Consider a normal distribution curve where the middle 5 % of the area under the curve lies above the interval ( 6 , 12 ). Find the mean and standard deviation.
Consider a normal distribution curve where 85-th percentile is at 12 and the 25-th percentile is at 6. Find the mean and standard deviation.
Here is the work I have done so far:
Can someone please explain what I am doing wrong in looking for the standard deviations? Any help would be very much appreciated!
Best Answer
The middle $5\%$ is between $z=\pm0.06270678$. I.e. $\Phi(-0.06270678)=0.475$ and $1-\Phi(0.6270678)=0.475$ so $2(0.475)=0.95$ is outside those bounds, and $0.05$ is inside.
So: \begin{align} \mu+0.06270678\sigma & = 12, \\ \mu-0.06270678\sigma & = 6. \end{align} Subtracting the second of these from the first gives $2(0.06270678\sigma)$ on the left and $12-6$ on the right. Having found $\sigma$, you can plug that into either of the two equations above and find $\mu$.