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

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 2 / 33

Outline

1 The Central Limit Theorem for Proportions

2 Applications

3 Assignment

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 3 / 33

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.

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 4 / 33

The Central Limit Theorem for Proportions

The Sample Sizen is large enough if

np ≥ 5 and n(1− p) ≥ 5.

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 5 / 33

Outline

1 The Central Limit Theorem for Proportions

2 Applications

3 Assignment

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 6 / 33

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?

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 7 / 33

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.

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 8 / 33

Applications

0.55 0.60 0.65 0.70

2

4

6

8

10

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 9 / 33

Applications

0.55 0.60 0.65 0.70

2

4

6

8

10

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 10 / 33

Applications

The probability that p̂ is greater than 0.65 is

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

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 11 / 33

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.

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 12 / 33

Applications

0.55 0.60 0.65 0.70

5

10

15

20

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 13 / 33

Applications

0.55 0.60 0.65 0.70

5

10

15

20

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 14 / 33

Applications

The probability that p̂ is greater than 0.65 is

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

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 15 / 33

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%?

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 16 / 33

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)?

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 17 / 33

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)?

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 18 / 33

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.

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 19 / 33

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.

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 20 / 33

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

2

4

6

8

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 21 / 33

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

2

4

6

8

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 21 / 33

Homework Problem

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

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

(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_____.

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 22 / 33

Homework Problem

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

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

(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 0.551and 0.649.

(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_____.

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 23 / 33

Homework Problem

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

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

(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 0.551and 0.649.

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

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

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 24 / 33

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.

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 25 / 33

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.

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 26 / 33

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.

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 26 / 33

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.

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 26 / 33

Homework Problem

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

proportions you could get based on a simple random sample of400 students.

0.50 0.55 0.60 0.65 0.70 0.75

5

10

15

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 27 / 33

Homework Problem

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

proportions you could get based on a simple random sample of400 students.

0.50 0.55 0.60 0.65 0.70 0.75

5

10

15

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 27 / 33

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.

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 28 / 33

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.

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 28 / 33

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.

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 28 / 33

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.

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 28 / 33

Outline

1 The Central Limit Theorem for Proportions

2 Applications

3 Assignment

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 29 / 33

Assignment

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

Exercises 7 - 14, page 526.

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 30 / 33

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.

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 31 / 33

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.

Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 32 / 33

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.

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