Two studies were completed in California. One study in northern California involved 1,000 patients; 74% of them experienced flulike symptoms during the month of December. The other study, in southern California, involved 500 patients; 34% of them experienced flulike symptoms during the same month. Which study has the smallest margin of error for a 98% confidence interval?

Question

Vocaloid · Answer

Moved to Statistics

sillybilly123 · Answer

According to the Central Limit Theorem, sample proportion, $\hat p$, is distributed normally as:

$~~~ \hat p \sim \mathbb N\left(\mu_{\hat p} = p, \sigma_{\hat p} = \sqrt{\dfrac{p(1-p)}{n}}  \right) $
 
... where $n$ is the related sample size

The MoE is:
 
 $~~~~~ z^{*} \cdot \sqrt{\dfrac{\hat p ( 1 - \hat p)}{n}}$
 
For $98 \%$ confidence level, $z^{*} = 2.33$ - from the standard normal distribution table 
 
For the first sample, MoE is:
 
  $ ~~~~~ 2.33 \cdot \sqrt{\dfrac{0.74 ( 1 - 0.74)}{1000}} = 3.2 \% ~ ~ [1dp]$
  
 (ie TBC, we conclude, with $98 \%$ confidence, that $74\% \pm 3.2 \%$ of that **population** will have experienced flu-like symptoms in Dec)
 
  For the second sample, MoE is:
 
  $~~~~~ 2.33 \cdot \sqrt{\dfrac{0.34 ( 1 - 0.34)}{500}} = 4.9 \% ~ ~ [1dp]$

---> The first study has the smallest MoE for **any** given confidence level.  Sample size is key .... more people you test, more likely you are to get to truth.