Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 1 Estimates and Sample Sizes...

Copyright © 1998, Triola, Elementary Statistics

Addison Wesley Longman 1

Estimates and Sample Estimates and Sample SizesSizesChapter 6Chapter 6

M A R I O F. T R I O L ACopyright © 1998, Triola, Elementary Statistics

Addison Wesley Longman



Chapter 6Estimate and Sample Sizes

6-1 Overview

6-2 Estimating a Population Mean: Large Samples

6-3 Estimating a Population Mean: Small Samples

6-4 Determining Sample Size

6-5 Estimating a Population Proportion

6-6 Estimating a Population Variance



6-1 Overview

This chapter presents:methods for estimating population means,

proportions, and variances

methods for determining sample sizes necessary to estimate the above parameters.



6-2 Estimating a Population Mean: Large Samples



Definitions Estimator

a sample statistic used to approximate a population parameter

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

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



Definitions Estimator

a sample statistic used to approximate a population parameter

Estimatea specific value or range of values used to approximate some

population parameter

Point Estimatea single volue (or point) used to approximate a popular

parameter

The sample mean x

population mean µ.is the best point estimate of the



Confidence Interval (or Interval Estimate)

a range (or an interval) of values likely to contain the true value of the population

parameter

Lower # < population parameter < Upper #



Confidence Interval (or Interval Estimate)

a range (or an interval) of values likely to contain the true value of the population

parameter

Lower # < population parameter < Upper #

As an exampleLower # < µ < Upper #



Definition Degree of Confidence

(level of confidence or confidence coefficient)

the probability 1 – (often expressed as the equivalent percentage value) that the confidence interval contains the true value of the population parameter

usually 95% or 99% ( = 5%) ( = 1%)



Confidence Intervals from Different Samples

98.00 98.50

• •x

98.08 98.32

• •• •

This confidence intervaldoes not contain µ

• •• •

µ = 98.25 (but unknown to us)

Figure 6-3



Definition Critical Valuethe 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. There is an area of

1 – between the vertical borderlines at

–z/2 and z/2.



If Degree of Confidence = 95%




–z2z2

95%

.95

.025.025

2 = 2.5% = .025 = 5%




–z2z2

95%

.95

.025.025

2 = 2.5% = .025 = 5%

Critical Values



95% Degree of Confidence




.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



Definition Margin of Error



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

population mean µ.




population mean µ.

denoted by E

µ x + Ex – E



x – E


population mean µ.

denoted by E

µ x + E

x – E < µ < x + E




population mean µ.

denoted by E

µ x + Ex – E

x – E < µ < x + E

lower limit




population mean µ.

denoted by E

µ x + Ex – E

x – E < µ < x +E

upper limitlower limit



Calculating the Margin of Error When Is Unknown



Calculating the Margin of Error When Is known

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.

E = z/2 • Formula 6-1n



Confidence Interval

• 1. If using original data, round to one more decimal place than used in data.

Round off Rules



Confidence Interval

• 1. If using original data, round to one more decimal place than used in data.

• 2. If given summary statistics (n, x, s), round to same number of decimal places

as in x.

Round off Rules



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 thepopulation standard deviation is unknown andn > 30, use the value of the sample standarddeviation s.






2. Evaluate the margin of error E = z2 • n . If thepopulation standard deviation is unknown andn > 30, use the value of the sample standarddeviation s.

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 thepopulation standard deviation is unknown andn > 30, use the value of the sample standarddeviation s.

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

x – E < µ < x + E 4. Round using the confidence intervals round off rules.



Confidence Intervals from Different Samples



6-3

Determining Sample Size







z/ 2 •E =

n




z/ 2 •E =

n

(solve for n by algebra)




If n is not a whole number, round it up to the next higher whole number.

z/ 2 •E =

n

(solve for n by algebra)

z/ 2 E

n =

2Formula 6-2



What happens when E is doubled ?




/ 2 z

1

n = =2

1/ 2

(z ) 2

E = 1 :




Sample size n is decreased to 1/4 of its original value if E is doubled.

Larger errors allow smaller samples.

Smaller errors require larger samples.

/ 2 z

1

n = =2

1/ 2

(z ) 2

/ 2 z

2

n = =

2

4/ 2

(z ) 2

E = 1 :

E = 2 :



What if is unknown ?

1. Use the range rule of thumb to estimate the standard deviation as follows: range / 4

or

2. Calculate the sample standard deviation s and use it in place of . That value can be refined as more sample data are obtained.

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