Question

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?

Answer 1

$$a.\ \ \ P(Z\leq z^*)=0.92, P(Z>z^*)=0.08$$ $$z^*=1.4051$$

$$b.\ \ \ P(Z\leq z^*)=0.28, P(Z>z^*)=0.72$$ $$z^*=-0.5828$$

$$c.\ \ \ P(Z\leq z_1^*)=0.5-0.29=0.21, P(Z<z_2^*)=0.5+0.29=0.79$$

$$z^*=\plusmn0.8064$$

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

$$d. \ \ \ P(Z\leq z_1^*)=0.5-0.225=0.275, P(Z<z_2^*)=0.5+0.225=0.725$$

$$z^*=\plusmn0.5978$$

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