A company manufactures soaps. The mean mass of the soaps is 300g with variance 52g. A sample of 100soaps was selected. a)Find the probability of selecting soaps more than masses 320g. b)How many soaps will have masses between 280g and 500g.

Let $X=$ mass of the soap: $X\sim N(\mu, \sigma^2).$

Given $\mu=300\ g, \sigma=\sqrt{52}\ g.$

a)

$P(X>320)=1-P(Z\le\dfrac{320-300}{\sqrt{52}})$

$\approx1-P(Z\le2.7735)\approx0.002773$

b)

$P(280<X<500)=P(Z<\dfrac{500-300}{\sqrt{52}})$

$-P(Z\le\dfrac{280-300}{\sqrt{52}})\approx P(Z\le27.7350)$

$-P(Z\le-2.7735)\approx1-0.002773$

$\approx0.997227$

$0.997227(100)=100$

100 soaps will have masses between 280g and 500g.