### 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 (๐ฅ)

6-10 11-15 16-20 21-25 26-30 31-35 36-40 41-45 46-50

Frequency (๐ฆ) 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

๐๐ฅยฒ ๐ฅ = 1,2,3,4

f(x) = ๐(8 โ ๐ฅ)2, ๐ฅ = 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 โค ๐ โค 6), (2 marks)

(d) the median of X. (2 marks)

Given that the random variable R is such that ๐
= 3๐ โ 2, find

(e) the mean and variance of ๐
(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 ๐ and ๐ are constants,

show that var(c ) var( ) X k c X ๏ซ ๏ฝ 2 . (4 marks)

4. The table below shows the inflation rate, ๐ฅ percent, and the unemployment rate, ๐ฆ percent, for 10 different countries in

the month of December 2010.

Inflation rate,

๐ฅ

13.9 21.4 9.6 1.5 31.7 23.1 18.4 34.4 27.6 5.6

Unemployment

rate, ๐ฆ

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 kendallโs coefficient of rank correlation between inflation rate and unemployment rate.

(4 marks)

You may use โ ๐ฅ = 187.2, โ ๐ฅ2 = 4599.12, โ ๐ฆ = 78.3, โ ๐ฆ2 = 746.69, โ ๐ฅ๐ฆ = 1766.18

5. (i) Two events A and B are such that

๐(๐ด) = 1

3

, ๐(๐ต) = 2

5

๐๐๐ ๐(๐ด/๐ต) = 1

12