Question

5 Gem stones from a certain mine have weights, X grams, which are normally distributed with mean 1.9 g and standard deviation 0.55 g. These gem stones are sorted into three categories for sale depending on their weights, as follows.

Small: under 1.2 g Medium: between 1.2 g and 2.5 g Large: over 2.5 g

(i) Find the proportion of gem stones in each of these three categories.

(ii) Find the value of k such that Pk < X < 2.5
=

Answer 1

$X\sim N(\mu,\sigma^2)$

Then $Z=\dfrac{X-\mu}{\sigma}\sim N(0,1)$

Given $\mu=1.9g,\sigma=0.55g$

$$P(X<1.2)=P(Z<\dfrac{1.2-1.9}{0.55})=$$ $$= P(Z<-\dfrac{14}{11})\approx0.101577$$

$$P(X < 2.5)=P(Z<\dfrac{2.5-1.9}{0.55})=$$

$$= P(Z<\dfrac{12}{11})\approx0.862344$$

$$P(1.2<X<2.5)=P(X<2.5)-P(X<1.2)\approx$$

$$\approx0.862344-0.101577\approx0.760767$$

$$P(X\geq2.5)=1-P(X<2.5)\approx$$

$$\approx1-0.862344\approx0.137656$$

$10.1577\%, \ 76.0767\%,\ 13.7656\%$

$$P(k<X<2.5)=P(X<2.5)-P(X<k)=0.8$$

$$P(X<k)=P(X<2.5)-0.8\approx$$

$$\approx0.862344-0.8=0.062344=P(Z<\dfrac{k-1.9}{0.55})$$

$$z=\dfrac{k-1.9}{0.55}\approx-1.53539$$

$$k\approx1.9-0.55\cdot1.53539\approx1.0555$$

$$k=1.0555$$