probability confidence level

8/12/2019 probability confidence level

1/58

1Copyright 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Chapter 7Confidence Intervals and

Sample Sizes7.2 Estimating a Proportion p

7.3 Estimating a Mean ( known)

7.4 Estimating a Mean ( unknown)

7.5 Estimating a Standard Deviation


2/58


Example 1

TRICK QUESTION!We only know the sample proportion s ,We do no t know the population proportion .

BUT The proportion of the sample (0.7) is ourbest point estimate (i.e. best guess).

In a recent poll, 70% or 1501 randomly selected adultssaid they believed in global warming.Q: What is the proportion of the adult population

that believe in global warming?


3/58


Definition

PopulationParameter

Best PointEstimate

Proportion

Mean

Std. Dev.

p

pxs

Point Estimate A single value (or point) used to approximatea population parameter


4/58


Definition

Confidence Interval : CIThe range (or interval ) of values to estimatethe true value of a population parameter.

It is abbreviated as CI

In Example 1, the 95% confidence interval for the population proportion p is CI = (0.677, 0.723)


5/58


Definition

Confidence Level : 1 The probability that the confidence intervalactually contains the population parameter.

The most common confidence levels usedare 90% , 95% , 99%

90% : = 0.1 95% : = 0.05 99% : = 0.01

In Example 1, the Confidence level is 95%


6/58


Definition

Margin of Error : EThe maximum likely difference betweenthe observed value and true value of the

population parameter (with probability is 1 )The margin of error is used to determine aconfidence interval (of a proportion or mean)

In Example 1, the 95% margin of error for the population proportion p is E = 0.023


7/587Copyright 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

A : 0.7 is the best point estimate of the proportion

of all adults who believe in global warming.The 95% confidence interval of the populationproportion p is:

CI = (0.677, 0.723)( with a margin of error E = 0.023 )

What does it mean, exactly?

In a recent poll, 70% or 1501 randomly selectedadults said they believed in global warming.

Q: What is the proportion of the adultpopulation that believe in global warming?

Example 1 Continued


8/58



Know the correct interpretation of aconfidence interval

It is incorrect to say

the probability that the population parameter belongs to the confidenceinterval is 95%

because the population parameter is nota random variable, its value cannot change

The population is set in stone

!!! Caution !!!



Do not confuse the two percentages

The proportion can be representedby percents (like 70% in Example 1)

The confidence level may be representedby percents (like 95% in Example 1)

Proportions can be any value from 0% to 100%

Confidence levels are usually 90%, 95%, or 99%

!!! Caution !!!



( y E , y + E )y = Best point estimate

E = Margin of Error

Confidence Interval Formula

Centered at the best point estimate

Width is determined by E

The value of E depends the critical value of the CI



Finding the Point Estimate and Efrom a Confidence Interval

Margin of Error : E

E = (upper confidence limit) (lower confidence limit) 2

Point estimate : y

y = (upper confidence limit) + (lower confidence limit)

2



Definition

Critical ValueThe number on the borderline separatingsample statistics that are likely to occur from

those that are unlikely to occur. A critical value is dependent on a probabilitydistribution the parameter follows and theconfidence level (1 ) .



Normal Dist. Critical ValuesFor a population proportion p and mean ( known), the critical values are found usingz -scores on a standard normal distribution

The standard normal distribution is divided intothree regions : middle part has area 1 andtwo tails (left and right) have area /2 each:



The z-scores z a /2 and z a /2 separate the values :

Likely values ( middle interval )

Unlikely values ( tails )

Use StatCrunch to calculate z-scores (see Ch. 6)

z a /2

z a /2

Normal Dist. Critical Values



The value z a /2

separates an area of a /2 in

the right tail of the z-dist.

The value z a /2 separates an area of a /2 in

the left tail of the z-dist. The subscript a /2 is simply a reminder that the z -score separates an area of a /2 in the tail.

Normal Dist. Critical Values



Section 7.2Estimating a Population Proportion

Objective

Find the confidence interval for a population

proportion p Determine the sample size needed to estimatea population proportion p



Definitions

The best point estimate for a populationproportion p is the sample proportion p

Best point estimate : p

The margin of error E is the maximumlikely difference between the observedvalue and true value of the populationproportion p (with probability is 1 )



Margin of Error for Proportions

2

pqE z

na

E = margin of error

p = sample proportion

q = 1 p

n = number sample values

1 = Confidence Level



Confidence Interval for aPopulation Proportion p

( p E , p + E ) where

2

pqE z n

a



Finding the Point Estimate and Efrom a Confidence Interval

Margin of Error:

E = (upper confidence limit) (lower confidence limit) 2

Point estimate of p :p = (upper confidence limit) + (lower confidence limit)

2


22/58


Round-Off Rule forConfidence Interval Estimates of p

Round the confidence interval limitsfor p to

three significant digits .


23/58


Find the 95%confidence interval for the populationproportion If a sample of size 100 has a proportion 0.67

Direct Computation

Example 1


24/58


Using StatCrunch

Stat Proportions One Sample with Summary

Example 1 Find the 95%confidence interval for the populationproportion If a sample of size 100 has a proportion 0.67


25/58


Using StatCrunch

Enter Values



26/58


Using StatCrunch

Click Next



27/58


Using StatCrunch

Select Confidence Interval



28/58


Using StatCrunch

Enter Confidence Level, then click Calculate



29/58


Using StatCrunch

From the output, we find the Confidence interval isCI = (0.578, 0.762)

Lower Limit

Upper Limit

Standard Error



30/58


Sample Size

Suppose we want to collect sample data inorder to estimate some populationproportion. The question is how many sample items must be obtained?


31/58


Determining Sample Size

(solve for n by algebra)

( )2 p qZn =

E 2

az E = p q n


32/58


Sample Size for EstimatingProportion p

When an estimate of p is known : ( )2 p q

n =

E 2a

z

When no estimate of p is known:

use p = q = 0.5( )2 0.25n =

E 2az


33/58


Round-Off Rule for DeterminingSample Size

If the computed sample size n is nota whole number, round the value of n up to the next larger whole number.

Examples:

n = 310.67 round up to 311n = 295.23 round up to 296n = 113.01 round up to 114

l


34/58


A manager for E-Bay wants to determine the current percentage of U.S. adults who now use the Internet.

How many adults must be surveyed in order to be95% confident that the sample percentage is in error

by no more than three percentage points when

(a ) In 2006, 73% of adults used the Internet.

(b ) No known possible value of the proportion.

Example 2

l


35/58


(a) Given:

Given a sample has proportion of 0.73,To be 95% confident that our sample proportionis within three percentage points of the true

proportion, we need at least 842 adults .

Example 2

E l 2


36/58


(b) Given:

For any sample,To be 95% confident that our sample proportionis within three percentage points of the true

proportion, we need at least 1068 adults .

Example 2

S


37/58


SummaryConfidence Interval of a Proportion

2

pqE z

na

( p E , p + E )

E = margin of error

p = sample proportion

n = number sample values


S


38/58


When an estimate of p is known :

( )2 p qn =

E 2

az

When no estimate of p is known (use p = q = 0.5)

( )2 0.25n = E 2

az

SummarySample Size for Estimating a Proportion


39/58


Section 7.3

Estimating a Population mean ( known)

Objective

Find the confidence interval for a population

mean when

is knownDetermine the sample size needed to estimatea population mean when is known


40/58


Best Point Estimation

The best point estimate for a populationmean ( known) is the sample mean x

Best point estimate : x


41/58


= population mean = population standard deviation

= sample mean

n = number of sample values

E = margin of error

z a /2 = z -score separating an area of /2 in theright tail of the standard normaldistribution

x

Notation


42/58


(1) The population standard deviation is known

(2) One or both of the following:The population is normally distributed

or

n > 30

Requirements


43/58


Margin of Error


44/58


Confidence Interval

( x E , x + E )where


45/58


Definition

The two values x E and x + E are

called confidence interval limits .


46/58


1. When using the original set of data , round theconfidence interval limits to one more decimalplace than used in original set of data.

2. When the original set of data is unknown andonly the summary statistics (n , x , s ) are used,

round the confidence interval limits to the samenumber of decimal places used for thesample mean.

Round-Off Rules for ConfidenceIntervals Used to Estimate

Example


47/58


Find the 90% confidence interval for the population mean If the population standard deviation is known to be 10 and a sample of

size 42 has a mean of 38.4

Direct Computation

Example

Example


48/58


Using StatCrunch

Stat Z statistics One Sample with Summary



Example

Example


49/58


Using StatCrunch

Enter Parameters



Example

Example


50/58


Using StatCrunch

Click Next



Example

Example


51/58


Using StatCrunch

Select Confidence Interval



Example

Example


52/58


Using StatCrunch

Enter Confidence Level, then click Calculate



Example

Example


53/58


Using StatCrunch

From the output, we find the Confidence interval isCI = (35.862, 40.938)

Lower LimitUpper Limit

Standard Error



Example

Sample Size for Estimating a


54/58


Sample Size for Estimating aPopulation Mean

(z a / 2) n = E

2

= population mean

= population standard deviation

= sample mean

E = desired margin of error

z /2 = z score separating an area of a /2 in the right tail ofthe standard normal distribution

x


55/58


Round-Off Rule for DeterminingSample Size

If the computed sample size n is nota whole number, round the value of n up to the next larger whole number.

Examples:

n = 310.67 round up to 311n = 295.23 round up to 296n = 113.01 round up to 114

Example


56/58


a /2 = 0.025

z a / 2 = 1.96 (using StatCrunch)

We want to estimate the mean IQ score for the population ofstatistics students. How many statistics students must be

randomly selected for IQ tests if we want 95% confidence thatthe sample mean is within 3 IQ points of the population mean?

Example

n = 1.96 15 = 96.04 = 973

2

With a simple random sample of only 97statistics students , we will be 95%confident that the sample mean is within3 IQ points of the true population mean .

What we know: a = 0.05 E = 3 = 15

Summary


57/58


SummaryConfidence Interval of a Mean

( known)

( x E , x + E )


x = sample mean

n = number sample values1 = Confidence Level

Summary


58/58

(z a / 2)

n = E 2

E = desired margin of error


x = sample mean


SummarySample Size for Estimating a Mean

( known)

probability confidence level

Documents