Advanced level 2024 CASPA mock mathematics with statistics 3

Advanced level 2024 CASPA mock mathematics with statistics 3

Advanced level 2024 CASPA mock mathematics with statistics 3

The table below shows the number of hours put in per week by workers of a certain factory;
Number of
hours (𝑥)
6-10 11-15 16-20 21-25 26-30 31-35 36-40 41-45 46-50
Frequency (𝑦) 2 4 8 11 16 5 2 1 1
(a) Calculate, to the nearest hour, the mean and standard deviation of the distribution. (5 marks)
(b) Draw a cumulative frequency curve for the data. From your curve, estimate the median hour of the
distribution (5 marks)
(c) Estimate, also, correct to two decimal places, the percentage of workers who put in between 30 to 40 hours of
work per week (3 marks)
2. A discrete random variable X has probability mass function f, defined by
𝑘𝑥² 𝑥 = 1,2,3,4
f(x) = 𝑘(8 − 𝑥)2, 𝑥 = 5,6,7,
0 elsewhere
Find :
(a) the value of the constant k, (2 marks)
(b) the mean and variance of X, (5 marks)
(c) P (3 ≤ 𝑋 ≤ 6), (2 marks)
(d) the median of X. (2 marks)
Given that the random variable R is such that 𝑅 = 3𝑋 − 2, find
(e) the mean and variance of 𝑅 (2 marks)
3. (a) 60% of the customers of a certain telephone booth buy credit cards. Given that the
number of customers buying credit cards is binomially distributed, find the
probability that in a random sample of 10 customers
(i) exactly 3 buy credit cards, (4 marks)
(ii) more than 7 buy credit cards. (5 marks)
(b) Given that X is a discrete random variable and that 𝑐 and 𝑘 are constants,
show that var(c ) var( ) X k c X   2 . (4 marks)
4. The table below shows the inflation rate, 𝑥 percent, and the unemployment rate, 𝑦 percent, for 10 different countries in
the month of December 2010.
Inflation rate,
𝑥
13.9 21.4 9.6 1.5 31.7 23.1 18.4 34.4 27.6 5.6
Unemployment
rate, 𝑦
2.9 11.3 5.2 6.1 9.0 8.8 5.9 15.6 9.8 3.7
Calculate, correct to three decimal places,
(a) the product moment correlation coefficient for this data, (7 marks)
(b) the least squares regression line of inflation rate on unemployment rate, (2 marks)
(c) the kendall’s coefficient of rank correlation between inflation rate and unemployment rate.
(4 marks)
You may use ∑ 𝑥 = 187.2, ∑ 𝑥2 = 4599.12, ∑ 𝑦 = 78.3, ∑ 𝑦2 = 746.69, ∑ 𝑥𝑦 = 1766.18
5. (i) Two events A and B are such that
𝑃(𝐴) = 1
3
, 𝑃(𝐵) = 2
5
𝑎𝑛𝑑 𝑃(𝐴/𝐵) = 1
12