Sampling Distribution of a Sample Proportionpeople.hsc.edu/faculty-staff/robbk/Math121/Lectures...Homework Problem Exercise 8.12, page 528. (b)Sketch a picture of the distribution

Sampling Distribution of a Sample ProportionLecture 26Section 8.4

Robb T. Koether

Hampden-Sydney College

Mon, Mar 1, 2010

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 1 / 33

Outline

1 The Central Limit Theorem for Proportions

2 Applications

3 Assignment


Outline


2 Applications

3 Assignment


The Central Limit Theorem for Proportions

Theorem (The Central Limit Theorem for Proportions)For any population, the sampling distribution of p̂ has the followingmean and standard deviation:

µp̂ = p

σp̂ =

√p(1− p)

n.

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


The Central Limit Theorem for Proportions

The Sample Sizen is large enough if

np ≥ 5 and n(1− p) ≥ 5.


Outline


2 Applications

3 Assignment


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

If p = 0.60 and our sample size is n = 150, then p̂ is normal withmean µp̂ = 0.60 and

σp̂ =

√(0.60)(0.40)

150=√

0.0016 = 0.04.

Want to know the probability that p̂ ≥ 0.65.


Applications

0.55 0.60 0.65 0.70

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Applications

0.55 0.60 0.65 0.70

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Applications

The probability that p̂ is greater than 0.65 is

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


Applications

What if our sample size were 600 instead of 150?Then p̂ is normal with mean µp̂ = 0.60 and

σp̂ =

√(0.60)(0.40)

600=√

0.0004 = 0.02.


Applications

0.55 0.60 0.65 0.70

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Applications

0.55 0.60 0.65 0.70

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Applications

The probability that p̂ is greater than 0.65 is

normalcdf(.65,E99,.60,.02) = 0.0062.


Applications

Suppose that the true proportion of Americans who stronglydisapprove of President Obama’s performance is 40%.Can we use a sample of 1500 Americans to disprove thehypothesis that the rate is 45%?


Applications

Which is the correct question to ask?I Given that we observed p̂ = 0.40 in a sample, what is the

probability that p = 0.45?I If p really is 0.45, what is the probability that we would observe

p̂ = 0.40 (or more extreme)?


Applications

Which is the correct question to ask?I Given that we observed p̂ = 0.40 in a sample, what is the

probability that p = 0.45?I If p really is 0.45, what is the probability that we would observe

p̂ = 0.40 (or more extreme)?


Hypothesis Testing

Let us test the hypotheses

H0: 45% of all Americans strongly disapprove of PresidentObama’s performance.

H1: Less than 45% of all Americans strongly disapprove ofPresident Obama’s performance.


Hypothesis Testing

Using the Central Limit Theorem, the null hypothesis predicts thatthe distribution of p̂ is

I Normal, withI Mean p, which is 0.45 (hypothetically).I Standard deviation

√(0.45)(0.55)

1500 = 0.0128.

That is, p̂ is N(0.45,0.0128).Find the probability that p̂ ≤ 0.40.Design a decision rule so that α = 0.05.


Homework Problem

Exercise 8.12, page 528.Suppose that 60% of all students at a large university access courseinformation using the Internet.(a) Sketch a picture of the distribution for the possible sample

proportions you could get based on a simple random sample of100 students.

0.50 0.55 0.60 0.65 0.70 0.75

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4

6

8


Homework Problem

Exercise 8.12, page 528.Suppose that 60% of all students at a large university access courseinformation using the Internet.(a) Sketch a picture of the distribution for the possible sample


0.50 0.55 0.60 0.65 0.70 0.75

2

4

6

8


Homework Problem

Exercise 8.12, page 528.(b) Sketch a picture of the distribution for the possible sample


(c) Use the 68–95–99.7 rule for normal distributions to complete thefollowing statements:

(i) There is a 68% chance that the sample proportion is between_____ and _____.

(ii) There is a 95% chance that the sample proportion is between_____ and _____.

(iii) It is almost certain that the sample proportion is between _____ and_____.


Homework Problem




(i) There is a 68% chance that the sample proportion is between 0.551and 0.649.

(ii) There is a 95% chance that the sample proportion is between_____ and _____.



Homework Problem




(i) There is a 68% chance that the sample proportion is between 0.551and 0.649.

(ii) There is a 95% chance that the sample proportion is between 0.502and 0.698.



Homework Problem

Exercise 8.12, page 528.(b) Use the 68–95–99.7 rule for normal distributions to complete the

following statements:(i) There is a 68% chance that the sample proportion is between 0.551

and 0.649.(ii) There is a 95% chance that the sample proportion is between 0.502

and 0.698.(iii) It is almost certain that the sample proportion is between 0.453 and

0.747.


Homework Problem

Exercise 8.12, page 528.(c) Would it be likely to observe a sample proportion of 0.50, based on

a simple random sample of size 100, if the population proportionwere 0.60? Explain.

(The question should ask how likely it is to observe a sampleproportion at least as low as 0.50.)No. A sample proportion of 0.50 is more than 2 standarddeviations from the mean.


Homework Problem


a simple random sample of size 100, if the population proportionwere 0.60? Explain.(The question should ask how likely it is to observe a sampleproportion at least as low as 0.50.)

No. A sample proportion of 0.50 is more than 2 standarddeviations from the mean.


Homework Problem


a simple random sample of size 100, if the population proportionwere 0.60? Explain.(The question should ask how likely it is to observe a sampleproportion at least as low as 0.50.)No. A sample proportion of 0.50 is more than 2 standarddeviations from the mean.


Homework Problem

Exercise 8.12, page 528.(d) Sketch a picture of the distribution for the possible sample


0.50 0.55 0.60 0.65 0.70 0.75

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10

15


Homework Problem

Exercise 8.12, page 528.(d) Sketch a picture of the distribution for the possible sample


0.50 0.55 0.60 0.65 0.70 0.75

5

10

15


Homework Problem

Exercise 8.12, page 528.(i) How does this picture differ from the one in part (a)?

It is only half as wide.How will the increased sample size affect the range of values yougave in (i)–(iii) of part (b)They will each be only half as wide.


Homework Problem

Exercise 8.12, page 528.(ii) (i) How does this picture differ from the one in part (a)?

It is only half as wide.

How will the increased sample size affect the range of values yougave in (i)–(iii) of part (b)They will each be only half as wide.


Homework Problem


It is only half as wide.How will the increased sample size affect the range of values yougave in (i)–(iii) of part (b)

They will each be only half as wide.


Homework Problem


It is only half as wide.How will the increased sample size affect the range of values yougave in (i)–(iii) of part (b)They will each be only half as wide.


Outline


2 Applications

3 Assignment


Assignment

Homework(ii) Read Sections 8.1 - 8.2, pages 499 - 508.

Exercises 7 - 14, page 526.


Assignment

Homework Answers8. (a) 0.0264.

(b) Yes. The probability of that is 3.117× 10−6.10. (a) The sampling distribution for the sample proportion of men is

normal with mean 0.49 and standard deviation 0.04999.(b) 0.0548.


Assignment

Homework Answers12. (a) Draw a normal curve with mean 0.60 and standard deviation

0.04899.(b) (i) 0.55101 and 0.64899.

(ii) 0.50202 and 0.69798.(iii) 0.45303 and 0.74697.

(c) No. According to part (b)(ii), a proportion as low as 0.50 has onlyabout a 2 1

2 % chance of occurring.(d) Sketch a normal curve with mean 0.60 and standard deviation

0.02449.(i) It is narrower.(ii) They will each be half as wide as before.


Assignment

Homework Answers14. (a) It is normal with mean 0.50 and standard deviation 0.02887.

(b) p̂ = 0.56. p-value of 0.56 is 0.0188.(c) Reject H0. A majority of shoppers favor longer shopping hours.


Sampling Distribution of a Sample Proportionpeople.hsc.edu/faculty-staff/robbk/Math121/Lectures...Homework Problem Exercise 8.12, page 528. (b)Sketch a picture of the distribution

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