m ario f . t riola

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


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


6-1 Overview

methods for estimating population means, proportions, and variances

methods for determining sample sizes

This chapter presents:


6-2

Estimating a Population Mean:Large Samples


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.


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.


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


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


DefinitionConfidence Interval

(or Interval Estimate)

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

parameter





parameter

Lower # < population parameter < Upper #





parameter

Lower # < population parameter < Upper #

As an exampleLower # < < Upper #


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)


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


Figure 6-1

Confidence Intervals from 20 Different Samples


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


The Critical Value

z=0

Figure 6-2Found from Table A-2

(corresponds to area of

0.5 - 2 )

z2

z2-z2

2 2


-z2z2

95%

.95

.025.025

2 = 2.5% = .025 = 5%

Finding z2 for 95% Degree of Confidence


-z2z2

95%

.95

.025.025

2 = 2.5% = .025 = 5%

Critical Values



.4750

.025

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

= 0.025 = 0.05




.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


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

mean µ.

denoted by E

Definition



mean µ.

denoted by E

µ x + Ex - E

Definition



mean µ.

denoted by E

µ x + Ex - E

x -E < µ < x +E

Definition



mean µ.

denoted by E

µ x + Ex - E

x -E < µ < x +Elower limit

Definition

upper limit


Definition Margin of Error

µ x + Ex - E

E = z/2 • Formula 6-1n


Definition Margin of Error

µ x + Ex - E

also called the maximum error of the estimate

E = z/2 • Formula 6-1n


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.


Confidence Interval (or Interval Estimate)

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

x - E < µ < x + E


x - E < µ < x + E

µ = x + E




x - E < µ < x + E

µ = x + E

(x + E, x - E)




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.





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






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



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.


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


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.


n = 106

x = 98.2o

s = 0.62o

= 0.05/2 = 0.025

z / 2 = 1.96



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



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



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



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



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


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 .


Finding the Point Estimate and E from a Confidence Interval

Point estimate of µ:

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

2


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

m ario f . t riola

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