Question

According to the Current Results website, the state of California has a mean annual rainfall of 25 inches, whereas the state of New York has a mean annual rainfall of 52 inches. Assume that the standard deviation for both states is 3 inches. A sample of 30 years of rainfall for California and a sample of 45 years of rainfall for New York has been taken. Use z-table.

a. Show the probability distribution of the sample mean annual rainfall for California. This is a graph of a normal distribution with E(x)=_____ and 0x=______ (round to 4 decimals).

b. What is the probability that the sample mean is within 1 inch of the population mean for California? (round to 4 decimals)

c. What is the probability that the sample mean is within 1 inch of the population mean for New York? (round to 4 decimals)

d. In which case, part (b) or part (c), is the probability of obtaining a sample mean within 1 inch of the population mean greater? Why?

Answer 1

a) E(X) = = 25

= = 3/ = 0.5477

b) P(24 < < 26)

= P((24 - )/() < ( - )/() < (26 -)/() )

= P((24 - 25)/0.5477 < Z < (26 - 25)/0.5477)

= P(-1.83 < Z < 1.83)

= P(Z < 1.83) - P(Z < -1.83)

= 0.9664 - 0.0336

= 0.9328

c) P(51 < < 53)

= P((51 - )/() < ( - )/() < (53 -)/())

= P((51 - 52)/(3/) < Z < (53 - 52)/(3/))

= P(-2.24 < Z < 2.24)

= P(Z < 2.24) - P(Z < -2.24)

= 0.9875 - 0.0125

= 0.9750

d) The probability for part (c) is greater. Because the sample size for part(c) is greater.