Question

1. draw a vertical line through a normal distribution for each of the following z - score locations. find the pro - portion of the distribution located between the mean and the z - score.

a. z = +1.60
b. z = +0.90
c. z = -1.50
d. z = -0.40

Explanation:

Step1: Recall the z - table property

The z - table gives the cumulative proportion from the left - hand side of the standard normal distribution (mean = 0, standard deviation = 1). The proportion between the mean ($z = 0$) and a positive $z$ - score $z$ is $P(0

Step2: Calculate for $z = + 1.60$

Using the z - table, $\Phi(1.60)=0.9452$. Then $P(0 < Z<1.60)=0.9452 - 0.5=0.4452$.

Step3: Calculate for $z = + 0.90$

From the z - table, $\Phi(0.90)=0.8159$. So $P(0 < Z<0.90)=0.8159 - 0.5=0.3159$.

Step4: Calculate for $z=-1.50$

From the z - table, $\Phi(-1.50)=0.0668$. Then $P(-1.50 < Z<0)=0.5 - 0.0668 = 0.4332$.

Step5: Calculate for $z=-0.40$

From the z - table, $\Phi(-0.40)=0.3446$. So $P(-0.40 < Z<0)=0.5 - 0.3446=0.1554$.

Answer:

a. $0.4452$
b. $0.3159$
c. $0.4332$
d. $0.1554$