I just want to share a problem that was shown as an example for confidence intervals that I am bit confused about. It is;

In a poll of 100 randomly selected voters, 35 of them expressed support for candidate A. We wish to estimate the proportion of voters that support candidate A from the entire population.

In the example, they wanted to use this:

$\widehat{p} \pm z_{\frac{\alpha}{2}}\cdot \left ( \sqrt{\frac{\hat{p} (1-\widehat{p})}{n}} \right)$

We are given:

confidence Interval = 95% (1.96)

$n = 100$

$\widehat{p} = \frac{35}{100}$

However, with the numbers plugged into its place, it was written like this:

${\frac{35}{100}} \pm 1.96\cdot \left ( \sqrt{\frac{{\frac{35}{100}} \cdot\frac{65}{100}}{100}} \right)$

Why was the $1-\widehat{p}$ replaced by $({\frac{35}{100}} \cdot\frac{65}{100})$ ?

  • $\begingroup$ 65/100 came from the number of voters who did not express support for candidate A. $\endgroup$ – LinePlay Jun 2 '17 at 19:05
  • 1
    $\begingroup$ The $1-p$ should have been $1-\hat{p}$ all along. $\endgroup$ – Ian Jun 2 '17 at 19:05
  • $\begingroup$ With the $1-p$ replaced by $1-\hat{p}$ it is just that $\hat{p}=0.35$ so $1-\hat{p}=0.65$. $\endgroup$ – Ian Jun 2 '17 at 19:38

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