Chapter 7

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Chapter 7Estimates and Sample Sizes

7-1 Review and Preview

7-2 Estimating a Population Proportion

7-3 Estimating a Population Mean: Known

7-4 Estimating a Population Mean: Not Known

7-5 Estimating a Population Variance

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Section 7-2

Estimating a PopulationProportion

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Definition

A point estimate is a single value (orpoint) used to approximate a populationparameter.

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The sample proportion is the bestpoint estimate of the populationproportion .

Definition

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Definition

A confidence interval (or intervalestimate) is a range (or an interval)of values used to estimate the truevalue of a population parameter.A confidence interval is sometimesabbreviated as CI.

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A confidence level is the probability (oftenexpressed as the equivalent percentage value)that the confidence interval actually does containthe population parameter, assuming that theestimation process is repeated a large number oftimes. (The confidence level is also called degreeof confidence, or the confidence coefficient.)

Most common choices are 90%, 95%, or 99%.

Definition

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We must be careful to interpret confidence intervalscorrectly. There is a correct interpretation and manydifferent and creative incorrect interpretations of theconfidence interval .

“We are 95% confident that the interval from 0.677 to0.723 actually does contain the true value of thepopulation proportion .”

This means that if we were to select many differentsamples of size 1501 and construct the correspondingconfidence intervals, 95% of them would actuallycontain the value of the population proportion .

Interpreting a Confidence Interval

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Definition

A critical value is the number on theborderline separating sample statisticsthat are likely to occur from those that areunlikely to occur. The number is acritical value that is a z score with theproperty that it separates an area of inthe right tail of the standard normaldistribution.

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The Critical Value

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Finding for a 95%Confidence Level

Critical Values

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Examples

a)Compute the critical value thatcorresponds to a 90% level of confidence.

b)Compute the critical value thatcorresponds to an 82% level of confidence.

c)Find the critical value that corresponds to = 0.02

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Margin of Error forProportions

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= population proportion

Confidence Interval for Estimatinga Population Proportion

= sample proportion

= number of sample values

= margin of error

= z score separating an area of in

the right tail of the standard normal

distribution

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Confidence Interval for Estimatinga Population Proportion

1. The sample is a simple random sample.

2. The conditions for the binomial distributionare satisfied: there is a fixed number oftrials, the trials are independent, there aretwo categories of outcomes, and theprobabilities remain constant for each trial.

3. There are at least 5 successes and 5failures.

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Confidence Interval for Estimatinga Population Proportion

where

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Confidence Interval for Estimatinga Population Proportion

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Round-Off Rule forConfidence Interval Estimates of

Round the confidence interval limitsfor to

three significant digits.

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Use the TI-83/84 calculator.

Stat > Tests > 1-PropZInt

Enter the values of x, n, C-level

Procedure for Constructinga Confidence Interval for

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Example:

In a survey of 1002 people, 701 said that theyvoted in a recent election.

a) Find a 99% confidence interval estimate for theproportion of people who say that they voted.

b) Voting records show that 61% of the eligiblevoters actually did vote. Are the survey resultsconsistent with the actual voter turnout?

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In a study of 1228 randomly selected medicalmalpractice lawsuits, it is found that 856 ofthem were later dropped or dismissed.

a)What is the best point estimate of theproportion of medical malpractice lawsuitsthat are dropped or dismissed?

b)Construct a 99% confidence intervalestimate of the proportion of medicalmalpractice lawsuits that are dropped ordismissed?

c)Does it appear that the majority of such suitsare dropped or dismissed?

Example:

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Analyzing Polls

When analyzing polls consider:

1.The sample should be a simple random sample,not an inappropriate sample (such as a voluntaryresponse sample).

2.The confidence level should be provided. (It isoften 95%, but media reports often neglect toidentify it.)

3.The sample size should be provided. (It is usuallyprovided by the media, but not always.)

4.Except for relatively rare cases, the quality of thepoll results depends on the sampling method andthe size of the sample, but the size of thepopulation is usually not a factor.

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Caution

Never follow the common misconceptionthat poll results are unreliable if thesample size is a small percentage of thepopulation size. The population size isusually not a factor in determining thereliability of a poll.

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Sample Size

Suppose we want to collect sampledata in order to estimate somepopulation proportion. The question ishow many sample items must beobtained?

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Determining Sample Size

(solve for n by algebra)

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Sample Size for EstimatingProportion

When an estimate of is known:

When no estimate of is known:

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Round-Off Rule for DeterminingSample Size

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

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Example:

The Internet is affecting us all in manydifferent ways, so there are many reasons forestimating the proportion of adults who useit. Assume that a manager for E-Bay wants todetermine the current percentage of U.S.adults who now use the Internet. How manyadults must be surveyed in order to be 95%confident that the sample percentage is inerror by no more than three percentagepoints?

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

b.No known possible value of the proportion.

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a)Use

To be 95% confidentthat our samplepercentage is withinthree percentagepoints of the truepercentage for alladults, we shouldobtain a simplerandom sample of 842adults.

Example:

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b)Use

To be 95% confident thatour sample percentageis within threepercentage points of thetrue percentage for alladults, we should obtaina simple random sampleof 1068 adults.

Example: