The remaining of the question goes like this:

Consider the formula:

Why is it not appropriate for Amy to use this formula for the standard deviation of Green – Pink?

The formula you did not mention is

\sigma _{\widehat{p}_{1} - \widehat{p}_{2}} = \sqrt{\frac{p_{1}(1-p_{1})}{n_{1}}+\frac{p_{2}(1-p_{2})}{n_{2}}}

where n_{1} and [math]n_{2}[math] are sizes of each sample.

This standard deviation formula works as long as we have:

- Independent observations
*between*the two samples. - Independent observations
*within*each sample*

Here p_{1} and p_{2} have to come from independent samples, but is not the case here because they are coming from the same sample.

The correct procedure would be to

- Randomly select a sample calculate the proportion of pink balls.
- Randomly select another sample, calculate the proportion of green balls

and then calculate the difference. So Amy cannot use this formula for the standard deviation of Green- Pink because both they are not independent and come from the same sample.

Here is a detailed explanation of the same concept: