I am trying to understand this example for normal approximations:
In a particular faculty 60% of students are men and 40% are women. In a random sample of 50 students what is the probability that more than half are women?
Let the random variable $X$ be the number of women in the sample.
Assume $X$ has the binomial distribution with $n = 50$ and $p = 0.4$.
Then $E(X)=np=50\times 0.4=20$
$\operatorname{var}(X) = npq = 50 \times 0.4 \times 0.6 = 12$
so approximately $X \sim N(20,12)$.
We need to find $P(X > 25)$. Note - not $P(X >= 25)$.
so $$ P(X > 25) = P(Z > 1.44)\\ = 1 - P(Z < 1.44)\\ = 1 - 0.9251\\ = 0.075 $$
I don't understand the line near the bottom beginning at $1-P(Z > 1.44)$. Where do you get the value for $P(Z > 1.44)$?
You should be using a continuity correction and finding $$p(X>25.5)$$
So if you are using a table of $z$ values, you will be finding $$1-\Phi(1.59)$$
Or if you have a new calculator with the Normal CD function, you can get a more accurate answer $0.05617559886$