Assume that 47% of Americans support Obama. We pick a random sample of size 100 from the population of all Americans. What is the probability, that the relative share of those who support Obama in the sample is less than 40 %?
Solution attempt:
Let $\widehat p$ denote the sample proportion of Americans that support Obama. The CLT tells us $${\widehat p_n} = {\widehat p_{100}}{ \approx _D}N\left( {0.47,\,\frac{{(0.47)(1 - 0.47)}}{{100}}} \right)$$
From the description of the problem we need to find: $$P({\widehat p_{100}} < 0.4) = P\left( {\frac{{{{\widehat p}_{100}} - 0.47}}{{\sqrt {\frac{{(0.47)(1 - 0.47)}}{{100}}} }} < \frac{{0.4 - 0.47}}{{\sqrt {\frac{{(0.47)(1 - 0.47)}}{{100}}} }}} \right) \approx P\left( {Z < - 1.40} \right)$$ $$P\left( {Z < - 1.40} \right) = 0.0808$$
Not sure if i have a right answer...