Question

The height that a certain type of plant reaches a month after being planted is normally distributed with a mean of 10cm and a standard deviation of 2cm.

(a) For a randomly selected plant, what is the probability that its height is
(i) more than 13cm? (5 marks)   (ii) less than 8cm? (5 marks)

(b) A random sample of 10 plants is taken.
(i) What is the expected value and standard error of Xbar , the mean height of these 10 plants? (6 marks)
(ii) Show that the probability of Xbar being within 1cm of the population mean height is 0.8858. (9 marks)

(c) Suppose that 5 independent batches of 10 plants are analyzed. A batch is considered acceptable if the mean height of its 10 plants is within 1cm of the population mean height. Let Y denote the number of the 5 batches that are considered acceptable.
(i) What distribution does Y follow? State the values of any parameters of this distribution. (3 marks)   (ii) Calculate Pr(Y ≤ 2). (7 marks)

Answer 1

mean = 10 and sd = 2

a)
i)
P(X > 13)
= P(z > (13 - 10)/2)
= P(z > 1.5)
= 0.0668

ii)
P(X < 8)
= P(z < (8 - 10)/2)
= P(z < -1)
= 0.1587

b)
for n = 10
i)
xbar = 10
se = 2/sqrt(10) = 0.6325

ii)
P(-1/0.6325 < z < 1/0.6325)
= P(-1.5810 < z < 1.5810)
= 0.7844

c)
i) Y follows binomial distribution with p = 0.7844 and n = 5

ii)
P(Y <= 2)
= P(Y = 0) + P(Y = 1) + P(Y = 2)
= (1 - 0.7844)^5 + 5C1*0.7844*(1-0.7844)^4 + 5C2*0.7844^2*(1-0.7844)^3
= 0.0706