Thread: What methods were used to calculate these results

Hi everyone. I'm trying to deconstruct the following example argument. Unfortunately without a background in math I'm struggling to infer what methods of calculation were used to identify the highlighted results:

62 percent of voters in a sample of 400 registered voters said that they favor John Kerry over George W. Bush for President in the 2004 election. This supports with a probability of at least .95 the hypothesis that between 57 percent and 67 percent of all registered voters favor Kerry over Bush for President.

Is Bayes' theorem being used to calculate the .95? Is there an undisclosed value of "all registered voters" that's being taken into account?
Thank you

Originally Posted by manikinesque

Hi everyone. I'm trying to deconstruct the following example argument. Unfortunately without a background in math I'm struggling to infer what methods of calculation were used to identify the highlighted results:

62 percent of voters in a sample of 400 registered voters said that they favor John Kerry over George W. Bush for President in the 2004 election. This supports with a probability of at least .95 the hypothesis that between 57 percent and 67 percent of all registered voters favor Kerry over Bush for President.

Is Bayes' theorem being used to calculate the .95? Is there an undisclosed value of "all registered voters" that's being taken into account?
Thank you

They are treating the percentage of voters that favored Kerry over Bush as a Normal random variable with

$\mu = 0.62$

$\sigma = \sqrt{\dfrac{(0.62)(1-0.62)}{400}}\approx 0.02427$

$\Phi\left(\dfrac{0.67 - \mu}{\sigma}\right) - \Phi\left(\dfrac{0.57 - \mu}{\sigma}\right) \approx 0.96 = 96\%$