Chapter 6 Sampling and Sampling Distributions

Chapter 6

Sampling and Sampling Distributions

6.1 Definitions

? A statistical population is a set or collection of all possible observations of some characteristic.

? A sample is a part or subset of the population.

? A random sample of size is a sample that is chosen in such a way as to ensure that every sample of size has the same probability of being chosen.

? A parameter is a number describing some (unknown) aspect of a population. (i.e. )

? A statistic is some function of the sample observations. (i.e. ? )

? The probability distribution of a statistic is known as a sampling distribution. (How is ? distributed)

? We need to distinguish the distribution of a random variable, say ? from the realization of the random variable (ie. we get data and calculate some sample mean say ? = 42) 1

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CHAPTER 6. SAMPLING AND SAMPLING DISTRIBUTIONS

Populations and Samples

A Population is the set of all items or individuals of interest

Examples:

All lik ely voters in the next election All parts produced today All sales receipts for November

A Sample is a subset of the population

Examples:

1000 voters selected at random for interview

A few parts selected for destructi ve testing Random receipts selected for audi t

Stati sti cs for Business and E conomi cs, 6e ? 2007 Pearson E ducation, I nc.

Figure 6.1:

Chap 7-4

6.1. DEFINITIONS

3

Population vs. Sample

Population

a b cd ef gh i jk l m n

o p q rs t u v w xy z

Sample

b c gi n o ru

y

Stati sti cs for Business and E conomi cs, 6e ? 2007 Pearson E ducation, I nc.

Figure 6.2:

Chap 7-5

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CHAPTER 6. SAMPLING AND SAMPLING DISTRIBUTIONS

Note on Statistics

? The value of the statistic will change from sample to sample and we can therefore think of it as a random variable with it's own probability distribution.

? ? is a random variable

? Repeated sampling and calculation of the resulting statistic will give rise to a distribution of values for that statistic.

6.1. DEFINITIONS

5

Sampling Distributions

A sampling distribution is a distribution of all of the possible values of a statistic for a given size sample selected from a population

Stati sti cs for Business and E conomi cs, 6e ? 2007 Pearson E ducation, I nc.

Figure 6.3:

Chap 7-10

6

CHAPTER 6. SAMPLING AND SAMPLING DISTRIBUTIONS

Chapter Outline

Sampling Distributions

Sampling

Distribution of

Sample Mean

Sampling

Distribution of

Sample Proportion

Stati sti cs for Business and E conomi cs, 6e ? 2007 Pearson E ducation, I nc.

Figure 6.4:

Sampling Distribution of

Sample Difference in

Means

Chap 7-11

6.2. IMPORTANT THEOREMS RECALLED

7

6.2 Important Theorems Recalled

Suppose 1 2 are independent with [] = and [] = 2 = 1 2 .

Suppose = 11 + 22 + + + , then: X

[ ] = [ + ] = 1[1] + 2[2] + ? ? ? + [] +

= X 11 + +

=

+

and X

[ ] = [ + ] = 21 [1] + 22 [2] + ? ? ? + 2 []

= X2121 + 2222 + ? ? ? + 22

=

2 2 because of independence

and if is normal i.e. ( 2 ) independently :

X

X

( + 2 2 )

6.3 Frequently used Statistics

6.3.1 The sample mean

? Let 1 2 be a random sample of size from a population with mean and variance 2. The sample mean is:

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CHAPTER 6. SAMPLING AND SAMPLING DISTRIBUTIONS

?

=

1

X

=1

1. The expected value of the sample mean is the population mean:

[? ]

=

( 1

X )

=

1

X []

=

1 (

+

+

)

=

=1

=1

2. The variance of the sample mean ( 0 independent):

[? ]

=

1 (

X )

=

1 2

X

()

=1

=1

=

1 2

X 2

=

2

=1

3. If we do not have independence it can be shown that

[? ]

=

2

?

? -

where

is

the

population

size

-1

?

?

- -1

is called the correction factor

and

if

is

large

relative

to

then

? - ?

-1

1

so

that

[? ]

=

2

Note on Sample Mean

1. The use of the formulas for expected values and variances of sums of random variables that we saw in chapter 5.

2. The variance of the sample mean is a decreasing function of the sample size.

3. The standard deviation of the sample mean (under independence)

?

=

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