# AP Statistics Curriculum 2007 Estim Proportion

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=== Estimating a Population Proportion=== | === Estimating a Population Proportion=== | ||

- | 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 | + | 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 05:12, 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 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** slightly and obtain the **corrected-sample-proportion** :

where is the normal critical value we saw earlier.

The standard error of also needs a slight modification

### Confidence intervals for proportions

The confidence intervals for the sample proportion and the corrected-sample-proportion are given by

### 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

- SOCR Home page: http://www.socr.ucla.edu

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