1Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

MARIO F. TRIOLAMARIO F. TRIOLA EIGHTHEIGHTH

EDITIONEDITION

ELEMENTARY STATISTICSChapter 6 Estimates and Sample Sizes

2Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Chapter 6Estimates and Sample Sizes

6-1 Overview

6-2 Estimating a Population Mean: Large Samples

6-3 Estimating a Population Mean: Small Samples

6-4 Sample Size Required to Estimate µ

6-5 Estimating a Population Proportion

6-6 Estimating a Population Variance

3Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

6-1 Overview

  methods for estimating population   means, proportions, and variances

  methods for determining sample sizes

This chapter presents:

4Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

6-2

Estimating a Population Mean:Large Samples

5Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Assumptions

n > 30The sample must have more than 30 values.

Simple Random SampleAll samples of the same size have an equal chance

of being selected.

6Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Assumptions

n > 30The sample must have more than 30 values.

Simple Random SampleAll samples of the same size have an equal chance

of being selected.

Data collected carelessly can be absolutely worthless, even if the sample

is quite large.

7Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Definitions Estimator

a formula or process for using sample data to estimate a population parameter

Estimatea specific value or range of values used to approximate some population parameter

Point Estimatea single value (or point) used to approximate a population parameter

8Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Estimatora formula or process for using sample data to estimate a population

parameter

Estimatea specific value or range of values used to approximate

some population parameter

Point Estimatea single value (or point) used to approximate a population

parameter

The sample mean x is the best point estimate of the population mean µ.

Definitions

9Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

DefinitionConfidence Interval

(or Interval Estimate)

a range (or an interval) of values used to estimate the true value of the population

parameter

10Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

DefinitionConfidence Interval

(or Interval Estimate)

a range (or an interval) of values used to estimate the true value of the population

parameter

Lower # < population parameter < Upper #

11Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

DefinitionConfidence Interval

(or Interval Estimate)

a range (or an interval) of values used to estimate the true value of the population

parameter

Lower # < population parameter < Upper #

As an exampleLower # < < Upper #

12Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

the probability 1 - (often expressed as the equivalent percentage value) that is the relative frequency of times the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times

usually 90%, 95%, or 99% ( = 10%), ( = 5%), ( = 1%)

DefinitionDegree of Confidence

(level of confidence or confidence coefficient)

13Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Interpreting a Confidence Interval

Correct: We are 95% confident that the interval from 98.08 to 98.32 actually does contain the true value of

. This means that if we were to select many different samples of size 106 and construct the confidence intervals, 95% of them would actually contain the

value of the population mean .

Wrong: There is a 95% chance that the true value of will fall between 98.08 and 98.32.

98.08o < µ < 98.32o

14Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Figure 6-1

Confidence Intervals from 20 Different Samples

15Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

the number on the borderline separating sample statistics that are likely to occur from those that

are unlikely to occur. The number z/2 is a critical

value that is a z score with the property that it separates an area /2 in the right tail of the standard normal distribution.

Definition

Critical Value

16Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

The Critical Value

z=0

Figure 6-2Found from Table A-2

(corresponds to area of

0.5 - 2 )

z2

z2-z2

2 2

17Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

-z2z2

95%

.95

.025.025

2 = 2.5% = .025 = 5%

Finding z2 for 95% Degree of Confidence

18Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

-z2z2

95%

.95

.025.025

2 = 2.5% = .025 = 5%

Critical Values

Finding z2 for 95% Degree of Confidence

19Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

.4750

.025

Use Table A-2 to find a z score of 1.96

= 0.025 = 0.05

Finding z2 for 95% Degree of Confidence

20Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Finding z2 for 95% Degree of Confidence

.025.025

- 1.96 1.96

z2 = 1.96

.4750

.025

Use Table A-2 to find a z score of 1.96

= 0.025 = 0.05

21Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Margin of Error is the maximum likely difference observed between sample mean x and true population

mean µ.

denoted by E

Definition

22Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Margin of Error is the maximum likely difference observed between sample mean x and true population

mean µ.

denoted by E

µ x + Ex - E

Definition

23Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Margin of Error is the maximum likely difference observed between sample mean x and true population

mean µ.

denoted by E

µ x + Ex - E

x -E < µ < x +E

Definition

24Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Margin of Error is the maximum likely difference observed between sample mean x and true population

mean µ.

denoted by E

µ x + Ex - E

x -E < µ < x +Elower limit

Definition

upper limit

25Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Definition Margin of Error

µ x + Ex - E

E = z/2 • Formula 6-1n

26Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Definition Margin of Error

µ x + Ex - E

also called the maximum error of the estimate

E = z/2 • Formula 6-1n

27Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Calculating E When Is Unknown

  If n > 30, we can replace in Formula 6-1   by the sample standard deviation s.

  If n 30, the population must have a   normal distribution and we must know   to use Formula 6-1.

28Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Confidence Interval (or Interval Estimate)

for Population Mean µ(Based on Large Samples: n >30)

x - E < µ < x + E

29Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

x - E < µ < x + E

µ = x + E

Confidence Interval (or Interval Estimate)

for Population Mean µ(Based on Large Samples: n >30)

30Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

x - E < µ < x + E

µ = x + E

(x + E, x - E)

Confidence Interval (or Interval Estimate)

for Population Mean µ(Based on Large Samples: n >30)

31Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Procedure for Constructing a Confidence Interval for µ

( Based on a Large Sample: n > 30 )

1. Find the critical value z2 that corresponds to the desired degree of confidence.

32Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Procedure for Constructing a Confidence Interval for µ

( Based on a Large Sample: n > 30 )

1. Find the critical value z2 that corresponds to the desired degree of confidence.

2. Evaluate the margin of error E = z2 • / n . If the population standard deviation is unknown,  use the value of the sample standard deviation s  provided that n > 30.

33Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Procedure for Constructing a Confidence Interval for µ

( Based on a Large Sample: n > 30 )

1. Find the critical value z2 that corresponds to the desired degree of confidence.

3. Find the values of x - E and x + E. Substitute thosevalues in the general format of the confidenceinterval: x - E < µ < x + E

2. Evaluate the margin of error E = z2 • / n . If the population standard deviation is unknown,  use the value of the sample standard deviation s  provided that n > 30.

34Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Procedure for Constructing a Confidence Interval for µ

( Based on a Large Sample: n > 30 )

1. Find the critical value z2 that corresponds to the desired degree of confidence.

3. Find the values of x - E and x + E. Substitute thosevalues in the general format of the confidenceinterval:

4. Round using the confidence intervals roundoff rules.

x - E < µ < x + E

2. Evaluate the margin of error E = z2 • / n . If the population standard deviation is unknown,  use the value of the sample standard deviation s  provided that n > 30.

35Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Round-Off Rule for Confidence

Intervals Used to Estimate µ1. When using the original set of data, round the     confidence interval limits to one more decimal     place than used in original set of data.

36Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

1. When using the original set of data, round the     confidence interval limits to one more decimal     place than used in original set of data.

2. When the original set of data is unknown and     only the summary statistics (n, x, s) are     used, round the confidence interval limits to     the same number of decimal places used for     the sample mean.

Round-Off Rule for Confidence

Intervals Used to Estimate µ

37Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval.

38Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

n = 106

x = 98.2o

s = 0.62o

= 0.05/2 = 0.025

z / 2 = 1.96

Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval.

39Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

n = 106

x = 98.20o

s = 0.62o

= 0.05/2 = 0.025

z / 2 = 1.96

E = z / 2 • = 1.96 • 0.62 = 0.12n 106

Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval.

40Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

n = 106

x = 98.20o

s = 0.62o

= 0.05/2 = 0.025

z / 2 = 1.96

E = z / 2 • = 1.96 • 0.62 = 0.12n 106

x - E < < x + E

Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval.

41Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

n = 106

x = 98.20o

s = 0.62o

= 0.05/2 = 0.025

z / 2 = 1.96

E = z / 2 • = 1.96 • 0.62 = 0.12n 106

x - E < < x + E98.20o - 0.12 < < 98.20o + 0.12

Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval.

42Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

n = 106

x = 98.20o

s = 0.62o

= 0.05/2 = 0.025

z / 2 = 1.96

E = z / 2 • = 1.96 • 0.62 = 0.12n 106

x - E < < x + E98.20o - 0.12 < < 98.20o + 0.12

98.08o < < 98.32o

Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval.

43Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

n = 106

x = 98.20o

s = 0.62o

= 0.05/2 = 0.025

z / 2 = 1.96

E = z / 2 • = 1.96 • 0.62 = 0.12n 106

x - E < < x + E98.08o < < 98.32o

Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval.

Based on the sample provided, the confidence interval for the

population mean is 98.08o < < 98.32o. If we were to select many different samples of the same size, 95% of the confidence intervals

would actually contain the population mean .

44Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Finding the Point Estimate and E from a Confidence Interval

Point estimate of µ:

x = (upper confidence interval limit) + (lower confidence interval limit)

2

45Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Finding the Point Estimate and E from a Confidence Interval

Point estimate of µ:

x = (upper confidence interval limit) + (lower confidence interval limit)

2

Margin of Error:

E = (upper confidence interval limit) - (lower confidence interval limit)

2