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?
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#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)#
#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)#.
