Sampling Distributions

Generally, we are interested in population parameters.. When the census is impossible, we draw a sample from the population, then construct sample statistics, that have close relationshi

Sampling Distributions

Chapter 8

1

Generally, we are interested in population

parameters

When the census is impossible, we draw a sample

from the population, then construct sample statistics, that have close relationship to the population

parameters.

2

Samples are random, so the sample statistic is a

random variable.

3

8.1 Sampling Distribution of the Mean

Example 1: A die is thrown infinitely many times Let

X represent the number of spots showing on any

throw The probability distribution of X is

4

x 1 2 3 4 5 6

p(x) 1/6 1/6 1/6 1/6 1/6 1/6

E(X) = 1(1/6) +2(1/6) + 3(1/6)+……….= 3.5

V(X) = (1-3.5) 2 (1/6) + (2-3.5) 2 (1/6) +….…… …= 2.92

Suppose we want to estimate  from the mean

of a sample of size n = 2.

What is the distribution of ?

5

x

Throwing a die twice – sample mean

6

1 1,1 1 13 3,1 2 25 5,1 3

2 1,2 1.5 14 3,2 2.5 26 5,2 3.5

3 1,3 2 15 3,3 3 27 5,3 4

4 1,4 2.5 16 3,4 3.5 28 5,4 4.5

5 1,5 3 17 3,5 4 29 5,5 5

6 1,6 3.5 18 3,6 4.5 30 5,6 5.5

7 2,1 1.5 19 4,1 2.5 31 6,1 3.5

8 2,2 2 20 4,2 3 32 6,2 4

9 2,3 2.5 21 4,3 3.5 33 6,3 4.5

10 2,4 3 22 4,4 4 34 6,4 5

11 2,5 3.5 23 4,5 4.5 35 6,5 5.5

12 2,6 4 24 4,6 5 36 6,6 6

Sample Mean Sample Mean Sample Mean

1 1,1 1 13 3,1 2 25 5,1 3

2 1,2 1.5 14 3,2 2.5 26 5,2 3.5

3 1,3 2 15 3,3 3 27 5,3 4

4 1,4 2.5 16 3,4 3.5 28 5,4 4.5

5 1,5 3 17 3,5 4 29 5,5 5

6 1,6 3.5 18 3,6 4.5 30 5,6 5.5

7 2,1 1.5 19 4,1 2.5 31 6,1 3.5

8 2,2 2 20 4,2 3 32 6,2 4

9 2,3 2.5 21 4,3 3.5 33 6,3 4.5

10 2,4 3 22 4,4 4 34 6,4 5

11 2,5 3.5 23 4,5 4.5 35 6,5 5.5

12 2,6 4 24 4,6 5 36 6,6 6

These are all the possible pairs of values for the 2 throws And these are the means of each pair

The distribution of when n = 2

x

7

Notice there are 36 possible pairs of values:

1,1 1,2 … 1,6 2,1 2,2 … 2,6

………

6,1 6,2 … 6,6

1 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

1 2 3 4 5 6 5 4 3 2 1

1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36

x

Calculating the relative frequency of each value

of we have the following results

Frequency

Relative freq

(2+1)/2 = 1.5

(1+3)/2 = 2 (2+2)/2 = 2 (3+1)/2 = 2

) 25

( 1167

5 3

25 n

2 x

2 x

x















) 10

( 2917

5 3

10 n

2 x

2 x

x















) 5

( 5833

5 3

5 n

2 x

2

x

x















As the sample size changes, the mean of the sample mean does not change!

) 25

( 1167

5 3

25 n

2 x

2 x

x















) 10

( 2917

5 3

10 n

2 x

2 x

x















) 5

( 5833

5 3

5 n

2 x

2

x

x















As the sample size increases, the

variance of the sample mean decreases!

Demonstration: Why is the variance of the

sample mean is smaller than the population

variance?

Mean = 1.5 Mean = 2. Mean = 2.5

Compare the range of the population

to the range of the sample mean.Let us take samplesof two observations

2

The Central Limit Theorem

If a random sample is drawn from any population, the sampling distribution of the sample mean is:

– Normal if the parent population is normal,

– Approximately normal if the parent population is

not normal, provided the sample size is

sufficiently large The larger the sample size, the more closely the sampling distribution of will resemble a normal distribution

11

x

x

μ 

n

σ σ

2 x

2

x 

The mean of X is equal to the mean of the parent

population

The variance of X is equal to the parent population variance divided by ‘n’.

n Sampling Distribution

1 Population distribution

2

5

30

50

90

120

Census

0

n Sampling Distribution

1 Normal

Pop distribution

2

5

Example 2: The amount of soda pop in each bottle is normally distributed with a mean of 32.2 ounces and

a standard deviation of 3 ounces

Find the probability that a bottle bought by a

customer will contain more than 32 ounces.

16

0.7486

 = 32.2

x = 32

32) P(x 

0.7486 67)

P(z

) 3

32.2

32 σ

μ

x P(

32)

P(x

x



















Find the probability that a carton of four bottles will have a mean of more than 32 ounces of soda per

bottle.

9082

0 )

33

1 z

( P

) 4

3

2 32 32

x ( P )

32 x

(

P

x























17

32

x  x  32 2

32) x

P( 

Example 3: The average weekly income of B.B.A

graduates one year after graduation is $600 Suppose the distribution of weekly income has a standard

deviation of $100

What is the probability that 35 randomly selected

graduates have an average weekly income of less

than $550?

18

0.0015 2.97)

P(z

) 35 100

600

550 σ

μ

x P(

550) x

P(

x



















19