Originally Posted by
iamnobody917 Suppose X1+X2+...+X80 follows a binomial distribution with n = 80, p = 0.74, q = 0.26, mean = 59.2, variance = 15.392
then we may say that X1 + X2 + ... + X80 follows a normal distribution(approximately) with mean = 59.2, variance = 15.392
Yi is another random variable such that Yi = 7Xi - 1 for i=1, 2, ..., 80
Y1 + Y2 + ... + Y80 = 7(X1 + X2 + ... + X80) - 80
so Mean(Y1 + Y2 + ... Y80) = 7Mean(X1+X2+...+X80) - 80 = 7*59.2 - 80 = 334.4
Var(Y1 + Y2 + ... + Y80) = 49Var(X1+X2+...+X80) = 49 * 15.392 = 754.208
so we may also say that Y1+Y2+...+Y80 follows a normal distribution with mean = 334.4, variance = 754.208
Is this correct up to this point?
Now suppose we want to calculate:
probability(Y1+Y2+...+Y80 >= 340)
=P( Z >= (339.5- 334.4)/sqrt(754.208) )
=P( Z >= 0.1857 )
On the other hand
probability(Y1+Y2+...+Y80 >= 340)
=P( 7(X1+X2+...+X80)-80 >= 340 )
=P( X1+X2+...+X80 >= 60 )
=P( Z >= (59.5-59.2)/sqrt(15.392) )
=P( Z >= 0.0765 )
which is completely different.
It seems the methods do not give consistent answers.
Now when we want to calculate
probability(Y1+Y2+...+Y80 >= 300)
=P( Z >= (299.5 - 334.4)/sqrt(754.208) )
=P( Z >= -1.2708 )
on the other hand
P(Y1+Y2+...+Y80 >= 300)
=P( 7(X1+X2+...+X80)-80 >= 300 )
=P( X1+X2+...+X80 >= 55 )
=P( Z >= (54.5-59.2)/15.392 )
=P( Z >= -1.1980 )
which is quite close to the answer from method 1
So which method is the correct one?
I also don't understand the reasons for such a large discrepancy in the first calculation.
Thanks in advance.