Sampling Distribution of a Sample Mean - Lecture 26 Section...

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Sampling Distribution of a Sample Mean Robb T. Koether Homework Review Introduction An Experiment Assignment Sampling Distribution of a Sample Mean Lecture 26 Section 8.4 Robb T. Koether Hampden-Sydney College Mon, Oct 13, 2008

Transcript of Sampling Distribution of a Sample Mean - Lecture 26 Section...

Page 1: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb

SamplingDistribution of

a SampleMean

Robb T.Koether

HomeworkReview

Introduction

AnExperiment

Assignment

Sampling Distribution of a Sample MeanLecture 26Section 8.4

Robb T. Koether

Hampden-Sydney College

Mon, Oct 13, 2008

Page 2: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb

SamplingDistribution of

a SampleMean

Robb T.Koether

HomeworkReview

Introduction

AnExperiment

Assignment

Outline

1 Homework Review

2 Introduction

3 An Experiment

4 Assignment

Page 3: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb

SamplingDistribution of

a SampleMean

Robb T.Koether

HomeworkReview

Introduction

AnExperiment

Assignment

Homework Review

Exercise 8.12, page 528.

Suppose that 60% of all students at a large universityaccess course information using the Internet.(a) Sketch a picture of the distribution for the possible

sample proportions you could get based on a simplerandom sample of 100 students.

(b) Use the 68–95–99.7 rule for normal distributions tocomplete the following statements:

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

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

(iii) It is almost certain that the sample proportion isbetween _____ and _____.

Page 4: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb

SamplingDistribution of

a SampleMean

Robb T.Koether

HomeworkReview

Introduction

AnExperiment

Assignment

Homework Review

Exercise 8.12, page 528.

(c) Would it be likely to observe a sample proportion of0.50, based on a simple random sample of size 100, ifthe population proportion were 0.60? Explain.

(d) Sketch a picture of the distribution for the possiblesample proportions you could get based on a simplerandom sample of 400 students.

(i) How does this picture differ from the one in part (a)?(ii) How will the increased sample size affect the range of

values you gave in (i)–(iii) of part (b)

Page 5: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb

SamplingDistribution of

a SampleMean

Robb T.Koether

HomeworkReview

Introduction

AnExperiment

Assignment

Homework Review

Solution(a) For n = 100 students, the sketch is

0.600.453 0.747

Page 6: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb

SamplingDistribution of

a SampleMean

Robb T.Koether

HomeworkReview

Introduction

AnExperiment

Assignment

Homework Review

Solution(b) First, compute µp̂ and σp̂.

µp̂ = p = 0.60.

σp̂ =

√p(1− p)

n=

√(0.60)(0.40)

100= 0.0490.

(i) There is a 68% chance that the sample proportion isbetween 0.60− 0.0490 = 0.551 and0.60 + 0.0490 = 0.649.

(ii) There is a 95% chance that the sample proportion isbetween 0.60− 2(0.0490) = 0.502 and0.60 + 2(0.0490) = 0.698.

(iii) It is almost certain that the sample proportion is between0.60− 3(0.0490) = 0.453 and 0.60 + 3(0.0490) = 0.747.

Page 7: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb

SamplingDistribution of

a SampleMean

Robb T.Koether

HomeworkReview

Introduction

AnExperiment

Assignment

Homework Review

Exercise 8.12, page 528.

(c) The question should ask how likely it is to observe asample proportion at least as low as 0.50. Theprobability is P(p̂ ≤ 0.50), which is

normalcdf(-E99,0.50,0.60,0.490) = 0.0206.

Page 8: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb

SamplingDistribution of

a SampleMean

Robb T.Koether

HomeworkReview

Introduction

AnExperiment

Assignment

Homework Review

Exercise 8.12, page 528.

(d) For n = 400 students, the sketch of p̂ is

0.600.5265 0.6735

Page 9: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb

SamplingDistribution of

a SampleMean

Robb T.Koether

HomeworkReview

Introduction

AnExperiment

Assignment

Homework Review

Exercise 8.12, page 528.

(d) (i) This distribution is only half as wide (and twice as tall).(ii) The standard deviation of p̂ is only half as much, so the

answers are(i) There is a 68% chance that the sample proportion is

between 0.60− 0.0245 = 0.5755 and0.60 + 0.0245 = 0.6245.

(ii) There is a 95% chance that the sample proportion isbetween 0.60− 2(0.0245) = 0.5510 and0.60 + 2(0.0245) = 0.6490.

(iii) It is almost certain that the sample proportion is between0.60− 3(0.0245) = 0.5265 and 0.60 + 3(0.0245) = 0.6735.

Page 10: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb

SamplingDistribution of

a SampleMean

Robb T.Koether

HomeworkReview

Introduction

AnExperiment

Assignment

Introduction

Recall the experiment we did in which we collected 100samples of size 5 and computed the sampleproportions.We graphed our results and saw a good approximationto the normal curve.Then we calculated the mean and standard deviation ofour distribution and found that we had goodapproximations to

µp̂ = p

and

σp̂ =

√p(1− p)

n.

Now we will do the same with sample means.

Page 11: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb

SamplingDistribution of

a SampleMean

Robb T.Koether

HomeworkReview

Introduction

AnExperiment

Assignment

Introduction

Recall the experiment we did in which we collected 100samples of size 5 and computed the sampleproportions.We graphed our results and saw a good approximationto the normal curve.Then we calculated the mean and standard deviation ofour distribution and found that we had goodapproximations to

µp̂ = p

and

σp̂ =

√p(1− p)

n.

Now we will do the same with sample means.

Page 12: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb

SamplingDistribution of

a SampleMean

Robb T.Koether

HomeworkReview

Introduction

AnExperiment

Assignment

Introduction

Recall the experiment we did in which we collected 100samples of size 5 and computed the sampleproportions.We graphed our results and saw a good approximationto the normal curve.Then we calculated the mean and standard deviation ofour distribution and found that we had goodapproximations to

µp̂ = p

and

σp̂ =

√p(1− p)

n.

Now we will do the same with sample means.

Page 13: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb

SamplingDistribution of

a SampleMean

Robb T.Koether

HomeworkReview

Introduction

AnExperiment

Assignment

Introduction

Recall the experiment we did in which we collected 100samples of size 5 and computed the sampleproportions.We graphed our results and saw a good approximationto the normal curve.Then we calculated the mean and standard deviation ofour distribution and found that we had goodapproximations to

µp̂ = p

and

σp̂ =

√p(1− p)

n.

Now we will do the same with sample means.

Page 14: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb

SamplingDistribution of

a SampleMean

Robb T.Koether

HomeworkReview

Introduction

AnExperiment

Assignment

The US Senate

There are 100 senators in the US Senate.Their tenures range from 1 year to 49 years.The mean and standard deviation of the population are

µ = 13.45 years

andσ = 11.18 years.

Page 15: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb

SamplingDistribution of

a SampleMean

Robb T.Koether

HomeworkReview

Introduction

AnExperiment

Assignment

The US Senate

The histogram:

0

5

10

15

25

20

0 10 50403020

Page 16: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb

SamplingDistribution of

a SampleMean

Robb T.Koether

HomeworkReview

Introduction

AnExperiment

Assignment

The US Senate

The boxplot:

0 10 50403020

Page 17: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb

SamplingDistribution of

a SampleMean

Robb T.Koether

HomeworkReview

Introduction

AnExperiment

Assignment

An Experiment

State Years State Years State Years State Years State Years State Years State YearsAL 11 DE 7 IA 23 MI 7 NH 5 OK 14 TX 15AL 10 DE 35 IA 27 MI 29 NH 15 OK 3 TX 6AK 40 FL 7 KS 11 MN 1 NJ 2 OR 11 UT 31AK 6 FL 3 KS 12 MN 5 NJ 5 OR 12 UT 15AZ 13 GA 5 KY 23 MS 1 NM 25 PA 1 VT 1AZ 21 GA 3 KY 9 MS 30 NM 35 PA 27 VT 33AR 5 HI 18 LA 11 MO 1 NY 7 RI 1 VA 1AR 9 HI 45 LA 3 MO 21 NY 9 RI 11 VA 29CA 16 ID 17 ME 13 MT 1 NC 5 SC 5 WA 7CA 15 ID 9 ME 11 MT 30 NC 3 SC 3 WA 15CO 11 IL 11 MD 1 NE 7 ND 16 SD 11 WV 49CO 3 IL 3 MD 21 NE 11 ND 16 SD 3 WV 23CT 19 IN 31 MA 46 NV 7 OH 1 TN 1 WI 19CT 27 IN 9 MA 21 NV 21 OH 9 TN 5 WI 15

WY 1WY 11

Page 18: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb

SamplingDistribution of

a SampleMean

Robb T.Koether

HomeworkReview

Introduction

AnExperiment

Assignment

An Experiment

State Years State Years State Years State Years State Years State Years State Years1 11 15 7 29 23 43 7 57 5 71 14 85 152 10 16 35 30 27 44 29 58 15 72 3 86 63 40 17 7 31 11 45 1 59 2 73 11 87 314 6 18 3 32 12 46 5 60 5 74 12 88 155 13 19 5 33 23 47 1 61 25 75 1 89 16 21 20 3 34 9 48 30 62 35 76 27 90 337 5 21 18 35 11 49 1 63 7 77 1 91 18 9 22 45 36 3 50 21 64 9 78 11 92 299 16 23 17 37 13 51 1 65 5 79 5 93 7

10 15 24 9 38 11 52 30 66 3 80 3 94 1511 11 25 11 39 1 53 7 67 16 81 11 95 4912 3 26 3 40 21 54 11 68 16 82 3 96 2313 19 27 31 41 46 55 7 69 1 83 1 97 1914 27 28 9 42 21 56 21 70 9 84 5 98 15

99 1100 11

Page 19: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb

SamplingDistribution of

a SampleMean

Robb T.Koether

HomeworkReview

Introduction

AnExperiment

Assignment

An Experiment

Work in pairs.Use the TI-83 to get 10 samples of 5 senators each.(Allow repetitions.)For each sample, find the number of years that eachsenator has been in the senate.Record the average (out of 5).When you are finished, report the 10 sample meansthat you found.

Page 20: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb

SamplingDistribution of

a SampleMean

Robb T.Koether

HomeworkReview

Introduction

AnExperiment

Assignment

An Example

For example,Sample Tenures Mean

{54, 38, 28, 70, 9} {11, 11, 9, 9, 16} 11.2{46, 84, 6, 72, 49} {5, 5, 21, 3, 1} 7.0{7, 4, 32, 26, 79} {5, 6, 12, 3, 5} 6.2{33, 18, 80, 56, 35} {23, 3, 3, 21, 11} 12.6{85, 54, 59, 25, 27} {15, 11, 2, 11, 31} 14.0{99, 73, 63, 82, 56} {1, 11, 7, 3, 21} 8.6{51, 20, 72, 46, 70} {1, 3, 3, 5, 9} 4.2{70, 1, 93, 87, 95} {9, 11, 7, 31, 49} 21.4{25, 5, 4, 28, 66} {11, 13, 6, 9, 3} 8.4{40, 73, 88, 1, 51} {21, 11, 15, 11, 1} 11.8

Page 21: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb

SamplingDistribution of

a SampleMean

Robb T.Koether

HomeworkReview

Introduction

AnExperiment

Assignment

An Example

0

1

2

3

4

0 10 50403020

5

Page 22: Sampling Distribution of a Sample Mean - Lecture 26 Section 8people.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-10-14 · Sampling Distribution of a Sample Mean Robb

SamplingDistribution of

a SampleMean

Robb T.Koether

HomeworkReview

Introduction

AnExperiment

Assignment

Assignment

HomeworkStudy the Central Limit Theorem.Catch up on past homework.