1.	The following frequency distribution of marks has mean 4.5.

Mark	1	2	3	4	5	6	7	
Frequency	2	4	6	9	x	9	4	
 
(a)	Find the value of x.
(4)
 
(b)	Write down the standard deviation.
(2)
(Total 6 marks)

2.	A scientist has 100 female fish and 100 male fish. She measures their lengths to the nearest cm. These are shown in the following box and whisker diagrams.

(a)	Find the range of the lengths of all 200 fish.
(3)

(b)	Four cumulative frequency graphs are shown below.
		
	Which graph is the best representation of the lengths of the female fish?
(2)
(Total 5 marks)

3.	A fisherman catches 200 fish to sell. He measures the lengths, l cm of these fish, and the results are shown in the frequency table below.

Length l cm	0 ≤ l < 10	10 ≤ l < 20	20 ≤ l < 30	30 ≤ l < 40	40 ≤ l < 60	60 ≤ l < 75	75 ≤ l < 100	
Frequency	30	40	50	30	33	11	6	
 
(a)	Calculate an estimate for the standard deviation of the lengths of the fish.
(3)

(b)	A cumulative frequency diagram is given below for the lengths of the fish.

	Use the graph to answer the following.
(i)	Estimate the interquartile range.
(ii)	Given that 40 % of the fish have a length more than k cm, find the value of k.
(6)
 
	In order to sell the fish, the fisherman classifies them as small, medium or large.
	Small fish have a length less than 20 cm.
Medium fish have a length greater than or equal to 20 cm but less than 60 cm.
Large fish have a length greater than or equal to 60 cm.
(c)	Write down the probability that a fish is small.
(2)

	The cost of a small fish is $4, a medium fish $10, and a large fish $12.
(d)	Copy and complete the following table, which gives a probability distribution for the cost $X.

Cost $X	4	10	12	
P(X = x)		0.565		
(2)
 
(e)	Find E(X).
(2)
(Total 15 marks)

4.	A test marked out of 100 is written by 800 students. The cumulative frequency graph for the marks is given below.

(a)	Write down the number of students who scored 40 marks or less on the test.
(2)
 
(b)	The middle 50 % of test results lie between marks a and b, where a < b.
Find a and b.
(4)
(Total 6 marks)

5.	Consider the four numbers a, b, c, d with a £ b £ c ≤ d, where a, b, c, d Î .
The mean of the four numbers is 4.
The mode is 3.
The median is 3.
The range is 6.
Find the value of a, of b, of c and of d.
(Total 6 marks)

6.	The mean of the population x1, x2, ........ , x25  is m. Given that = 300 and
 = 625, find
(a)	the value of m;
(b)	the standard deviation of the population.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

7.	A survey is carried out to find the waiting times for 100 customers at a supermarket.
waiting time
(seconds)	number of 
customers	
0–30	5	
30– 60	15	
60– 90	33	
90 –120	21	
120–150	11	
150–180	7	
180–210	5	
210–240	3	
(a)	Calculate an estimate for the mean of the waiting times, by using an appropriate approximation to represent each interval.
(2)
(b)	Construct a cumulative frequency table for these data.
(1)
(c)	Use the cumulative frequency table to draw, on graph paper, a cumulative frequency graph, using a scale of 1 cm per 20 seconds waiting time for the horizontal axis and 1 cm per 10 customers for the vertical axis.
(4)
(d)	Use the cumulative frequency graph to find estimates for the median and the lower and upper quartiles.
(3)
(Total 10 marks)