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Sampling Distributions

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Page 1: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

Sampling Distributions

Page 2: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

Parameter & Statistic

Parameter• Summary measure

about population

Sample Statistic• Summary measure

about sample

• P in Population & Parameter

• S in Sample & Statistic

Page 3: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

Common Statistics & Parameters

Sample Statistic Population Parameter

Variance S2 2

StandardDeviation S

Mean X

Binomial Proportion

pp̂

Page 4: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

1. Theoretical probability distribution

2. Random variable is sample statistic• Sample mean, sample proportion,

etc.

3. Results from drawing all possible samples of a fixed size

4. List of all possible [x, p(x)] pairs•Sampling distribution of the sample

mean

Sampling Distribution

Page 5: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

Sampling from Normal Populations

Page 6: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

定理 1

平均數

變異數

),(~ , , ),(~ 222 bbaNYthenbxaYNxif 若

( ) ( ) ( ) ( ) ( )E Y E a bx E a E bx a bE x a b

2 2 2( ) ( ) ( ) ( ) 0 ( )V Y V a bx V a V bx b V x b

Page 7: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

定理 2

2 2 X ~ ( , ) , ~ ( , ) , X and Y are independentx x Y Yif N Y N

2 2 2 2 , then ~ ( , )X Y X Yif w aX bY w N a b a b

Page 8: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

定理: Let Y1,Y2,…,Yn be a random sample of size n from a normal distribution with mean μand varianceσ2. Then

is normally distribution with mean And variance

n

iiYn

Y1

1

Y 2 2 /Y n

Page 9: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

Proof:

niYVYE ii ,...,2,1for )( and )( 2

1 21

1 1 1 1( ) ( ) ( )

n

i ni

Y Y Y Y Yn n n n

1 1 2 2

where 1/ , 1, 2,...,n n

i

a Y a Y a Y

a n i n

Page 10: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

1 2

1 1 1( ) ( ) ( ) ( )

1 1 1 ( ) ( ) ( )

nE Y E Y Y Yn n n

n n n

Page 11: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

1 2

2 2 22 2 2

1 1 1( ) ( ) ( ) ( )

1 1 1 ( ) ( ) ( )

nV Y V Y Y Yn n n

n n n

22

2

1( )n

n n

Page 12: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

Properties of the Sampling Distribution of x

Page 13: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

xn

3. Formula (sampling with replacement)

2. Less than population standard deviation

1. Standard deviation of all possible sample means, x

● Measures scatter in all sample means, x

Standard Error of the Mean

Page 14: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

Central Tendency

Dispersion

Sampling with replacement

m = 50

s = 10

X

n =16

X = 2.5

n = 4

X = 5

mX = 50- X

Sampling Distribution

Population Distributionx

xn

Sampling from Normal Populations

Page 15: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

Standardizing the Sampling Distribution of x

Standardized Normal Distribution

m = 0

s = 1

Z

x

x

X XZ

n

Sampling Distribution

XmX

sX

Page 16: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

You’re an operations analyst for AT&T. Long-distance telephone calls are normally distribution with = 8 min. and = 2 min. If you select random samples of 25 calls, what percentage of the sample means would be between 7.8 & 8.2 minutes?

© 1984-1994 T/Maker Co.

Thinking Challenge

Page 17: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

Sampling Distribution

8

s `X = .4

7.8 8.2 `X 0

s = 1

–.50 Z.50

.3830

Standardized Normal Distribution

.1915.1915

7.8 8.50

225

8.2 8.50

225

XZ

n

XZ

n

Sampling Distribution Solution*

Page 18: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

Sampling from Non-Normal Populations

Page 19: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

Population size, N = 4

Random variable, x

Values of x: 1, 2, 3, 4

Uniform distribution

© 1984-1994 T/Maker Co.

Suppose There’s a Population ...

Developing Sampling Distributions

Page 20: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

1 2.5

N

ii

X

N

2

1 1.12

N

ii

X

N

Population Distribution

Summary Measures

.0

.1

.2

.3

1 2 3 4

P(x)

x

Population Characteristics

Page 21: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

Sample with replacement

1.0 1.5 2.0 2.5

1.5 2.0 2.5 3.0

2.0 2.5 3.0 3.5

2.5 3.0 3.5 4.0

16 Samples

1stObs

1,1 1,2 1,3 1,4

2,1 2,2 2,3 2,4

3,1 3,2 3,3 3,4

4,1 4,2 4,3 4,4

2nd Observation1 2 3 4

1

2

3

4

2nd Observation1 2 3 4

1

2

3

4

1stObs

16 Sample Means

All Possible Samples of Size n = 2

Page 22: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

1.0 1.5 2.0 2.5

1.5 2.0 2.5 3.0

2.0 2.5 3.0 3.5

2.5 3.0 3.5 4.0

2nd Observation1 2 3 4

1

2

3

4

1stObs

16 Sample Means Sampling Distribution of the

Sample Mean

.0

.1

.2

.3

1.0 1.5 2.0 2.5 3.0 3.5 4.0

P(x)

x

Sampling Distribution of All Sample Means

Page 23: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

1 1.0 1.5 ... 4.02.5

16

N

ii

X

X

N

2

1

N

i Xi

X

X

N

2 2 2(1.0 2.5) (1.5 2.5) ... (4.0 2.5).79

16

Summary Measures of All Sample Means

Page 24: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

Population Sampling Distribution

2.5x .79x

.0

.1

.2

.3

1 2 3 4

2.5 1.12

.0

.1

.2

.3

1.0 1.5 2.0 2.5 3.0 3.5 4.0

P(x)

x

P(x)

x

Comparison

Page 25: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

A fair die is thrown infinitely many times,with the random variable X = # of spots on any throw.

The probability distribution of X is:

…and the mean and variance are calculated as well:

9.25

x 1 2 3 4 5 6P(x

)1/6 1/6 1/6 1/6 1/6 1/6

Sampling Distribution of the Mean…

Page 26: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

Sampling Distribution of Two Dice

A sampling distribution is created by looking atall samples of size n=2 (i.e. two dice) and their means…

While there are 36 possible samples of size 2, there are only 11 values for , and some (e.g. =3.5) occur more frequently than others (e.g. =1). 9.26

Page 27: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

9.279.27

1.0 1/361.5 2/362.0 3/362.5 4/363.0 5/363.5 6/364.0 5/364.5 4/365.0 3/365.5 2/366.0 1/36

P( )

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

6/36

5/36

4/36

3/36

2/36

1/36

P(

)

Sampling Distribution of Two Dice…

Page 28: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

9.28

Compare…

Compare the distribution of X…

…with the sampling distribution of .

As well, note that:

1 2 3 4 5 6 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

Page 29: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

Sampling from Non-Normal Populations

Page 30: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

Law of Large Numbers

The law of large numbers states that, under general conditions, will be near with very high probability when n is large.

The conditions for the law of large numbers are Yi , i=1, …, n, are i.i.d. The variance of Yi , , is finite.

Page 31: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population
Page 32: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

9.32

Central Limit Theorem…

The sampling distribution of the mean of a random sample drawn from any population is approximately normal for a sufficiently large sample size.

The larger the sample size, the more closely the sampling distribution of X will resemble a normal distribution.

Page 33: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

9.33

Central Limit Theorem…

If the population is normal, then X is normally distributed for all values of n.

If the population is non-normal, then X is approximately normal only for larger values of n.

In most practical situations, a sample size of 30 may be sufficiently large to allow us to use the normal distribution as an approximation for the sampling distribution of X.

Page 34: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population
Page 35: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population
Page 36: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

Central Tendency

Dispersion

Sampling with replacement

Population Distribution

Sampling Distributionn =30

X = 1.8n = 4

X = 5

m = 50

s = 10

X

mX = 50- X

x

xn

Sampling from Non-Normal Populations

Page 37: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

X

As sample size gets large enough (n 30) ...

sampling distribution becomes almost normal.

x

xn

Central Limit Theorem

Page 38: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

Central Limit Theorem Example

The amount of soda in cans of a particular brand has a mean of 12 oz and a standard deviation of .2 oz. If you select random samples of 50 cans, what percentage of the sample means would be less than 11.95 oz?

SODA

Page 39: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

Sampling Distribution

12

s `X = .03

11.95 `X 0

s = 1

–1.77 Z

.0384

Standardized Normal

Distribution

.4616

11.95 121.77

.250

XZ

n

Shaded area exaggerated

Central Limit Theorem Solution*

Page 40: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

9.40

Example

One survey interviewed 25 people who graduated one year ago and determines their weekly salary.

The sample mean to be $750.

To interpret the finding one needs to calculate the probability that a sample of 25 graduates would have a mean of $750 or less when the population mean is $800 and the standard deviation is $100.

After calculating the probability, he needs to draw some conclusions.

Page 41: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

We want to find the probability that the sample mean is less than $750. Thus, we seek

The distribution of X, the weekly income, is likely to be positively skewed, but not sufficiently so to make the distribution of nonnormal. As a result, we may assume that is normal with mean

and standard deviation9.41

Example

)750X(P

X

800x

2025/100n/x

X

Page 42: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

9.42

Example

Thus,

The probability of observing a sample mean as low as $750 when the population mean is $800 is extremely small. Because this event is quite unlikely, we would have to conclude that the dean's claim is not justified.

0062.

4938.5.

)5.2Z(P

20

800750XP

)750X(P

x

x

Page 43: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

9.43

Using the Sampling Distribution for Inference

Here’s another way of expressing the probability calculated from a sampling distribution.

P(-1.96 < Z < 1.96) = .95Substituting the formula for the sampling distribution

With a little algebra

95.)96.1n/

X96.1(P

95.)n

96.1Xn

96.1(P

Page 44: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

9.44

Using the Sampling Distribution for Inference

Returning to the chapter-opening example where µ = 800, σ = 100, and n = 25, we compute

or

This tells us that there is a 95% probability that a sample mean will fall between 760.8 and 839.2. Because the sample mean was computed to be $750, we would have to conclude that the dean's claim is not supported by the statistic.

95.)25

10096.1800X

25

10096.1800(P

95.)2.839X8.760(P

Page 45: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

9.45

Using the Sampling Distribution for Inference

For example, with µ = 800, σ = 100, n = 25 and α= .01, we produce

01.1)n

zXn

z(P 005.005.

99.)25

100575.2800X

25

100575.2800(P

99.)5.851X5.748(P

Page 46: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

Sampling Distributions The Proportion

size sample

interest of sticcharacteri thehaving sample in the ofnumber

n

X itemsp

The proportion of the population having some characteristic is denoted π.

Page 47: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

Standard error for the proportion:

Z value for the proportion:

Sampling Distributions The Proportion

n

)(1σp

n

)(1σZ

p

pp

Page 48: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

If the true proportion of voters who support Proposition A is π = .4, what is the probability that a sample of size 200 yields a sample proportion between .40 and .45?

In other words, if π = .4 and n = 200, what is

P(.40 ≤ p ≤ .45) ?

Sampling Distributions The Proportion: Example

Page 49: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

Sampling Distributions The Proportion: Example

.03464200

.4).4(1

n

)(1σ

p

1.44)ZP(0

.03464

.40.45Z

.03464

.40.40P.45)P(.40

p

Find :

Convert to standardized normal:

Page 50: Sampling Distributions. Parameter & Statistic Parameter Summary measure about population Sample Statistic Summary measure about sample P P PP in Population

Sampling Distributions The Proportion: Example

Use cumulative normal table:

P(0 ≤ Z ≤ 1.44) = P(Z ≤ 1.44) – 0.5 = .4251

Z.45

1.44

.4251

Standardize

Sampling Distribution

Standardized Normal

Distribution

.40

0p