# AP Statistics Curriculum 2007 Estim Proportion

(Difference between revisions)
 Revision as of 18:58, 14 June 2007 (view source)IvoDinov (Talk | contribs)← Older edit Revision as of 05:11, 4 February 2008 (view source)IvoDinov (Talk | contribs) Newer edit → Line 2: Line 2: === Estimating a Population Proportion=== === Estimating a Population Proportion=== - Example on how to attach images to Wiki documents in included below (this needs to be replaced by an appropriate figure for this section)! -
[[Image:AP_Statistics_Curriculum_2007_IntroVar_Dinov_061407_Fig1.png|500px]]
- ===Approach=== + 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 - Models & strategies for solving the problem, data understanding & inference. + : $\hat{p}={y\over n} \longrightarrow \tilde{y}={y+0.5z_{\alpha \over 2}^2 \over n+z_{\alpha \over 2}^2},$ + where [[AP_Statistics_Curriculum_2007_Normal_Critical | $z_{\alpha \over 2}$ is the normal critical value we saw earlier]]. - * TBD + 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=== ===Model Validation===

## 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 $\hat{p}$ slightly

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

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.

• TBD

• TBD

### Examples

Computer simulations and real observed data.

• TBD

### Hands-on activities

Step-by-step practice problems.

• TBD

• TBD