lecture 26 - sampling distribution proportion.pdf

8/14/2019 Lecture 26 - Sampling Distribution Proportion.pdf

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Outline

1 Computing the Sampling Distribution ofp

2 The Central Limit Theorem for Proportions

3 Applications

4 Assignment

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 5, 2012 3 / 35
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Sampling Distributions

Definition (Sampling Distribution of a Statistic)

Thesampling distribution of a statisticis the distribution of values of

that statistic over all possible samples of a given size nfrom the

population.

We may sample with or without replacement.

For our purposes, it will be simpler to sample with replacement.

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The Sample Proportion

We will work out the sampling distribution for pfor sample sizes of1, 2, and 3.

Then I will show you the sampling distribution for pfor samplesizes of 4, 5, and 10.

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Example

Suppose that 45%of all people approve of President Obamasperformance.

Suppose that we select one person at random.

We may diagram the 2 possibilities.



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Example

Y

N

0.55

0.45

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Example

Y

N

0.55

0.45

NY

NN

1

0

(0.55)(0.45) = 0.2475

(0.55)2= 0.3025

Y

N0.55

0.45

Y

N0.55

0.45

YN 1 (0.45)(0.55) = 0.2475

YY 2 (0.45)2= 0.2025

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Example

Now we take a sample of 3 people, sampling with replacement.

Find the sampling distribution of p.

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Example

Y

N

0.55

0.45

NYY

NNY

2

1

(0.55)(0.45)2= 0.1114

(0.55)2

(0.45) = 0.1114

Y

N0.55

0.45

YNY 2 (0.45)(0.55)(0.45) = 0.1114

YYY 3 (0.45)3= 0.0911Y

N0.55

0.45

Y

N0.55

0.45

Y

N0.55

0.45

Y

N0.55

0.45

Y

N0.55

0.45

NYN

NNN

1

0

(0.55)(0.45)(0.55) = 0.1361

(0.55)3= 0.1664

YNN 1 (0.45)(0.55)2= 0.1361

YYN 2 (0.45)2(0.55) = 0.1114

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Example

Letpbe the sample proportion of people who strong disapproveof President Obamas performance.

The sampling distribution of pis

p P(p)0 0.1664

1/3 0.40842/3 0.3341

1 0.0911


S S
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Samples of Sizen= 4

If we sample 4 people, then the sampling distribution of the

sample proportion is

p P(p)

0 0.09151/4 0.29952/4 0.36753/4 0.2005

1 0.0410


S l f Si 5
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Samples of Sizen= 5



p P(p)

0 0.05031/5 0.20592/5 0.33693/5 0.27574/5 0.1128

1 0.0185


S l f Si 6
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Samples of Sizen= 6



p P(p)

0 0.0277

1/6 0.13592/6 0.27803/6 0.30324/6 0.18615/6 0.0609

1 0.0083


S l f Si 10
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Samples of Sizen= 10



p P(p)

0.00 0.00250.10 0.0207

0.20 0.07630.30 0.16650.40 0.23840.50 0.23400.60 0.15960.70 0.07460.80 0.02290.90 0.00421.00 0.0003


O tli
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Outline



3 Applications

4 Assignment


The Central Limit Theorem for Proportions


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Theorem (The Central Limit Theorem for Proportions)

For any population and any sample size, the sampling distribution

ofp has the following mean and standard deviation:

p = p

p =

p(1 p)

n .

Furthermore, the sampling distribution ofp is approximatelynormal, provided n is large enough.




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The Sample Size

nis large enough if

np 5 andn(1 p) 5.

Ifnis small, then we have to work out the distribution by hand.


Outline


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Outline



3 Applications

4 Assignment


Applications
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Applications

Suppose that 60% of all high-school students own a cell phone.

If we survey 3 high-school students, how likely is it that we will findthat at least 2 of them own a cell phone?


Applications
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Applications

Suppose that 60% of all high-school students own a cell phone.

If we survey 150 high-school students, how likely is it that we willfind that at least 65% of them own a cell phone?


Applications
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Applications

Ifp=0.60 and our sample size isn=150, thenpis normal withmeanp=0.60 and

p=

(0.60)(0.40)

150 =

0.0016= 0.04.

We want to know the probability that p 0.65.


Applications
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Applications

0.55 0.60 0.65 0.70

2

4

6

8

10


Applications
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Applications

0.55 0.60 0.65 0.70

2

4

6

8

10


Applications
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Applications

The probability that pis greater than 0.65 is

normalcdf(.65,E99,.60,.04) =0.1056.


Applications
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Applications

What if our sample size were 600 instead of 150?

Thenpis normal with meanp=0.60 and

p=

(0.60)(0.40)

600 =

0.0004= 0.02.


Applications
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Applications

0.55 0.60 0.65 0.70

5

10

15

20


Applications
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pp

0.55 0.60 0.65 0.70

5

10

15

20

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Guessing on a Test


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g

Example (Guessing on a Test)

A student takes a math placement test with 25 multiple-choice

questions.



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Guessing on a Test


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Example (Guessing on a Test)

A student takes a math placement test with 25 multiple-choice

questions.

Each question has 5 choices.If a student guesses at each answer, what will his score most

likely be?

If a student scores 10 out of 25, can we be sure that he did not

guess at all 25 answers?


Guessing on a Test
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Example (Guessing on a Test)If he guesses at all 25 answers, then the probability of getting any

one answer correct isp=0.20.

The proportion that he actually gets correct isp.


Guessing on a Test
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The distribution ofpis normal with mean 0.20 and standard

deviation

(0.20)(0.80)

25 =0.04.


Guessing on a Test


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deviation

(0.20)(0.80)

25 =0.04.

If we allow three standard deviations each way, then pis almostcertainly at least 0.08 and at most 0.32.


Guessing on a Test
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deviation

(0.20)(0.80)

25 =0.04.

If we allow three standard deviations each way, then pis almostcertainly at least 0.08 and at most 0.32.

That represents from 2 to 8 correct answers.


Outline
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3 Applications

4

Assignment


Assignment
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Homework

Read Sections 8.1 - 8.2, pages 499 - 508.

Exercises 7 - 14, page 526.

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lecture 26 - sampling distribution proportion.pdf

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