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?

1 Answer
Jan 27, 2017

Answer:

#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)#.