Question

The population mean and standard deviation are given below. Find the required probability and determine whether the given sample mean would be considered unusual.

For a sample of

nequals=75

find the probability of a sample mean being greater than

220

if

muμequals=219

and

sigmaσequals=3.8

The heights of fully grown trees of a specific species are normally​ distributed, with a mean of

55.5

feet and a standard deviation of

5.50

feet. Random samples of size

13

are drawn from the population. Use the central limit theorem to find the mean and standard error of the sampling distribution. Then sketch a graph of the sampling distribution.

The mean of the sampling distribution is

mu Subscript x overbarμxequals=nothing.

The standard error of the sampling distribution is

sigma Subscript x overbarσxequals=nothing.

​(Round to two decimal places as​ needed.)

Answer 1

µ =    219                                      
σ =    3.8                                      
n=   75                                      
                                          
X =   220                                      
                                          
Z =   (X - µ )/(σ/√n) = (   220   -   219   ) / (    3.8   / √   75   ) =   2.279  
                                          
P(X ≥   220   ) = P(Z ≥   2.28   ) =   P ( Z <   -2.279   ) =    0.0113           (answer)
yes,  given sample mean would be considered unusual because probability is less than 0.05

==============

2)

The mean of the sampling distribution is

µx = = 55.5

The standard error of the sampling distribution is=σ/√n= 5.5/√13 = 1.53