Question

find two z values so that 78% of the middle area is bounded by them. use the standard normal distribution table and enter the answers rounded to 2 decimal places, separated by a comma if needed.

Explanation:

Step1: Calculate the tail area

The total area under the standard normal curve is \(1\). If the middle area is \(78\%=0.78\), then the total tail area is \(1 - 0.78=0.22\). Since the distribution is symmetric, the area in each tail is \(\frac{0.22}{2}=0.11\).

Step2: Find the \(z\) - value

We want to find the \(z\) - value such that \(P(Z < z)=1 - 0.11 = 0.89\). Looking up in the standard normal distribution table (the \(z\) - table), the \(z\) - value corresponding to an area of \(0.89\) is approximately \(1.23\). Due to symmetry, the two \(z\) - values are \(- 1.23\) and \(1.23\).

Answer:

\(-1.23,1.23\)