Hopefully you guys can help me. I'm completely stumped by normal distribution and the like. No matter what I do, I can't get it right. Hopefully someone here can save me!
This is my question:
A curtain rail is manufactured to be approximately normally distributed with a mean $28$mm and a standard deviation of $2.7$mm. The Rails are required to be in lengths that range from $25.5$mm to $31$mm.
A) What proportion of curtain rails will be shorter than $27$mm?
B) What length is exceeded by $15\%$ of the rails?
C) What proportion of curtain rails will meet the specification?
D) What value of the standard deviation is required so that less than $8\%$ of the rods have a length smaller than $25.5$mm?
I have (possibly) figured out A.
So it would be that $P(Z<2.7) = 1-0.9965 = 0.0035$. So $0.35\%$ of rods will be the required length. Is that correct?
I can't even begin to figure out the others. Please help!
The question asked in A is what proportion of rails will be shorter than 27mm , not 2.7 mm., so the answer is actually 35.5%. Strictly speaking the question should should be "what is the probability that a rail will be shorter than 27 mm?". That this will be the actual proportion is a broad inference.
Question B likewise can be rephrased as the probability of a rail being in the 85th percentile or above and is the value of $x$ that satisfies the condition
$$ 1-\int_{-\infty}^x p(\mu,\sigma,\xi)d\xi=1-D(\mu,\sigma,x)=0.15 $$
where $p()$ and $D()$ are the probability density and cumulative distribution functions. The answer is $x=$ 30.8 mm, using EXCEL's NORMINV function.
The answer to C is $D(\mu,\sigma,x_{max})-D(\mu,\sigma,x_{min})$ which you can calculate and figure out D on your own.