Rainfall in the state of California

Question

According to the Current Results website, the state of California has a mean annual rainfall of 24 inches, whereas the state of New York has a mean annual rainfall of 47 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(line above x)  and ox=  (to 4 decimals).

b. What is the probability that the sample mean is within 1 inch of the population mean for California?

 (to 4 decimals)

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

 (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?

The probability of being within 1 inch is - Select your answer -greater or lower for New York in part (c) because the sample size is - Select your answer -larger or smaller

Answer 1

a) This is the wrong venue for asking questions which demand graphics. I can not graph a normal distribution with an Xbar of 24 and a σ_Xbar of 0.5477 into this answer. You can get a graphic of the distribution at this location using μ=24 and σ = 0.5477 (htps://homepage.divms.uiowa.edu/~mbognar/applets/normal.html)

b)Poorly written questions. There is no sample mean given, only a population mean. Assuming the question being asked is actually "Assuming that there is no trend in annual rainfall and a random sample of size 30 is taken from some unspecified population of yearly measurements and that the yearly measurements are normally distributed with a variance equal to 9 inches, what is the probability that a random sample of size 30 will have a value equal to the given population mean of 24, plus or minus 1 (23 to 25)? Well, using a μ of 24 and a σ_Xbar of 0.5477 i.e.(9/30)^.5 . Using the CDF φ(P 23 < X < 25) we get an answer of .9321. for the record, the same answer can be gotten from a Z table showing cumulative probabilities less than or equal to the z score by getting p value for a z score of -1/.5477, multiplying it by two and subtracting that from 1.

c)This one is even weirder. What sample mean? The one for California, the one for New York If from NY, and we make the same assumptions as in b, only for NY, the answer is calculated the same, only with a standard deviation of the mean (σ_Xbar ) being (9/45)^.5 or 0.4472. and the resulting probability would be .9747

d) Making comparisons using the standard error or standard deviation of the mean of a random sample will always result in a tighter (smaller) confidence interval for larger sample size as the standard deviation of the mean gets smaller as the sample size gets larger.

The probability of being within 1 inch is greater  for New York in part (c) because the sample size is larger