A random sample of size 36 is to be selected from a population that has a mean m

Question

A random sample of size 36 is to be selected from a population that has a mean mu = 50 and a standard deviation sigma of 10. a. This sample of 36 has a mean value of x, which belongs to a sampling distribution. Find the shape of this sampling distribution. b. Find the mean of this sampling distribution. c. Find the standard error of this sampling distribution. d. What is the probability that this sample mean will be between 45 and 55? e. What is the probability that the sample mean will have a value greater than 48? f. What is the probability that the sample mean will be within 3 units of the mean?

Explanation / Answer

mean = 50 and sigma = 10

(a) The shape of the sampling distribution is apporximately normal

(b) Mean of sample i.e. xbar = 50

(c) Standard error = sigma/sqrt(n) = 10/sqrt(36) = 1.6667

(d)
P(45 < xbar < 55) = P(xbar < 55) - P(xbar < 45) = P(z < (55 - 50)/1.667) - P(z < (45 - 50)/1.667) = P(z < 2.999) - P(z<-2.999) = 0.9973

(e)
P(xbar > 48) = P(z > (48 - 50)/1.667) = P(z > -1.1998) = 0.8849

(f) 3/1.667 = 1.7996
P(-1.7996 < z < 1.7996) = P(z<1.7996) - P(z<-1.7996) = 0.9281

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