Question
If probabilities for the value of standard normal variable $Z$ are as under, then estimate the value of $Z-$score$ (z_1) (i)$ Area to the left of $Z = (z_1)$ is $0.9928$. $(ii)$ Area to the right of $Z = (z_1)$ is $0.0250.$
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Answer
(i) Area to the left of $Z = z_1 $ is $0.9928.$
$ P\left[Z \leq z_1\right]=0.9928 $
$ \therefore P [0 \leq Z \leq z_1 ]=P [-\infty] $
$=0.9928-0.5000$
$ =0.4928$
From the table, for the probability $0.4927, z_1=2.44$ and for the probability
$0.4929, z_1=2.45 \text {. }$
For average probability $=\frac{0.4927+0.4929}{2}$
$ =0.4928 $
$ z_1=\frac{2.44+2.45}{2}=2.445$
(ii) Area to the right of $Z = z_1$ is $0.0250.$
$P [Z \geq z_1] = 0.0250$
$\therefore P [0 \leq Z \leq z_1] = 0.5000 – 0.0250$
$= 0.4750$
From the table, for $0.4750, z_1 = 1.96$
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