Chapter 11 – 1

Lecture 7How certain are we?

Sampling and the normal distribution

Chapter 11 – 2

Lecture 6: Summary• If we take a simple random sample

– from a well-defined population

• we expect– that the sample mean– is “probably” “close” to the population mean

• By “close” we mean “within ~2 standard errors”

Lecture 7: Preview• Today, we’ll learn that “probably” means

– in 95% of all samples

Chapter 11 – 3

Overview

• Review of sampling distributions

• Sampling distributions have a “normal” shape

• Properties of the “normal” distribution, e.g.:– In 95% of all samples,

• the sample mean

• is within 1.96 standard errors

• of the population mean

Chapter 11 – 4

Repeated sampling

Y YY Y

Y Y Y Y Y Y Y YY Y Y Y Y Y Y Y YY Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y YY Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y

Each Y represents the number of children in a household

Population: All US households

Y YY Y

…

All possible samples

25.1YN=4

50.2YN=4

Y=1.75

Chapter 11 – 5

Notation

Mnemonics:Population measures are called Parameters.Sample measures are called Statistics.The P words and S words go together.

Population parameters use Greek lettersSample statistics use Roman letters

=Greek m=Greek p=Greek s

The population is the source of the sample.Greek culture was the source of Roman culture.

Chapter 11 – 6

Population

Population US households

Variable Y (# of children)

population mean Y=1.75

population standard deviation Y=1.62

Chapter 11 – 7

Sample

CHILDS1022

sample mean 1.25

Sample size N=4

Variable Y (# of children)

sample mean

sample standard deviation sY=.92

25.1Y

Within sample…

Chapter 11 – 8

Population of samples

CHILDS2404

sample mean 2.50

CHILDS1022

sample mean 1.25

…# of samples: infinite

“Variable”

Mean

Standard error

YYY here 1.75

NYY / here 1.62 / 41/2 = 1.62 / 2 = 0.81

Y

Y

(Std. dev. of sample means)

Across samples…but just N=4 adults per sample

Chapter 11 – 9

As sample size (N) grows……standard error shrinks!…shape of sampling distribution gets closer to “normal”!

1.75.81

1.75.405

1.75.2025

Chapter 11 – 10

Normal distribution

• symmetric

• bell-shaped

• very specific numeric properties

Chapter 11 – 11

“Margin of error”

This means: In 95% of all samples, the sample mean is within 1.96 standard errors of the population mean.

+/- 1.96 (or 2) standard errors often called “margin of error”

Confidence z94% 1.8895% 1.9696% 2.05

In your course binder,find the “z (standard normal)…table”.Look for this line.

Chapter 11 – 12

Example 1Again “population”: US adultsVariable: Y: “How many children have you ever had?”Y=1.75, Y=1.62.

Consider samples of size N=16.

95% of all sample means are within 1.96 standard errors of pop. mean—i.e., in

.542 to96.0

79.75.1

)4/62.1(96.175.1

)16/62.1(96.175.1

/96.1

96.1

N

Z

YY

YY

YY

95%

Y

Chapter 11 – 13

More on sampling error

This means: In 99% of all samples, the sample mean is within 2.58 standard errors of the population mean.

(1% of samples have means that are further away.)

Look for this line.

Confidence z98% 2.3399% 2.5899.9% 3.29

Chapter 11 – 14

Example 2Variable: Y: “How many children have you ever had?”Y=1.75, Y=1.62.

Consider samples of size N=45.

99% of all sample means are within 2.58 standard errors of the population mean

—i.e., in

37.2 to13.1

62.75.1

)45/62.1(58.275.1

/58.2

58.2

NYY

YY

99%

Chapter 11 – 15

Sampling error: Exercise

Complete the following:90% of all samples have means within _______SE’s of the population mean.

Complete the following:If researchers take samples of 100 US adults,90% of the time the sample will average between _______ and _________ children.

Chapter 11 – 16

• The sampling distribution of – has mean– and standard error

• As the sample size N gets larger,– the standard error gets smaller– and the sampling distribution gets closer to “normal.”

• So– larger samples give

• closer

• more predictable

– approximations to the population mean

NYY /

Summary: Central Limit Theorem (CLT)

Y

YY

Chapter 11 – 17

Summary• Lecture 6 (Law of Large Samples)

– If we take simple random samples• from a well-defined population

– we expect• that the sample means• is “usually” “close” to the population mean

• Lecture 7 (Central Limit Theorem)– If by “close”

• we mean “within 1.96 standard errors”– then by “usually”

• we mean “in 95% of all samples”– For other definitions of “close” and “usually,”

• see the “z (standard normal)…table” in your course binder

Chapter 11 – 18

Teaser: Lecture 8 (Confidence intervals)

• So if we take– just one sample

• we can guess– that the population statistic is “close”

• and we’ll “usually” be right