# AP Statistics Curriculum 2007 Estim Proportion

(Difference between revisions)
 Revision as of 05:11, 4 February 2008 (view source)IvoDinov (Talk | contribs)← Older edit Revision as of 05:12, 4 February 2008 (view source)IvoDinov (Talk | contribs) (→Estimating a Population Proportion)Newer edit → Line 3: Line 3: === 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 [[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 $\hat{p}$ slightly + When the sample size is large, the sampling distribution of the sample proportion $\hat{p}$ 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''' $\hat{p}$ slightly and obtain the '''corrected-sample-proportion''' $\tilde{p}$: : $\hat{p}={y\over n} \longrightarrow \tilde{y}={y+0.5z_{\alpha \over 2}^2 \over n+z_{\alpha \over 2}^2},$ : $\hat{p}={y\over n} \longrightarrow \tilde{y}={y+0.5z_{\alpha \over 2}^2 \over n+z_{\alpha \over 2}^2},$ where [[AP_Statistics_Curriculum_2007_Normal_Critical | $z_{\alpha \over 2}$ is the normal critical value we saw earlier]]. where [[AP_Statistics_Curriculum_2007_Normal_Critical | $z_{\alpha \over 2}$ is the normal critical value we saw earlier]]. Line 9: Line 9: The standard error of $\hat{p}$ also needs a slight modification The standard error of $\hat{p}$ also needs a slight modification : $SE_{\hat{p}} = \sqrt{\hat{p}(1-\hat{p})\over n} \longrightarrow SE_{\tilde{p}} = \sqrt{\tilde{p}(1-\tilde{p})\over n+z_{\alpha \over 2}^2}.$ : $SE_{\hat{p}} = \sqrt{\hat{p}(1-\hat{p})\over n} \longrightarrow SE_{\tilde{p}} = \sqrt{\tilde{p}(1-\tilde{p})\over n+z_{\alpha \over 2}^2}.$ - ===Confidence intervals for proportions=== ===Confidence intervals for proportions===

## 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 $\tilde{p}$:

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

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.

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### Examples

Computer simulations and real observed data.

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### Hands-on activities

Step-by-step practice problems.

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