Question

Determine the value of z* such that it satisfies the conditions below. (Round your answers to two decimal places.)

(a) −z* and z* separate the middle 94% of all z values from the most extreme 6%.
z* =  

(b) −z* and z* separate the middle 92% of all z values from the most extreme 8%.
z* =  

(c) −z* and z* separate the middle 97.8% of all z values from the most extreme 2.2%.
z* =  

(d) −z* and z* separate the middle 82% of all z values from the most extreme 18%.
z* =

You may need to use the appropriate table in Appendix A to answer this question.

Answer 1

(a)

Here we need z scores that have 0.94 area between them. That is area at both tails is

(1 - 0.94) /2 = 0.03

From z table, z-score -1.88 has approximately 0.03 area to its left. By symmetry z-score 1.88 has 0.03 area to its right. So area between -1.88 and 1.88 is 0.96.

Answer: z* = 1.88

(b)

Here we need z scores that have 0.92 area between them. That is area at both tails is

(1 - 0.92) /2 = 0.04

From z table, z-score -1.75 has approximately 0.04 area to its left. By symmetry z-score 1.75 has 0.04 area to its right. So area between -1.75 and 1.75 is 0.92.

Answer: z* = 1.75

(c)

Here we need z scores that have 0.978 area between them. That is area at both tails is

(1 - 0.978) /2 = 0.011

From z table, z-score -1.23 has approximately 0.011 area to its left. By symmetry z-score 1.23 has 0.011 area to its right. So area between -1.23 and 1.23 is 0.978.

Answer: z* = 1.23

(d)

Here we need z scores that have 0.82 area between them. That is area at both tails is

(1 - 0.82) /2 = 0.09

From z table, z-score -1.34 has approximately 0.09 area to its left. By symmetry z-score 1.34 has 0.09 area to its right. So area between -1.34 and 1.34 is 0.82.

Answer: z* = 1.23