Question

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?

Answer 1

`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 and sigma = 10.4`

`z=(x- mu)/sigma`

`z=(78-68.2)/10.4=0.9423`

`P(z >=0.9423) = 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 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,...,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(X>=20) = 1=P(X<20)`.
`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)`.