Answer to Question #107866 in Statistics and Probability for feng jiaqi

Question

Alex is running a recycling business. The weight of collected used Iron in a day is a variable, X, which is normally distributed with mean 203 kg and standard deviation 40 kg. If the cost to recycle each kg of used Iron is $15 and the other cost for running the recycling business per day is $2300, the total expense per day, T, is given by T = 15X + 2300.

Answer 1

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

Given that $\mu_X=203\ kg, \sigma_X=40\ kg$

Let $T=15X+2300$

a) Find mean and the standard deviation of T.

$$\mu_T=15\mu_X+2300=15(203)+2300=5345$$

$$Var(T)=\sigma_T^2=15^2 \cdot\sigma_X^2=360000$$

$$\sigma_T=\sqrt{\sigma_T^2}=15(40)=600$$

(b) There is a 13.8% chance that the total daily expense of the business is more than $\$K.$ Find the value of $K.$

$$T\sim N(5345, 360000)$$

Then $Z=\dfrac{T-5345}{600}\sim N(0, 1)$

$$P(T>K)=0.138$$

$$P(T\leq K)=1-0.138=0.862$$

$$P(Z\leq Z^* )=0.862=>Z^*\approx1.089349$$

$${T^*-5345\over 600}=1.089349$$

$$T^*=\$5410.36$$

(c) Suppose each kg of collected used Iron can be sold for $35 after the recycling. Find the mean, median and the standard deviation of the daily profit of the business.

$$D=35X-T=20X-2300$$

$$\mu_D=20\mu_X-2300=15(203)+2300=745$$

$$Var(D)=\sigma_D^2=20^2 \cdot\sigma_X^2=640000$$

$$\sigma_D=\sqrt{\sigma_D^2}=20(40)=800$$

(d) Alex can apply for government allowance if there is more than 10% chance that his recycling business has negative profit in a day. Can he apply for the government allowance? Explain your answer with calculation.

$$D\sim N(745, 640000)$$

Then $Z=\dfrac{D-745}{800}\sim N(0, 1)$

$$P(Z\leq Z^* )=0.1=>Z^*\approx-1.28155$$

$${D^*-745\over 800}=-1.28155$$

$$D^*=-\$280.24<0$$

Yes, he can apply for the government allowance.