ch08-SamplingDist

Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 8-1

Chapter 8

Sampling Distribution


Chapter Goals

After completing this chapter, you should be able to:

Define the concept of a sampling distribution

Determine the mean and standard deviation for the sampling distribution of the sample mean, X

Sampling distribution of means

Central Limit Theorem

Sampling distribution of S2

Sampling distribution of ps

__


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



Assume there is a population …

Population size N=4

Random variable, X,

is age of individuals

Values of X: 18, 20,

22, 24 (years)

A B C D


.3

.2

.1

0 18 20 22 24

A B C D

Uniform Distribution

P(x)

x

(continued)

Summary Measures for the Population Distribution:


214

24222018

N

Xμ i

2.236N

μ)(Xσ

2i


1st 2nd Observation Obs 18 20 22 24

18 18,18 18,20 18,22 18,24

20 20,18 20,20 20,22 20,24

22 22,18 22,20 22,22 22,24

24 24,18 24,20 24,22 24,24

16 possible samples (sampling with replacement)

Now consider all possible samples of size n=2


18 18 19 20 21

20 19 20 21 22

22 20 21 22 23

24 21 22 23 24

(continued)


16 Sample Means



18 18 19 20 21

20 19 20 21 22

22 20 21 22 23

24 21 22 23 24

Sampling Distribution of All Sample Means

18 19 20 21 22 23 240

.1

.2

.3 P(X)

X

Sample Means Distribution

16 Sample Means

_

Sampling Distribution(continued)

(no longer uniform)

_


Summary Measures of this Sampling Distribution:

Sampling Distribution(continued)

2116

24211918

N

Xμ i

X

1.5816

21)-(2421)-(1921)-(18

N

)μX(σ

222

2Xi

X


Comparing the Population with its Sampling Distribution

18 19 20 21 22 23 240

.1

.2

.3 P(X)

X 18 20 22 24

A B C D

0

.1

.2

.3

PopulationN = 4

P(X)

X _

1.58σ 21μXX

2.236σ 21μ

Sample Means Distributionn = 2

_





of the Mean


of the Proportion


of the Variance


Sampling Distributions of the Mean



of the Mean


of the Proportion


of the Variance


Standard Error of the Mean

Different samples of the same size from the same population will yield different sample means

A measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean:

Note that the standard error of the mean decreases as the sample size increases

n

σσ

X


If the Population is Normal

If a population is normal with mean μ and

standard deviation σ, the sampling distribution

of is also normally distributed with

and

(This assumes that sampling is with replacement or sampling is without replacement from an infinite population)

X

μμX

n

σσ

X


Z-value for Sampling Distributionof the Mean

Z-value for the sampling distribution of :

where: = sample mean

= population mean

= population standard deviation

n = sample size

Xμσ

n

σμ)X(

σ

)μX(Z

X

X

X


Finite Population Correction

Apply the Finite Population Correction if: the sample is large relative to the population

(n is greater than 5% of N)

and… Sampling is without replacement

Then

1NnN

n

σ

μ)X(Z


Normal Population Distribution

Normal Sampling Distribution (has the same mean)

Sampling Distribution Properties

(i.e. is unbiased )xx

x

μμx

μ

xμ


Sampling Distribution Properties

For sampling with replacement:

As n increases,

decreasesLarger sample size

Smaller sample size

x

(continued)

xσ

μ


If the Population is not Normal

We can apply the Central Limit Theorem:

Even if the population is not normal, …sample means from the population will be

approximately normal as long as the sample size is large enough.

Properties of the sampling distribution:

andμμx n

σσx


n↑

Central Limit Theorem

As the sample size gets large enough…

the sampling distribution becomes almost normal regardless of shape of population

x


Population Distribution

Sampling Distribution (becomes normal as n increases)

Central Tendency

Variation

(Sampling with replacement)

x

x

Larger sample size

Smaller sample size

If the Population is not Normal(continued)

Sampling distribution properties:

μμx

n

σσx

xμ

μ


How Large is Large Enough?

For most distributions, n > 30 will give a sampling distribution that is nearly normal

For fairly symmetric distributions, n > 15

For normal population distributions, the sampling distribution of the mean is always normally distributed


Example

Suppose a population has mean μ = 8 and standard deviation σ = 3. Suppose a random sample of size n = 36 is selected.

What is the probability that the sample mean is between 7.8 and 8.2?


Example

Solution:

Even if the population is not normally distributed, the central limit theorem can be used (n > 30)

… so the sampling distribution of is approximately normal

… with mean = 8

…and standard deviation

(continued)

x

xμ

0.536

3

n

σσx


Example

Solution (continued):(continued)

0.38300.5)ZP(-0.5

363

8-8.2

nσ

μ- μ

363

8-7.8P 8.2) μ P(7.8 X

X

Z7.8 8.2 -0.5 0.5


Standard Normal Distribution .1915

+.1915

Population Distribution

??

??

?????

??? Sample Standardize

8μ 8μX

0μz xX


Sampling Distributions of the S2

where:

the distribution with v = n – 1 degrees of freedom.

n

i

iXXSn

1 2

2

2

2

2)()1(

If S2 is the variance of a random sample of size n takenfrom a normal population having the variance σ2, then Chi-square statistic is:


Sampling Distributions of the S2

2α

The 2 test statistic approximately follows a chi-squared distribution with one degree of freedom

0


Sampling Distributions of Proportion



of the Mean


of the Proportion


of the Variance


Sampling Distributions of p

p = the proportion of the population having some characteristic

Sample proportion ( ps ) provides an estimate of p:

0 ≤ ps ≤ 1

ps has a binomial distribution

(assuming sampling with replacement from a finite population or without replacement from an infinite population)

size sample

interest ofstic characteri the having sample the in itemsofnumber

n

Xps


Sampling Distribution of p

Approximated by a

normal distribution if:

where

and

(where p = population proportion)

Sampling DistributionP( ps)

.3

.2

.1 0

0 . 2 .4 .6 8 1 ps

pμsp

n

p)p(1σ

sp

5p)n(1

5np

and


Z-Value for Proportions

If sampling is without replacement

and n is greater than 5% of the

population size, then must use

the finite population correction

factor:

1N

nN

n

p)p(1σ

sp

np)p(1

pp

σ

ppZ s

p

s

s

Standardize ps to a Z value with the formula:

pσ


Example

If the true proportion of voters who support

Proposition A is p = .4, what is the probability

that a sample of size 200 yields a sample

proportion between .40 and .45?

i.e.: if p = .4 and n = 200, what is

P(.40 ≤ ps ≤ .45) ?


Example

if p = .4 and n = 200, what is

P(.40 ≤ ps ≤ .45) ?

(continued)

.03464200

.4).4(1

n

p)p(1σ

sp

1.44)ZP(0

.03464

.40.45Z

.03464

.40.40P.45)pP(.40 s

Find :

Convert to standard normal:

spσ


Example

Z.45 1.44

.4251

Standardize

Sampling DistributionStandardized

Normal Distribution

if p = .4 and n = 200, what is

P(.40 ≤ ps ≤ .45) ?

(continued)

Use standard normal table: P(0 ≤ Z ≤ 1.44) = .4251

.40 0ps


t-Distribution

Student’s t-Distribution

Population standard deviation is unknown

Sample size is not large enough

t value depends on degrees of freedom (d.f.)

The t-distribution is like the normal, but with greater spread since both and have fluctuations due to sampling.

nSX

T/

d.f. = v = n - 1


If the mean of these three values is 8.0, then X3 must be 9 (i.e., X3 is not free to vary)

Degrees of Freedom (df)

Here, n = 3, so degrees of freedom = n – 1 = 3 – 1 = 2

(2 values can be any numbers, but the third is not free to vary for a given mean)

Idea: Number of observations that are free to vary after sample mean has been calculated

Example: Suppose the mean of 3 numbers is 8.0

Let X1 = 7

Let X2 = 8

What is X3?


t-Distribution

t0

t (df = 5)

t (df = 13)t-distributions are bell-shaped and symmetric, but have ‘fatter’ tails than the normal

Standard Normal

(t with df = )

Note: t Z as n increases


t Table

Upper Tail Area

df .25 .10 .05

1 1.000 3.078 6.314

2 0.817 1.886 2.920

3 0.765 1.638 2.353

t0 2.920The body of the table contains t values, not probabilities

Let: n = 3 df = n - 1 = 2 = .10 /2 =.05

/2 = .05


t distribution values

With comparison to the Z value

Confidence t t t Z Level (10 d.f.) (20 d.f.) (30 d.f.) ____

.80 1.372 1.325 1.310 1.28

.90 1.812 1.725 1.697 1.64

.95 2.228 2.086 2.042 1.96

.99 3.169 2.845 2.750 2.57

Note: t Z as n increases


F-Distribution

The F statistic is the ratio of two 2 random variables, each divided by its number of degrees of freedom.

If S12 and S2

2 are the variances of samples of size n1 and n2 taken from normal populations with variances 1

2 and 22,

then

has an F-distribution with 1 = n1 - 1 and 2 = n2 - 1 degrees of freedom.

For the F-distribution table

Note that since the ratio is inverted, 1 and 2 are reversed.

2

2

2

2

2

1

2

1

SS

F

),(1

),(12

211

ff


F-Distribution Usage

The F-distribution is used in two-sample situations to draw inferences about the population variances.

In an area of statistics called analysis of variance, sources of variability are considered, for example: Variability within each of the two samples. Variability between the two samples (variability between

the means). The overall question is whether the variability within the

samples is large enough that the variability between the means is not significant.


Ex.8.54

PulI-strength tests on 10 soldered leads for a semiconductor device yield the following results in pounds force required to rupture the bond:

19.8 12.7 13.2 16.9 10.6

18.8 11.1 14.3 17.0 12.5

Another set of 8 leads was tested after encapsulation to determine whether the pull strength has been increased by encapsulation of the device, with the following results:

24.9 22.8 23.6 22.1 20.4 21.6 21.8 22.5

Comment on the evidence available concerning equality of the two population variances.


Solution


Sampling Distribution Summary

Normal distribution: Sampling distribution of Xbar when is known for any population distribution.

t-distribution: Sampling distribution of Xbar when is unknown and S is used. Population must be normal.

Chi-square (2) distribution: Sampling distribution of S2. Population must be normal.

F-distribution: The distribution of the ratio of two 2 random variables. Sampling distribution of the ratio of the variances of two different samples. Population must be normal.

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