3) Suppose that a 100-point test (scores are whole numbers) is administered to every high school student in the

USA at the start of their senior year and that the scores on this test are normally distributed with a mean of 70

and standard deviation of 10. If 5 scores are selected at random, what is the probability that exactly 3 of these

scores are between 65 and 75, inclusive?

4) Let X be a normal random variable with mean μ and standard deviation σ. Show that the expected value and

variance of the quantity x-μ/σ (this is x-μ over σ) are 0 and 1, respectively.

2. Originally Posted by MiyuCat
3) Suppose that a 100-point test (scores are whole numbers) is administered to every high school student in the

USA at the start of their senior year and that the scores on this test are normally distributed with a mean of 70

and standard deviation of 10. If 5 scores are selected at random, what is the probability that exactly 3 of these

scores are between 65 and 75, inclusive?

You need to find the probability $a$ such that $P(65\leq X\leq 75) = P\left(\frac{65-100}{10}\leq Z\leq \frac{75-100}{10}\right)= \dots = a$

Then find $P(Y=3)$ where $Y$ is binomial with $p= a, n= 5$

3. Originally Posted by MiyuCat
[snip]
4) Let X be a normal random variable with mean μ and standard deviation σ. Show that the expected value and

variance of the quantity x-μ/σ (this is x-μ over σ) are 0 and 1, respectively.
Set up the required integrals, then make the substitution z=x-μ/σ and use standard results.

Alternatively, use the following well known theorems:

E(aX + b) = aE(X) + b and Var(aX + b) = a^2Var(X).