Question

5. Determine the z-score value in each of the following scenarios:
a. What z-score value separates the top 8% of a normal distribution from the bottom
92%?
b. What z-score value separates the top 72% of a normal distribution from the bottom
28%?
c. What z-score value form the boundaries for the middle 58% of a normal
distribution?
d. What z-score value separates the middle 45% from the rest of the distribution?
2. Annual salaries for a large company are approximately normally distributed with a mean of
R50,000 and a standard deviation of R20,000.
a. What salary would an employee need to get in order to be in the lowest 30%?
b. What is the probability of having an above average salary range of between R60000 to
R80000.

Answer 1

a.

$$P(Z<z^*)=0.92$$ $$z^*=1.4051$$

b.

$$P(Z<z^*)=0.28$$

$$z^*=-0.5828$$

c.

$$P(z^*_1<Z<z^*_2)=0.58$$

$$0.5-0.58/2=0.21$$

$$P(Z<z^*_1)=0.21$$

$$z^*_1=-0.8064$$

$$z^*=\pm 0.8064$$

So the interval is $(-0.8064, 0.8064)$  

d.

$$0.5-0.45/2=0.275$$

$$P(Z<z^*_1)=0.275$$

$$z^*_1=-0.5978$$

$$z^*=\pm 0.5978$$

So the interval is $(-0.5978, 0.5978).$

2.

$$\mu=R50,000$$ $$\sigma=R20,000$$

a. What salary would an employee need to get in order to be in the lowest 30%? 

$$Z={X-\mu \over \sigma}={X-\mu \over \sigma}$$

$$P(Z<z^*)=0.3=>z^*=-0.5244$$

$$x=z^*\sigma+\mu$$

$$x=-0.5244(R20000)+R50000=R39512$$

b. What is the probability of having an above average salary range of between R60000 to

R80000.

$$z_1={R60000-R50000 \over R20000}=0.5$$

$$z_2={R80000-R50000 \over R20000}=1.5$$

$$P(R60000<X<R80000)=P(0.5<Z<1.5)=$$

$$=P(Z<1.5)-P(Z<0.5)=$$

$$=0.93319280-0.69146246=0.24173034$$