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Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics


Statistics for EconomistStatistics for Economist

1.1. IntroductionIntroduction

2.2. Confidence IntervalsConfidence Intervals

3.3. Interpreting a Confidence Interpreting a Confidence

IntervalsIntervals

Ch. 16 The Accuracy of PercentagesCh. 16 The Accuracy of Percentages

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STATISTISTATISTICSCS

INDEX

1 IntroductionIntroduction

2 Confidence IntervalsConfidence Intervals

3Interpreting a Confidence Interpreting a Confidence IntervalsIntervals

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1. Introduction

SE for the sample percentage

SE for the sample percentage

EX) Estimating the percentage of DemocratsPopulation: a district with 100,000 eligible voters

Sample: a simple random sample of 2,500 voters

In sample percentage: 53% (1328 out of 2500)

The difference between the estimated Democrats and the population percentage?

- Sample percentage is different from the population percentage due to the probability error.

- SE explains the difference of between the two.

- Sample percentage is different from the population percentage due to the probability error.

- SE explains the difference of between the two.

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1. Introduction

Calculation of SE

Calculation of SE

Is it likely to be off by as much as 3 percentage points? Population: A box including 100,000 ballots of 1 votes from each eligible voters. (Democrat=1, Otherwise=0) Sample: drawing 2,500 ballots at randomSD of box =SE of the # of eligible voters who is Democrats in sample=SE of sample percentage=25/2500=1% (3% points is 3 SEs)

50.047.053.0 25250050.0

When sampling from a 0-1 box whose composition is unknown, the SD of the box can be estimated by substituting the fraction of 0’s and 1’s in the sample for the unknown fractions in the Box. The estimate is good when the sample is reasonably large

When sampling from a 0-1 box whose composition is unknown, the SD of the box can be estimated by substituting the fraction of 0’s and 1’s in the sample for the unknown fractions in the Box. The estimate is good when the sample is reasonably large

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INDEX



3Interpreting a Confidence Interpreting a Confidence IntervalsIntervals

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2. 신뢰구간

confidence level & confidence interval

confidence level & confidence interval

What happens with a cutoff at 2 SEs?

51% 53% 55%

2SE 2SE

Sample percentage

The interval ‘sample percentage 1SE’ is 68%-confidence interval for the population percentage


The interval ‘sample percentage 3SE’ is 99.7%-confidence interval for the population percentage



The interval ‘sample percentage 3SE’ is 99.7%-confidence interval for the population percentage

Sample percentage = population percentage + probability error

in population Democrats percentage = 53%1%

(51%, 55%): confidence interval of confidence level 95%

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2. 신뢰구간

[Example 1] Estimating Democrats Percentage

[Example 1] Estimating Democrats Percentage

Population : a town with 25,000 electoral voters

Sample : drawing at random 1,600 electoral voters

Democrats percentage in sample: 57% (917 out of 1,600)

SD of box =

SE of in sample Democrats =

SE of in sample Democrats percentage = 20/1,600=1.25(%)

95% confidence interval of in population Democrats percentage

= 57% 21.25%

50.043.057.0 20160050.0

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2. 신뢰구간

Interpreting a confidence level

Interpreting a confidence level

confidence levels are often quoted as being “about” so

much. SEs have been estimated from the data.

Since it is based on central limit theorem, large samples justify it. A sample percentage near 0% or 100% suggests that the box is

lopsided and a very large number of draws will be needed before the normal approximation takes over.

If the sample percentage is near 50%, the normal approximation Should be satisfactory when there are only a hundred draws of so

The normal approximation has been used.

If the sample size gets larger, the more accurate the confidence level becomes

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INDEX



3 Interpreting a Confidence Interpreting a Confidence IntervalsIntervals

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3. Interpreting a Confidence Intervals

Interpreting a Confidence Interpreting a Confidence IntervalsIntervals


%25.12%57 The probability of population percentage of Democrats to be between 54.5% and 59.5% is 95%?

Among all possible samples, 95% does include the

population percentage in the confidence interval of

‘sample percentage2SE’, but not 5%

Among all possible samples, 95% does include the

population percentage in the confidence interval of

‘sample percentage2SE’, but not 5%

Confidence intervals changes from sample to sample.

The center and length of the confidence intervals change.

[Example 1]the confidence interval for the percentage of Democrats is

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The 95%-confidence interval of red marbles in the box is shown for 100 different samples.

The interval covers the population percentage, marked by a vertical line.

The ratio of red marbles in the box = 80%

Each takes a simple random sample of 2,500 marbles

The 95%-confidence interval of red marbles in the box is shown for 100 different samples.

The interval covers the population percentage, marked by a vertical line.

The ratio of red marbles in the box = 80%

Each takes a simple random sample of 2,500 marblesThe confidence interval changes from sample to

sample. 94 of the samples covers the population percentage.

The confidence interval changes from sample to sample. 94 of the samples covers the population percentage.

3. Interpreting a Confidence Intervals



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