Question

Use the normal model ​n(11371137​,9696​) for the weights of steers. ​a) what weight represents the 3737thth ​percentile? ​b) what weight represents the 9999thth ​percentile? ​c) what's the iqr of the weights of these​ steers?

Answer 1

Denote by $x_k$ the $100k$-th percentile of the distribution followed by $X$.

$\mathbb P(X\le x_{0.37})=\mathbb P\left(\dfrac{X-1137}{96}\le\dfrac{x_{0.37}-1137}{96}\right)=\mathbb P(Z\le z_{0.37})\approx0.37$
$\implies z_{0.37}=\dfrac{x_{0.37}-1137}{96}\approx-0.3319\implies x_{0.37}\approx1105.14$

$\mathbb P(X\le x_{0.99})=\mathbb P\left(\dfrac{X-1137}{96}\le\dfrac{x_{0.99}-1137}{96}\right)=\mathbb P(Z\le z_{0.99})\approx0.99$
$\implies z_{0.99}=\dfrac{x_{0.99}-1137}{96}\approx2.3264\implies x_{0.37}\approx1360.33$

$\mathrm{IQR}=x_{0.75}-x_{0.25}$
$z_{0.25}\approx-0.6745\implies x_{0.25}\approx1072.249$
$z_{0.75}\approx0.6745\implies x_{0.75}\approx1201.751$
$\implies\mathrm{IQR}\approx129.502$