Question

How do you find the z-score for which $76$ of the distribution's area lies between –z and z?

Answer 1

Hint : To solve this question we need to find at what parts there is the standard deviation that we aren’t calculating and checking with. Now once you find those percentages we aren’t dealing with, once we find the point for which it doesn’t lie in the part we need, we get our answer needed. We also need to remember that standard deviation is symmetrical.

Complete step-by-step answer :
Now we know that for z score where $76$ lies between –z and z when we need to solve. Now we can explain $\alpha$ to be part of the distribution we are not looking for in this question. Now $76$ can be written as a different way to be able to solve this question. Therefore we write it as $\dfrac{76}{100}$ which can be written as $0.76$ . Now we need to find $\alpha$ which is
$\alpha =1-0.76$
$\alpha =0.24$
Now we know that standard normal deviation is usually symmetric so we can explain it that we must divide what we got from two so that the part which we don’t have to calculate for is one either sides of our deviation that we need
$\dfrac{\alpha }{2}=\dfrac{0.24}{2}=0.12$
Now we find a correlating z score from the table of standard normal probabilities for negative z scores for $0.12$ which gives us that it lies from –z to z from $\left[ -1.175,1.175 \right]$
Therefore we can say that the $76$ of the deviation lies from $\left[ -1.175,1.175 \right]$ on the axis and the $12$ lies on either side of the standard deviation

Note : The z-score gives you an idea of how far from the mean a data point is. But more technically it can be understood as a measure of how many standard deviations below or above the population mean a raw score is.