Question

A random sample of size 40 is selected from a population with the mean of 482 and standard deviation of 18. This sample of 40 has a mean, which belongs to a sampling distribution.

a) Determine the shape of the sampling distribution b) Find the mean and standard error of the sampling distribution

c) Find the probability that the sample mean will be between 475 and 495?

d) Find the probability that the sample mean will have a value less than 478?

e) Find the probability that the sample mean will be within 5 units of the mean?

Answer 1

Solution :

Given that,

mean = = 482

standard deviation = = 18

n = 40

a) = = 482

= / n = 18/ 40 = 2.846

b) P(475 < < 495)  

= P[(475 - 482) / 2.846 < ( - ) / < (495 - 482) / 2.846)]

= P(-2.46 < Z < 4.57)

= P(Z < 4.57) - P(Z < -2.46)

Using z table,  

= 1 - 0.0069

= 0.9931

c) P( < 478) = P(( - ) / < (478 - 482) / 2.846)

= P(z < -1.41)

Using z table

= 0.0793

d) P(477 < < 487)  

= P[(477 - 482) / 2.846 < ( - ) / < (487 - 482) / 2.846)]

= P(-1.76 < Z < 1.76)

= P(Z < 1.76) - P(Z < -1.76)

Using z table,  

= 0.9608 - 0.0392  

= 0.9216