# Scores on a test are normally distributed with a mean of 68.2 and a standard deviation of 10.4. What is the probability that among 75 randomly selected students, at least 20 of them score greater than 78?

Jan 27, 2017

P(X>=20)=1-(""^75C_0*0.1728^0*0.9423^75 + ^75C_1*0.1728^1*0.8272^74+...+^75C_19*0.1728^19*0.8272^56)

#### Explanation:

$x = 78 , \mu = 68.2 \mathmr{and} \sigma = 10.4$

$z = \frac{x - \mu}{\sigma}$

$z = \frac{78 - 68.2}{10.4} = 0.9423$

$P \left(z \ge 0.9423\right) = 0.1728$ from Normal Distribution Table

Let say, $p$ is a probability student score more than $78$ and $q$ less than $78$,
therefore,
$p = 0.1728 \mathmr{and} q = 1 - 0.1728 = 0.8272$

To find a probability that at least more than 20 of 75 students score greater than 78 marks,
P(X>=r)=""^n C_r*p^r*q^(n-r)
where $n = 75$ and $r = 20 , 21 , 22 , \ldots , 75$

P(X>=20)=""^n C_r*p^r*q^(n-r)
P(X>=20)=""^75C_20*0.1728^20*0.9423^55 + ^75C_21*0.1728^21*0.8272^54+...+^75C_75*0.1728^75*0.8272^0

We also can calculate as $P \left(X \ge 20\right) = 1 = P \left(X < 20\right)$.
P(X>=20)=1-(""^75C_0*0.1728^0*0.9423^75 + ^75C_1*0.1728^1*0.8272^74+...+^75C_19*0.1728^19*0.8272^56).