Question

Scores on a test are normally distributed with a mean of 68.2 and a standard deviation of 10.4. Estimate the probability that among 75 randomly selscted students, at least 20 of them score greater that 78.

Answer 1

First, find the probability of scoring higher than 78. Scores are normally distributed, so you have

$\mathbb P(X>78)=\mathbb P\left(\dfrac{X-68.2}{10.4}>\dfrac{78-68.2}{10.4}\right)\approx\mathbb P(Z>0.9423)\approx0.173$

Now, the event that any given student scores higher than 78 follows a binomial distribution. Here you have 75 total students (so $n=75$) with success probability $p=0.173$.

So the probability of getting 20 students that fit the criterion is

$\mathbb P(Y=20)=\dbinom{75}{20}p^{20}(1-p)^{75-20}\approx0.0134$