Question

find the z - scores for which 76% of the distributions area lies between -z and z. click to view page 1 of the table. click to view page 2 of the table. the z - scores are - 1.15, 1.15 (use a comma to separate answers as needed. round to two decimal places as needed.)

Explanation:

Step1: Determine the area in the tails

Since 76% of the area lies between -z and z, the area in the two - tails combined is \(1 - 0.76=0.24\). So the area in each tail is \(\frac{0.24}{2}=0.12\).

Step2: Find the z - score for the left - tail area

We want to find the z - score such that the area to the left of it is \(0.12\). Looking up this value in the standard normal distribution table (the cumulative distribution function of the standard normal distribution \(\varPhi(z)\)), we find the z - score corresponding to an area of \(0.12\) is approximately \(z=- 1.15\).

Step3: Use the symmetry of the standard normal distribution

The standard normal distribution is symmetric about \(z = 0\). So if the z - score for the left - tail is \(z=-1.15\), the z - score for the right - tail (such that the area between them is 76%) is \(z = 1.15\).

Answer:

\(-1.15,1.15\)