6.1-6.3 Inference About a Single Proportion

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# 6.1-6.3 Inference About a Single Proportion
### E. Nordmoe
### Math 260

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# Outline

* Standard Error

* Sampling Distribution

* Confidence Interval

* Sample Size for Desired Margin of Error

* Hypothesis Test

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## Standard Error for Sample Proportions

* For random samples of size `$n$` from a population with proportion `$p$`, the **standard error of the sample proportion** `$\hat p$` is

$$
SE=\sqrt{\frac{p(1-p)}{n}}
$$

* The larger the sample size `$n$`, the smaller the SE.

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# The Central Limit Theorem for Proportions

* If `$n$` is sufficently large,
$$
\hat p \sim N\left(p,\sqrt{\frac{p(1-p)}{n}}\right)
$$

* A normal distribution is a good approximation as long as

+ `$np\ge 10$` **and**

+ `$n(1-p)\ge 10$`

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# Confidence Interval for a Single Proportion

A confidence interval for a popoulation proportion `$p$` can be computed using  
$$
\hat p \pm z^* \sqrt{\frac{\hat p(1-\hat p)}{n}}
$$

if `$n$` is large enough so that

* `$n\hat p\ge 10$` and

* `$n (1-\hat p)\ge 10$`

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# Sample Size for a Desired Margin of Error

To estimate a population proportion to within a desired margin of error, `$\text{ME}$` with a given level of confidence, choose `$n$` such that  
$$
n=\left(\frac{z^*}{\text{ME}}\right)^2 \tilde p (1-\tilde p)
$$

where `$\tilde p$` is either

* A guessed value for `$p$` (if one is available) OR  
--
* `$\tilde p=0.5$` if no other guess is available

**Note:** Setting `$\tilde p=0.5$` will ensure that `$n$` is large enough no matter what `$p$` turns out to be.

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# Hypothesis Testing for a Single Proportion

To test `$H_0:p=p_0$`, we use the standardized test statistic  
$$
z=\frac{\hat p-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}
$$
where `$\hat p$` is the proportion in a random sample of size `$n$`.

* Check sample size conditions:  
  + `$np_0\ge 10$`  
  + `$n(1-p_0)\ge 10$`  
  
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## Computing the P-value

* The appropriate `$p$`-value depends on the alternative hypothesis: