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((X-1137)/(96)\le(x_(0.37)-1137)/(96)\right)=\mathbb P(Z\le z_(0.37))\approx0.37`

`\implies z_(0.37)=(x_(0.37)-1137)/(96)\approx-0.3319\implies x_(0.37)\approx1105.14`


`\mathbb P(X\le x_(0.99))=\mathbb P\left((X-1137)/(96)\le(x_(0.99)-1137)/(96)\right)=\mathbb P(Z\le z_(0.99))\approx0.99`

`\implies z_(0.99)=(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`