Sections 7.1 and 7.2 Sections 7.1 and 7.2

This chapter presents the beginning of inferential statistics.

The two major applications of inferential The two major applications of inferential statisticsstatistics

Estimate a population parameter: proportion, meanEstimate a population parameter: proportion, mean

Test some claim (or hypothesis) about a population.Test some claim (or hypothesis) about a population.

Point estimate: a single number Point estimate: a single number

Interval estimate: interval of numbers.Interval estimate: interval of numbers.

Confidence IntervalConfidence Interval

Why?: point estimate is not reliable under Why?: point estimate is not reliable under rere--sampling.sampling.

A A confidence interval (CI):confidence interval (CI): an interval of an interval of values used to estimate the true population values used to estimate the true population parameter.parameter.

Point EstimatePoint Estimate

p = population proportion

p =ˆ xn sample proportion

(pronounced ‘p-hat’)

of x successes in a sample of size n.Unbiased estimate (best estimate)

q = 1 - p = sample proportion of failures in a sample size of n

ˆ ˆ

Example: PhotoExample: Photo--Cop Survey ResponsesCop Survey Responses

829 adult Minnesotans were surveyed, and 51% of them are 829 adult Minnesotans were surveyed, and 51% of them are opposed to the use of the photoopposed to the use of the photo--cop for issuing traffic cop for issuing traffic tickets. Using these survey results, find the best estimate of tickets. Using these survey results, find the best estimate of the proportion of all adult Minnesotans opposed to photothe proportion of all adult Minnesotans opposed to photo--cop use.cop use.

Best point estimate=sample proportion=51%. Best point estimate=sample proportion=51%.

αα: between 0 and 1: between 0 and 1

A A confidence level: confidence level: 11 -- αα or 100(1or 100(1-- αα)%. E.g. 95%.)%. E.g. 95%.

This is the proportion of times that the confidence This is the proportion of times that the confidence interval actually does contain the population parameter, interval actually does contain the population parameter, assuming that the estimation process is repeated a assuming that the estimation process is repeated a large number of times.large number of times.

Other names: Other names: degree of confidencedegree of confidence or the or the confidence coefficientconfidence coefficient..

Confidence LevelConfidence Level

The Critical ValueThe Critical Value(z(z--score)score)

Given α

Finding Finding zzα/α/22 for for 100(1100(1-- αα))%% Confidence LevelConfidence Level

α/2 = 2.5% = .025

α =5%

The sampling distribution of sample proportion can be The sampling distribution of sample proportion can be approximated by a normal distribution if npapproximated by a normal distribution if np≥≥15 and15 and

nqnq ≥≥15 15 : : phatphat is approximately N(p, is approximately N(p, pq/npq/n), q=1), q=1--p.p.

p

Sampling Distribution of p̂

pp^

nqpppzˆˆ

ˆ −=

Margin of Error of pMargin of Error of p

E = α / 2 z p q̂ˆ

^

the maximum likely (with probability 1 the maximum likely (with probability 1 –– αα) ) difference between the observed proportion difference between the observed proportion pp and the true population proportion and the true population proportion pp..^

n

Standard Error of p=se

^

Finding the 95% Confidence Interval Finding the 95% Confidence Interval for a Population Proportionfor a Population Proportion

A 95% confidence interval for a population A 95% confidence interval for a population proportion p is:proportion p is:

100(1100(1--αα)% confidence interval for p is)% confidence interval for p is

n)p̂-(1p̂ se with 1.96(se), p̂ =±

nppsewithsezp )ˆ1(ˆ

)(ˆ 2/−

=± α

Example: Would You Pay Higher Example: Would You Pay Higher Prices to Protect the Environment?Prices to Protect the Environment?

In 2000, the GSS asked: In 2000, the GSS asked: ““Are you willing to Are you willing to pay much higher prices in order to protect pay much higher prices in order to protect the environment?the environment?””

Of n = 1154 respondents, 518 were willing to Of n = 1154 respondents, 518 were willing to do sodo so

Find and interpret a 95% confidence interval Find and interpret a 95% confidence interval for the population proportion of adult for the population proportion of adult Americans willing to do so at the time of the Americans willing to do so at the time of the surveysurvey

Example: Would You Pay Higher Example: Would You Pay Higher Prices to Protect the Environment?Prices to Protect the Environment?

0.48) (0.42, 0.03 0.45 Ep̂03.0)015.0(96.11.96(se)

015.01154

)55.0)(45.0(

45.01154518p̂

=±=±===

==

==

E

se

What is the Error Probability for the What is the Error Probability for the Confidence Interval Method?Confidence Interval Method?

Summary: Effects of Confidence Level Summary: Effects of Confidence Level and Sample Size on Margin of Errorand Sample Size on Margin of Error

The The margin of errormargin of error for a confidence interval:for a confidence interval:

Increases as the confidence level increasesIncreases as the confidence level increases

Decreases as the sample size increasesDecreases as the sample size increases

Determining Sample SizeDetermining Sample Size

α / 2zE = p qnˆ ˆ

(solve for n by algebra)

2 ˆp qα / 2zn = ˆE2

Recall :

Sample Size for Estimating Sample Size for Estimating Proportion Proportion pp

When an estimate p of p is known: ˆˆ( )2 p qn = ˆ

E2α / 2 z

When no estimate of p is known:

( )2 0.25n = E2α / 2 z

ExampleExample:: Suppose a sociologistSuppose a sociologist wants to determine the wants to determine the current percentage of U.S. households using ecurrent percentage of U.S. households using e--mail. How many mail. How many households must be surveyed in order to be households must be surveyed in order to be 95% confident95% confident that the that the sample percentage is in error by no more thansample percentage is in error by no more than four percentage four percentage pointspoints??

a) Use this result from an earlier study: In 1997, 16.9% of U.Sa) Use this result from an earlier study: In 1997, 16.9% of U.S. . households used ehouseholds used e--mail (based on data from mail (based on data from The World AlmanacThe World Almanacand Book of Factsand Book of Facts).).

b) Assume that we have no prior information suggesting a possiblb) Assume that we have no prior information suggesting a possibleevalue ofvalue of pp. .

a) Use this result from an earlier study: In 1997, 16.9% of U.Sa) Use this result from an earlier study: In 1997, 16.9% of U.S. . households used ehouseholds used e--mail (based on data from mail (based on data from The World AlmanacThe World Almanacand Book of Factsand Book of Facts).).

nn = [= [zza/2 a/2 ]]22 p qp q

EE22ˆˆ

0.042

= [1.96]2 (0.169)(0.831)

= 337.194= 338 households

To be 95% confident that our sample percentage is within four percentage points of the true percentage for all households, we should randomly select and survey 338 households.

b) Assume that we have no prior information suggesting a possiblb) Assume that we have no prior information suggesting a possibleevalue ofvalue of pp. .

0.042

nn = [= [zza/2 a/2 ]]22 •• 0.250.25

EE22

= (1.96)2 (0.25)

= 600.25 = 601 households

With no prior information, we need a larger sample to achieve the same results with 95% confidence and an error of no more than 4%.

Finding the Point Estimate Finding the Point Estimate and E from a and E from a

Confidence IntervalConfidence Interval

Margin of Error:E = (upper confidence limit) — (lower confidence limit)

2

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

2ˆ