Question

Use Table A to find the value $z$from the standard Normal distribution that satisfies each of the following conditions. In each case, sketch a standard Normal curve with your value of $z$marked on the axis. Use your calculator or the Normal Curve applet to check your answers.

Working backward

(a) The $63$rd percentile.

(b) $75$% of all observations are greater than $z$.

Short answer

From the given information

a) The value of $z$corresponding to $63$rd percentile is $0.33$

b) The value of $z$such that $75$% of observations are above z is $-0.67$

Step-by-step solution

Part (a) Step 1: Given Information

The$63$rd percentile.

Part (a) Step 2: Explanation

It's assumed that we find the value of $z$that meets the following conditions.

a) We are expected to find a value that equals $0.63$.

The $63$percentile refers to the fact that $63\%$of occurrences fall below this value.

We can find $z$to be $0.63$by using the following standard normal table.

Part (a) Step 3: Explanation

The above table shows that the $z$value for $0.63$is $0.33$.

The following graph shows what the $63rd$percentile looks like along with the corresponding$z$value:

So, the value of$z$corresponding to localid="1649782327484" $63$rd percentile is $0.33$

Part (b) Step 1: Given Information

It is given in the question that, $75$% of all observations are greater than $z$.

Part (b) Step 2: Explanation

As a result of finding the value of $z$with a $75\%$probability that $75\%$of observed data is above $z$, we assume that $25\%$of observations are below $z$.

Part (b) Step 3: Explanation

Observing the table above, we can deduce that the value corresponds to the approximate value of$0.25$is $-0.67$.

Hence, the value of $z$such that $75$% observations are above z is $-0.67$