Question

2. (a) Yuk Ping belongs to an athletics club. In javelin throwing competitions her throws
are normally distributed with mean 41.0 m and standard deviation 2.0 m.
(i) What is the probability of her throwing between 40 m and 46 m?
(ii) What distance will be exceeded by 60% of her throws?

(b) Gwen belongs to the same club. In competitions 85% of her javelin throws exceed
35 m and 70% exceed 37.5 m. Her throws are normally distributed.
(i) Find the mean and s.d. of Gwen's throws, each correct to two decimal places.
(ii) What is the probability that, in a competition in which each athlete takes a
single throw, Yuk Ping will beat Gwen?
(iii) The club has to choose one of these two athletes to enter a major competition.
In order to qualify for the final rounds it is necessary to achieve a throw of at
least 48 m in the preliminary rounds. Which athlete should be chosen? Why?

Answer 1

2(a)

Mean, = 41

Standard deviation, = 2

Let X denote the distance of her throw

(i) The required probability = P(40 <= X <= 46)

Conversion to Z score gives the required probability = P(-0.5 <= Z <= 2.5)

= 0.6873

(ii) For the top 60% throws, the critical z value = -0.253

Thus, the required distance = 41 - 0.253*2 = 40.494 m

(b) Let Y be a random variable which denotes the distance of her javelin throws

u and s denotes the mean and standard deviation respectively

Corresponding to top 85% throws, the z score = -1.0365

and corresponding to top 70% throws, the z score = -0.5245

Thus, u - 1.0365*s = 35

and u - 0.5245*s = 37.5

Thus, s = 4.88 m

and u = 40.06 m

(i) The mean and SD of Gwen's throws are 40.06 m and 4.88 m respectively

(ii) Let D denote the distribution of the difference in distance of throws between Yuk Ping and Gwen

Mean of D = 41 - 40.06 = 0.94 m

Standard deviation of D = sqrt{2^2 + 4.88^2} = 5.274

Thus, the probability that Yuk Ping will beat Gwen = P(D > 0)

The corresponding z value = (0 - 0.94)/5.274 = -0.178

Thus, the required probability = P(Z > -0.178) = 0.5707

(iii) Z value corresponding to 48 m for Yuk Ping = (48 - 41)/2 = 3.5

Z value corresponding to 48 m for Gwen = (48 - 40.06)/4.88 = 1.627

As P(Z > 1.627) > P(Z > 3.5), Gwen should be chosen