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BIOL499A
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Blaine Mullins
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Chapter 18

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Biology (Biological Sciences)

BIOL499A

Blaine Mullins

Winter

Description

Chapter 18: Sampling Distribution Models
In this chapter, we will talk about:
Sampling distribution of the sample mean,
Central limit theorem,
Sampling distribution of the sample proportion, when sample size is large.
Definition: The probability distribution of a statistic is called the sampling
distribution of the statistic or the distribution of the statistic. Consider a population with numbers 2, 3, and 4. Suppose n=2 numbers are
selected with replacement.
Let Y= selected number. Then the probability distribution of population is:
y 2 3 4 Total
P(y) 1/3 1/3 1/3 1
The possible samples and corresponding sample averages are:
Sample (2,2) (2,3) (2,4) (3,2) (3,3) (3,4) (4,2) (4,3) (4,4)
y 22 2 2 2.5
2 2
Therefore, the probability distribution of sample meanis:
y
2 2.5 3 3.5 4 Total
P() 1 Recall that the probability distribution of population was:
y 2 3 4 Total
P(y) 1/3 1/3 1/3 1
Therefore, the mean and standard deviation of Y are:
1 1 1
Y)2)(3(43p y 3 3 3
1 1 1
Var( ) (y )2p( )(2 3) (3 3) (4 3)
3 3 3
121 1
( 1) 2 (0) (1)
333 3
S..) Y 2
3
In addition, the probability distribution of sample mean was:
y 2 2.5 3 3.5 4 Total
P() 1/9 2/9 3/9 2/9 1/9 1
Hence, the mean and standard deviation of sample mean are:
1 23 21
Y EY() p y 2 4 3
9 99 99 2 2 2 2 1 2 3
Var( ) p(y ) (2 3) (2.5 3) (3 3)
9 9 9
2 2 1
5 3) (4 3)
3 9
1 12 3
2 2 ( ) (0)
9 29 9
12 11
)2 2 (1)
29 93
..) 1
Y 3
Theorem: The distribution of the sample mean, based on a random
sample of size n from a population with mean and standard deviatio ,
has a mean ofY EY and a standard deviation of
Y D()Y (S)E Y
n Example 18.1: An automatic grinding machine in an auto parts plant
prepares axles with the target diameter 40.125millimeters (mm). The
machine has some variability, so the standard deviation of the diameters
is 0.002 mm. A sample of 4 axles is inspected each hour for process
control purposes, and records are kept of the sample mean diameter. What
will be the mean and standard deviation of the numbers recorded?
Theorem: In a random sampling from a normal population with mean
and standard deviation , the sample mean has the normal distribution with
mean and standard deviation / n . Example 18.2: Suppose the electric bill for the month of July for a single-
family home in a city has a normal distribution with mean $63 and standard
deviation $25, respectively.
(a) What percentage of single-families wi ll receive an electric bill greater
than $70?
(b) What percentage of all samples of 12 of single-families will have a
mean of electric bills greater than $70? Example 18.3: The weights of pears in an orchard are normally distributed
with mean 0.32 pound and standard deviation 0.08 pound.
(a) If one pear is selected at random, what is the probabili

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