a mean annual rainfall of 22 inches, whereas the state ofNew York has a mean annual rainfall of 42 inches (Current Results website, October 27,2012). Assume that the standard deviation for both states is 4 inches. A sample of 30 yearsof rainfall for California and a sample of 45 years of rainfall for New York has been taken.a. Show the probability distribution of the sample mean annual rainfall for California.b. What is the probability that the sample mean is within 1 inch of the population meanfor California?
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
The state of California has a mean annual rainfall of 22 inches, whereas the state of
New York has a mean annual rainfall of 42 inches (Current Results website, October 27,
2012). Assume that the standard deviation for both states is 4 inches. A sample of 30 years
of rainfall for California and a sample of 45 years of rainfall for New York has been taken.
a. Show the probability distribution of the sample mean annual rainfall for California.
b. What is the probability that the sample mean is within 1 inch of the population mean
for California?
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