### cameroon gce A level June 2024 pure mathematics with statistics 3

cameroon gce A level June 2024 pure mathematics with statistics 3

I. (i) Two independent events A and U me such tlml

Find

(a) P( A/ B),

(b) P(An B),

(e) P(A U /?). (7 murks)

(ii) 60% of the population of a eertain country have been vacillated against COVID-19.

During an outbreak of the COVID-19 epidemic

2-

of those who had not taken the vneine were infected.

Find the probability that an individual selected at random from this country

(d) is infected.

(e) is infected or had taken the vneine.

. – of those who had taken the vneine were infected and

5

(6 marks)

2. The lengths, to the nearest cm.of some randomly selected 140 leaves from a certain fruit tree were measured

and recorded as follows:

Length in cm 5 – 9 10 – 14 15 – 19 20 – 24 25 – 29 30 •34 35 – 39 40 44 45 – 49

Number of leaves 13 17 20 25 oo ** A* 16 12 4

(a) State the modal class of this distribution of lengths.

(b) Calculate the mean and variance of the lengths of the leaves, to 3 decimal places.

(c) Construct the cumulative frequency table and draw the cumulative frequency curve for these data.

(d) Use your curve to estimate the interquartile range.

(I, 6, 4, 2 marks)

3. A random variable X takes its values in the set {0, 1, 2, 3, 4} and has the distribution

0 2 3 4

P( X = x ) a a ( i l ) b

Given that P{ X > 2) = 3P( X < 2).

calculate

(a) the values of the probabilities a and b,

(b) the expected value of X ,

(c) the standard deviation of X ,

(d) the expected value ofY where Y = 5X + 3.

(5, 3, 3, 2 marks)

5. (i) In an experiment the length / of a metal bar is measured many times and the observed values recorded.

Ihe recorded values arc approximately normally distributed with mean 1.340m and standard deviation 0.021m.

Find the probability that a recorded value

(a) exceeds 1.37 m,

(b) lies between 1.31m and 1.37m. (3,3 marks)

Given that 10% of the recorded values are less than l,

(c) find, to three decimal places, the value of /.

(ii) In a train station, the time T in hours between the arrival of two trains is a random variable with an exponential

distribution having the density function

f(t) = 3e~3t , t > 0.

A train arrived at 9:00 am. What is the probability that the next train will arrive in less than 3 hours. (3 marks)

(4 marks)

6. A company manufactures memory cards and one out of 100 cards manufactured is known to be defective.

The memory cards are packed in packs of 10.

What is the probability that

(a) there will be at least a defective card in a randomly chosen pack?

(b) exactly two defective cards are found in a pack? (6,3 marks)

The company offers a replacement guarantee if a pack contains at least two defective cards,

(c) What percentage of packs will the company have to replace? (4 marks)

7. (i) A large population consists of equal numbers of the digits 2 and 4.

(a) Find the mean /u and the variance a2 of this population.

(h) Construct the probability distribution of the mean, x,of all possible samples of size 3 taken from this

population.

(ii) A machine is set to produce sweets whose masses are normally distributed with mean 15.21g and standard

deviation 0.5g. To check the setting of the machine a sample of 50 sweets is taken from the production line of

the machine and the mean mass is found to be 15g. Test at 3% level of significance whether the machine is

correctly set.

(4,3 marks)

(6 marks)