Answer to Question #304118 in Statistics and Probability for Eddie

Question

1.     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.

 

Answer 1

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.