Question

what relationship exists between the standard normal distribution and the box - plot methodology for describing distributions of data by means of quartiles? the answer depends on the true underlying probability distribution of the data. assume for the remainder of this exercise that the distribution is normal. complete parts a through e below.
a. calculate the values $z_l$ and $z_u$, the lower and upper values of the standard normal random variable $z$ respectively, for the lower and upper quartiles $q_l$ and $q_u$ of the probability distribution.
$z_l = square$
$z_u = square$
(round to two decimal places as needed.)

Explanation:

Step1: Recall z - score for quartiles

For a standard - normal distribution, the lower quartile $Q_{l}$ corresponds to a cumulative probability of $P(Z\leq z_{l}) = 0.25$ and the upper quartile $Q_{u}$ corresponds to a cumulative probability of $P(Z\leq z_{u})=0.75$.

Step2: Use z - table or calculator

Using a standard normal distribution table (z - table) or a calculator with a normal - distribution function (e.g., in a TI - 84 Plus: invNorm(0.25,0,1) for $z_{l}$ and invNorm(0.75,0,1) for $z_{u}$).
For $P(Z\leq z_{l}) = 0.25$, $z_{l}\approx - 0.67$.
For $P(Z\leq z_{u}) = 0.75$, $z_{u}\approx0.67$.

Answer:

$z_{l}=-0.67$, $z_{u}=0.67$