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The Scholastic Aptitude Test mathematics test scores across the population of high school seniors follow a normal distribution with mean 500 and standard deviation 100. If five seniors are randomly chosen, find the probability that all of them scored below 600.

My answer so far is currently

Let $X1, X2, X3, X4, X5$ correspond to the five seniors. Assuming independence, $E(X1+X2+X3+X4+X5) = 500(5) = 2,500.$

$Variance = Var(X1+X2+X3+X4+X5) = 5(100)^2 = 50,000.$

Find $P(X1+X2+X3+X4+X5 < 600)$

$P(600-\cfrac{2500}{\sqrt{50,000}})$

$= \cfrac{-1,900}{223.607}$

$= \phi(-8.497)$

$= 1 - \phi(8.497).$

I know I must have done something wrong since you're unable to find $\phi(8.497)$. Thank You

kayush
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agelakute
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1 Answers1

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You found the probability that the total score of these $5$ students is less than $600$. You are not asked anything about the total score, so there is no need to find the distribution of this (by calculating the expectations and variances as you did).

To find the probability they ask for, you need to first find the probability of a single student scoring less than $600$. This is given by $$P(X\le600)=\Phi\left(\frac{600-500}{100}\right)=\Phi(1)$$ Then the probability that all $5$ students satisfy this is $[P(X\le600)]^5$.

Note: I denoted $X\sim N(500,100^2)$ to represent one of the $X_i$'s you had.

John Doe
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