Question

How do you find the $z$- score for which $80\%$ of the distribution’s area lies between $-z$ and $z$?

Answer 1

In this question we have to find the $z-$ score for $80\%$ of the distribution therefore we will first calculate the value of $\alpha$which will be the unwanted area and since we are working on the normal distribution which is symmetric on both sides, we will only take the value of $\alpha$on one side of the graph and then use the $z$- score table to get the $z$- score for the given value of $\alpha$.

Complete step by step solution:
We have to find the $z$- score for which $80\%$ of the distribution’s area lies between $-z$ and $z$.
Therefore, first we have to find the alpha of the region we don’t want. Since we have to find the $z$- score for $80\%$ of the area, the unused area can be calculated as:
$\Rightarrow \alpha =100\%-80\%$
Now on expanding the percentages and writing it as a fraction, we get:
$\Rightarrow \alpha =\dfrac{100}{100}-\dfrac{80}{100}$
Now on taking the lowest common multiple of both the fractions, we get:
$\Rightarrow \alpha =\dfrac{100-80}{100}$
Which can be simplified as:
$\Rightarrow \alpha =\dfrac{20}{100}$
On simplifying the fraction, we get:
$\Rightarrow \alpha =0.2$
Now this value of $\alpha$represents both sides of the standard normal distribution. And since the distribution is symmetric, we can divide the value of $\alpha$ by $2$.
On dividing $\alpha$ by $2$, we get:
$\Rightarrow \dfrac{\alpha }{2}=\dfrac{0.2}{2}$
On simplifying, we get:
$\Rightarrow \dfrac{\alpha }{2}=0.1$
Now we will use the $z$- score table to get the $z$- score for the given value of $\alpha /2$.
Now from the $z$- score table we can see that for the value $0.1$, the value of $z$ is $-1.28$, which is the negative $z$ value therefore the positive $z$ value will be $1.28$ which is the required range.

Therefore, the $z$- score for $80\%$ of the distribution’s area lies between $-1.28$ and $1.28$.

Note: It is to be remembered that the $z$ score table is used in the normal distribution to find the probability. The probability of any event can never be negative of greater than one.
There also exist other types of probability distributions such as the binomial distribution, Bernoulli distribution etc.