Question

SAT scores are normally distributed with a mean of 2200 and a variance of 1600

a. what is the probability that a random sample of 64 scores will yield a mean of score between 2205 and 2210?

Answer 1

We have to calculate the probability of means of the elements. Hence, we first calculate the mean and S.D. of means.

New Mean = 2200

$New \ S.D. =\frac{\sigma}{\sqrt{n}}=\frac{1600}{\sqrt{64}}=200$

Now, using the new distribution,

$P(2205\leq x\leq 2210)=P(x\leq 2210)-P(x\leq 2205)$

$P(X \leq 2205)=P(z\leq \frac{X-\mu}{\sigma})=P(z\leq\frac{2205-2200}{200})$

$P(X \leq 2205)=P(z\leq0.025)=0.51=51\%$

$P(X \leq 2210)=P(z\leq \frac{X-\mu}{\sigma})=P(z\leq\frac{2210-2200}{200})$

$P(X \leq 2210)=P(z\leq0.05)=0.5199=51.99\%$

Hence,

$P(2205\leq x\leq 2210)=51.99\%-51\%=0.99\%$

I hope this solves your doubt.

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