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

Use z-scores.

z = (x - mean)/sd

With your data:

z = (78 - 68.2)/10.4 = 0.94

.1736 is the probability using a z-table for a single student with a score greater than 78.

Now we can use a normal approximation to the binomial distribution.

mean = np = (75)(.1736) = 13
standard deviation = √np(1-p) = √(75)(.1736)(.8264) = 3.28

Again, use z-scores.
z = (20 - 13)/3.28 = 2.13

Use the z-table to find the probability. (Remember the problem says "at least 20" which means 20 or more.)

I hope this will help.