Question

The state of California has a mean annual rainfall of 23 inches, whereas the state of New York has a mean annual rainfall of 53 inches ( Current Results website, October 27,2012). Assume that the standard deviation for both states is 2 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 1 inch of the population mean for California?

c. What is the probability that the sample mean is within 1 inch of the population mean for New York?

Answer 1

(a)

The probability distribution of the sample mean annual rainfall for California is obtained as shown below:

Let the random variable X denote the amount of rainfall in the state of California. The mean rainfall for state of California is 22 inches and the standard deviation is 4 inches. That is, . Also, a sample of 30 years is taken for California.

The probability distribution of the sample mean annual rainfall for California is,

= =

By the central limit theorem the mean of the sampling distribution is same as mean of the population distribution for the large sample size. The variance of the sampling distribution is obtained by taking the ratio of the population variance and the sample size.

(b)

The probability that the sample mean is 1 inch of the population mean for California is obtained as shown below:

The mean for the California is 23, this implies that if sample mean is 1 inch of the population mean, then the probability value between 22 and 24 has to be obtained.

The required probability is,

(c)

The probability that the sample mean is 1 inch of the population mean for New York is obtained as shown below:

The mean rainfall for the New York is 53; this implies that if sample mean is 1 inch of the population mean, then the probability value between 52 and 54 has to be obtained.

The required probability is,