Practice section 3 of 7

Independent random variables satisfy $X\sim N(100,49)$ and $Y\sim N(110,576)$. Find (i) $P(3X<250)$, (ii) $P(X+Y>200)$, (iii) $P(Y>X)$, (iv) $P(2X+Y<300)$.

In a country, men’s weights are normal with mean 70 kg and variance 18 kg², and children’s weights are normal with mean 30 kg and variance 9 kg². The weights of men and children are independent. Find the probability that

(i) a randomly selected man weighs more than two randomly selected children combined,

(ii) a randomly selected man weighs more than twice one randomly selected child. A certain HDB lift has maximum load 340 kg:

(iii) find the probability it is overloaded by 3 men and 4 children;

(iv) find the smallest a such that the probability that the average weight of three randomly chosen men exceeds a kg is less than 0.7.

Melons cost $1.50 per kilogram and their masses are normal with mean 0.8 kg and standard deviation 0.1 kg. Pumpkins cost $0.50 per kilogram and their masses are normal with mean 1.2 kg and standard deviation 0.2 kg. Find the probability that the total price of 5 randomly chosen melons and 3 randomly chosen pumpkins exceeds $8.

In a nation, men’s heights are normal with mean 1.70 m and standard deviation 10 cm; women’s heights are normal with mean 1.60 m and standard deviation 7.5 cm.

(i) Three men are selected at random: find the probability that one is taller than 1.83 m, one is shorter than 1.7 m and the other is between 1.7 m and 1.83 m.

(ii) Five men are selected at random: find the probability that exactly 3 are taller than 1.83 m.

(iii) Find the probability that the average height of 3 men and 2 women chosen randomly exceeds 1.68 m.

(iv) Find the probability that a husband and wife have not more than 5 cm difference in heights.

(v) State the assumptions made for (iii) and (iv).

Packages from a packing machine have a mass which is normally distributed with mean 200 g, and standard deviation 2 g. Find the probability that a package from the machine weighs

(i) between 198.5 and 199.5 g,

(ii) less than 197 g,

(iii) more than 200.5 g.

An examination is marked out of 100 and taken by a large number of candidates. The mean mark is 72.1 and standard deviation 15.2. Give a reason why a normal distribution with this mean and standard deviation would not give a good approximation to the marks.