Question

The population mean and standard deviation are given below. Find the indicated probability and determine whether a sample mean in the given range below would be considered unusual. If​ convenient, use technology to find the probability. For a sample of n=40​, find the probability of a sample mean being less than 12751 or greater than 12754, when mean = 12751 and standard deviation = 1.6.

A. For the given​ sample, the probability of a sample mean being less than 12751 or greater than 12754 is ____. ​(Round to four decimal places as​ needed.)

B. Would the given sample mean be considered​ unusual?

a. The sample mean would not be considered unusual because there is a probability greater than 0.05 of the sample mean being within this range.

b. The sample mean would be considered unusual because there is a probability less than 0.05 of the sample mean being within this range.

c. The sample mean would be considered unusual because there is a probability greater than 0.05 of the sample mean being within this range.

d. The sample mean would not be considered unusual because there is a probability less than 0.05 of the sample mean being within this range.

Answer 1

A)

Given,

= 12751 , = 1.6

Using central limit theorem,

P( < x) = P(Z < x - / ( / sqrt(n) ) )

So,

P( < 12751 OR > 12754 ) = 1 - P(12751 < < 12754)

Now ,

P(12751 < < 12754) = P( < 12754 ) - P( < 12751)

= P(Z < 12754 - 12751 / (1.6 / sqrt(40) ) ) - P(Z < 12751 - 12751 / (1.6 / sqrt(40) ) )

= P(Z < 11.86) - P(Z < 0)

= 1 - 0.5

= 0.5

Therefore,

P( < 12751 OR > 12754 ) = 1 - 0.5

= 0.5

B)

The sample mean would not be considered unusual because there is a probability greater than 0.05 of the sample

mean being within this range.