AP Statistics Curriculum 2007 Estim Proportion

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(Estimating a Population Proportion)
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=== Estimating a Population Proportion===
=== Estimating a Population Proportion===
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When the sample size is large, the sampling distribution of the sample proportion <math>\hat{p}</math> is approximately Normal, by [[AP_Statistics_Curriculum_2007_Limits_CLT |CLT]], as the sample proportion may be presented as a [[AP_Statistics_Curriculum_2007_Limits_Norm2Bin |sample average or Bernoulli random variables]]. When the sample size is small, the normal approximation may be inadequate. To accommodate this we will modify <math>\hat{p}</math> slightly
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When the sample size is large, the sampling distribution of the sample proportion <math>\hat{p}</math> is approximately Normal, by [[AP_Statistics_Curriculum_2007_Limits_CLT |CLT]], as the sample proportion may be presented as a [[AP_Statistics_Curriculum_2007_Limits_Norm2Bin |sample average or Bernoulli random variables]]. When the sample size is small, the normal approximation may be inadequate. To accommodate this we will modify the '''sample-proportion''' <math>\hat{p}</math> slightly and obtain the '''corrected-sample-proportion''' <math>\tilde{p}</math>:
: <math>\hat{p}={y\over n} \longrightarrow \tilde{y}={y+0.5z_{\alpha \over 2}^2 \over n+z_{\alpha \over 2}^2},</math>
: <math>\hat{p}={y\over n} \longrightarrow \tilde{y}={y+0.5z_{\alpha \over 2}^2 \over n+z_{\alpha \over 2}^2},</math>
where [[AP_Statistics_Curriculum_2007_Normal_Critical | <math>z_{\alpha \over 2}</math> is the normal critical value we saw earlier]].
where [[AP_Statistics_Curriculum_2007_Normal_Critical | <math>z_{\alpha \over 2}</math> is the normal critical value we saw earlier]].
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The standard error of <math>\hat{p}</math> also needs a slight modification
The standard error of <math>\hat{p}</math> also needs a slight modification
: <math>SE_{\hat{p}} =  \sqrt{\hat{p}(1-\hat{p})\over n} \longrightarrow SE_{\tilde{p}} =  \sqrt{\tilde{p}(1-\tilde{p})\over n+z_{\alpha \over 2}^2}.</math>
: <math>SE_{\hat{p}} =  \sqrt{\hat{p}(1-\hat{p})\over n} \longrightarrow SE_{\tilde{p}} =  \sqrt{\tilde{p}(1-\tilde{p})\over n+z_{\alpha \over 2}^2}.</math>
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===Confidence intervals for proportions===
===Confidence intervals for proportions===

Revision as of 05:10, 4 February 2008

Contents

General Advance-Placement (AP) Statistics Curriculum - Estimating a Population Proportion

Estimating a Population Proportion

When the sample size is large, the sampling distribution of the sample proportion \hat{p} is approximately Normal, by CLT, as the sample proportion may be presented as a sample average or Bernoulli random variables. When the sample size is small, the normal approximation may be inadequate. To accommodate this we will modify the sample-proportion \hat{p} slightly and obtain the corrected-sample-proportion \tilde{p}:

\hat{p}={y\over n} \longrightarrow \tilde{y}={y+0.5z_{\alpha \over 2}^2 \over n+z_{\alpha \over 2}^2},

where z_{\alpha \over 2} is the normal critical value we saw earlier.

The standard error of \hat{p} also needs a slight modification

SE_{\hat{p}} =  \sqrt{\hat{p}(1-\hat{p})\over n} \longrightarrow SE_{\tilde{p}} =  \sqrt{\tilde{p}(1-\tilde{p})\over n+z_{\alpha \over 2}^2}.

Confidence intervals for proportions

The confidence intervals for the sample proportion \hat{p} and the corrected-sample-proportion \tilde{p} are given by

\hat{p}\pm z_{\alpha\over 2} SE_{\hat{p}}
\tilde{p}\pm z_{\alpha\over 2} SE_{\tilde{p}}

Model Validation

Checking/affirming underlying assumptions.

  • TBD

Computational Resources: Internet-based SOCR Tools

  • TBD

Examples

Computer simulations and real observed data.

  • TBD

Hands-on activities

Step-by-step practice problems.

  • TBD

References

  • TBD



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