Question

The weight of competition pumpkins at the Circleville Pumpkin Show in Circleville, Ohio, can be represented by a normal distribution with a mean of 703 pounds and a standard deviation of 347 pounds.

a. Find the probability that a randomly selected pumpkin weighs at least 1,622 pounds.

b. Find the probability that a randomly selected pumpkin weighs between 465.1 and 1,622

pounds.

Answer 1

Let r. v. $X$ be the weight of randomly choosen pumpkin. Then $X\in N (703;347^2).$

a. We should find $P\{X\geq 1622\}.$

$P\{X\geq 1622\}=1-P\{X<1622\}.$

$P\{X<1622\}=F(1622)$ where $F(x)=\frac{1}{\sqrt{2\pi}347}\int_{-\infty}^x e^{-\frac{(t-703)^2}{2\cdot 347^2}}dt.$

$F(1622)=\Phi(\frac{1622-703}{347})\approx\Phi(2.648)\approx 0.9960$ where

$\Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-\frac{z^2}{2}}dz.$

So $P\{X\geq 1622\}=1-P\{X<1622\}\approx 1-0.9960=0.004$.

b. We should find $P\{465.1<X<1622\}.$

$P\{465.1<X<1622\}=F(1622)-F(465.1).$

$F(465.1)=\Phi(\frac{465.1-703}{347})\approx \Phi(-0.686)\approx 0.2464.$

$\text{So }P\{465.1<X<1622\}=F(1622)-F(465.1)\approx\\ \approx 0.9960-0.2464=0.7496.$