Question

find the value of z so that the area under the standard normal curve in the right tail is (approximately). 1053

Explanation:

Step1: Use the property of the standard normal distribution

The total area under the standard normal curve is \(1\). If the area in the right - tail is \(A = 0.1053\), then the area to the left of \(z\) is \(1 - A\). So, the area to the left of \(z\) is \(1-0.1053=0.8947\).

Step2: Look up the \(z\) - value in the standard normal table

We look for the value \(0.8947\) in the body of the standard normal table (the \(z\) - table).
In the standard normal table, when we look for the value closest to \(0.8947\), we find that \(z = 1.25\) (by cross - referencing the row and column values in the \(z\) - table. For example, in a standard \(z\) - table, the row for \(1.2\) and the column for \(0.05\) gives a cumulative probability of \(P(Z\leq z)=0.8944\) (very close to \(0.8947\)), and with more precise tables or using a calculator with a normal distribution function \(P(Z\leq z)=\Phi(z)\), we can confirm the value).

Answer:

\(z = 1.25\)