Chapter 18 -- Part 1

Sampling Distribution Models for p̂

Sampling Distribution Models

Population

Sample

Population Parameter?

Sample Statistic

Inference

Objectives

Describe the sampling distribution of a sample proportion

Understand that the variability of a statistic depends on the size of the sample Statistics based on larger samples are less

variable

Review

Chapter 12 – Sample Surveys Parameter (Population Characteristics)

(mean)• p (proportion)

Statistic (Sample Characteristics)• (sample mean)• (sample proportion)p̂y

Review

Chapter 12 “Statistics will be different for each sample. These

differences obey certain laws of probability (but only for random samples).”

Chapter 14 Taking a sample from a population is a random

phenomena. That means:• The outcome is unknown before the event occurs• The long term behavior is predictable

Example

Who? Stat 101 students in Sections G and H. What? Number of siblings. When? Today. Where? In class. Why? To find out what proportion of students’

have exactly one sibling.

Example

Population Stat 101 students in sections G and H.

Population Parameter Proportion of all Stat 101 students in

sections G and H who have exactly one sibling.

Example

Sample 4 randomly selected students.

Sample Statistic The proportion of the 4 students who have

exactly one sibling.

Example

Sample 1

Sample 2

Sample 3

p̂

p̂

p̂

What Have We Learned

Different samples produce different sample proportions.

There is variation among sample proportions.

Can we model this variation?

Example

Senators Population Characteristics p = proportion of Democratic Senators

Take SRS of size n = 10 Calculate Sample Characteristics

• = sample proportion of Democratic Senatorsp̂

Example

0.75

0.34

0.63

0.52

0.21

Sample p̂

SRS characteristics

Values of and are random Change from sample to sample Different from population characteristics

p = 0.50

p̂

Imagine

Repeat taking SRS of size n = 10 Collection of values for and ARE

DATA Summarize data – make a histogram

Shape, Center and Spread Sampling distribution for

p̂

p̂

Sampling Distribution for

Mean (Center)

We would expect on average to get p. Say is unbiased for p.

p̂

pp ˆ

p̂

Sampling Distribution for

Standard deviation (Spread)

As sample size n gets larger, gets smaller

Larger samples are more accurate

p̂

n

pq

n

ppp

)1(ˆ

p̂

Example

50% of people on campus favor current academic calendar.

1. Select n people. 2. Find sample

proportion of people favoring current academic calendar.

3. Repeat sampling. 4. What does sampling

distribution of sample proportion look like?

n=2

n=5

n=10

n=25

35.0

50.0

ˆ

ˆ

p

p

22.0

50.0

ˆ

ˆ

p

p

16.0

50.0

ˆ

ˆ

p

p

10.0

50.0

ˆ

ˆ

p

p

Example

10% of all people are left handed.

1. Select n people. 2. Find sample

proportion of left handed people.

3. Repeat sampling. 4. What does sampling

distribution of sample proportion look like?

n=2

n=10

n=50

n=100

214.0

10.0

ˆ

ˆ

p

p

096.0

10.0

ˆ

ˆ

p

p

043.0

10.0

ˆ

ˆ

p

p

030.0

10.0

ˆ

ˆ

p

p

Sampling Distribution for

Shape• Normal Distribution

Two assumptions must hold in order for us to be able to use the normal distribution

• The sampled values must be independent of each other

• The sample size, n, must be large enough

p̂

Sampling Distribution for

It is hard to check that these assumptions hold, so we will settle for checking the following conditions

• 10% Condition – the sample size, n, is less than 10% of the population size

• Success/Failure Condition – np > 10, n(1-p) > 10

These conditions seem to contradict one another, but they don't!

p̂

Sampling Distribution for

Assuming the two conditions are true (must be checked for each problem), then the sampling distribution for is

p̂

p̂

npq

pN ,

Sampling Distribution for

But the sampling distribution has a center (mean) of p (a population proportion) often times we don’t know p. Let be the center.

p̂

nqp

pNˆˆ

,ˆ

p̂

Example

Senators Check assumptions (p = 0.50)

1. 10(0.50) = 5 and 10(0.50) = 5

2. n = 10 is 10% of the population size. Assumption 1 does not hold. Sampling Distribution of ????p̂

Example #1

Public health statistics indicate that 26.4% of the U.S. adult population smoked cigarettes in 2002. Use the 68-95-99.7 Rule to describe the sampling distribution for the sample proportion of smokers among 50 adults.

Example #1

Check assumptions:1. np = (50)(0.264) = 13.2 > 10

nq = (50)(0.736) = 36.8 > 10

1. n = 50, less than 10% of population Therefore, the sampling distribution for

the proportion of smokers is

062.0,264.0N

Example # 1

About 68% of samples have a sample proportion between 20.2% and 32.6%

About 95% of samples have a sample proportion between 14% and 38.8%

About 99.7% of samples have a sample proportion between 7.8% and 45%

Example #2

Information on a packet of seeds claims that the germination rate is 92%. What's the probability that more than 95% of the 160 seeds in the packet will germinate?

Check assumptions:1. np = (160)(0.92) = 147.2 > 10 nq = (160)(0.08) = 12.8 > 102. n = 160, less than 10% of all seeds?

Review - Standardizing

You can standardize using the formula

npq

ppz

pz

p

p

ˆ

ˆ

ˆ

ˆ

Review

Chapter 6 – The Normal Distribution Y~ N(70,3)

•

•

•

Do you remember the 68-95-99.7 Rule?

6293.0)33.()3

7071()71(

ZPZPYP

7486.)67.(1)3

7068()68(

ZPZPYP

3779.2514.6293.)68()71()7168( YPYPYP

Example #2

Therefore, the sampling distribution for the proportion of seeds that will germinate is

02.0,92.0N

0668.0

)50.1(

50.1

02.0

92.095.095.0ˆ

ZP

ZP

ZPpP

Big Picture

Population

Sample

Population Parameter?

Sample Statistic

Inference

Big Picture Before we would take one random sample and compute

our sample statistic. Presently we are focusing on:

This is an estimate of the population parameter p. But we realized that if we took a second random sample

that from sample 1 could possibly be different from the we would get from sample 2. But from sample 2 is also an estimate of the population parameter p.

If we take a third sample then the for third sample could possibly be different from the first and second s. Etc.

p̂ number of outcomes

Total sample size

p̂p̂

p̂p̂ '

p̂

Big Picture So there is variability in the sample statistic . If we randomized correctly we can consider

as random (like rolling a die) so even though the variability is unavoidable it is understandable and predictable!!! (This is the absolutely amazing part).

p̂

p̂

Big Picture

So for a sufficiently large sample size (n) we can model the variability in with a normal model so:

p̂ ~ N p,pq

n

p̂

Big Picture The hard part is trying to visualize what is going

on behind the scenes. The sampling distribution of is what a histogram would look like if we had every possible sample available to us. (This is very abstract because we will never see these other samples).

So lets just focus on two things:

p̂

Take Home Message

1. Check to see that A. the sample size, n, is less than 10% of the

population size B. np > 10, n(1-p) > 10

2. If these hold then can be modeled with a normal distribution that is:

p̂ ~ N p,pq

n

p̂

Example #3

When a truckload of apples arrives at a packing plant, a random sample of 150 apples is selected and examined for bruises, discoloration, and other defects. The whole truckload will be rejected if more than 5% of the sample is unsatisfactory (i.e. damaged). Suppose that actually 8% of the apples in the truck do not meet the desired standard. What is the probability of accepting the truck anyway?

Example #3

What is the sampling distribution?1. np = (150)(0.08) = 12>10

nq = (150)(0.92) = 138>102. n = 150 > 10% of all apples

So, the sampling distribution is N(0.08,0.022).What is the probability of accepting the truck anyway?

0869.0

)36.1(

)022.0

08.005.0()05.0ˆ(

ZP

ZPpP