Answer to Question #222178 in Statistics and Probability for pesky

Question

The mean weight of cocoa produced by 350 farmers is 650kg and the standard deviation is 35kg. If the weights are normally distributed, find how many farmers that produced,

i. Less than 550kg

ii. Between 570 and 670kg

iii. More than 700kg

Answer 1

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

Given $\mu=650\ kg, \sigma=35\ kg.$

i.

$$P(X<550)=P(Z<\dfrac{550-650}{35})$$

$$\approx P(Z<-2.857143)\approx0.002137$$

$$n=0.002137(350)=1$$

ii.

$$P(570<X<670)=P(X<670)-P(Z\leq 570)$$

$$=P(Z<\dfrac{670-650}{35})-P(Z\leq \dfrac{570-650}{35})$$

$$\approx P(Z<0.571429)-P(Z\leq -2.2857149)$$

$$\approx0.7161456-0.0111355\approx0.705010$$

$$n=0.705010(350)=248$$

iii.

$$P(X>700)=1-P(Z\leq700)$$

$$=1-P(Z\leq \dfrac{700-650}{35})$$

$$\approx 1-P(Z\leq 1.428571)$$

$$\approx0.076564$$

$$n=0.076564(350)=27$$