Question

Question 2 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. (a) Find mean and the standard deviation of T. (b) There is a 13.8% chance that the total daily expense of the business is more than $K. Find the value of K (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) 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

Answer 1

X ~ N(203, 404)

:: E(X) = 203
     $\text {and}$  
Var(X) = 40% = 1600

(a)

T = 15X + 2300

:: E(T) = E(15X + 2300) = 15E(X) + 2300 = 15* 203 + 2300 = 5345

:: Var(T) = Var(15X + 2300) = 152V ar(X) = 225 * 1600 = 360000

::SD = 360000 = 600

Hence, mean is $5345 and standard deviation is $600.

(b)

T N (5345, 6002)

PT>K) = 0.138

T - 5345 Pl K - 5345 600 ) = 0.138 600 DUU

   [Using R-code 'qnorm(1-0.138)']

PZ> K - 5345 600 = P(Z > 1.089349)

K - 5345 P= 1.089349 600

K= 5345 + 600 * 1.089349

::K = 5998.609

Hence, there is 13.8% chance that the total daily expense of business is more than $5998.609.

(c)

Suppose, random variable Y denotes amount of daily profit (in dollar).

Y = 35X – (15X + 2300) = 20X – 2300

:: E(Y) = E(20X – 2300) = 20E(X) – 2300 = 20 * 203 - 2300 = 1760

:: Var(Y) = Var(20X – 2300) = 20°Var(X) = 400* 1600 = 640000

1. SD = V640000 = 800

For normally distributed random variable mean=median.

Hence, for daily profit of the business mean is $1760, median is $1760 and standard deviation is $800.

(d)

For random variable Y denoting daily profit (in dollar) we have .

Y ~ N(1760, 8002)

Probability that Alex has negative profit in a day is given by

Y – 17600 P(Y < 0) = P - 1760 800 8003-

      [Using R-code 'norm(-2.2)']

= P(Z < -2.2) = 0.01390345 = 1.390345%

As the chance that Alex's recycling business has a negative profit in a day is 1.390345% (i.e. lower than 10%), he cannot apply for government allowance.