2. In a speed test, the mean and standard deviation of the speeds of the cars are 65 km/h and
5 km/h respectively. The speed of Vivian’s car is 63 km/h in the speed test.
(a) Find the standard score of Vivian’s car in the speed test. (4 marks) (b) It is known that data obtained in the speed test should be adjusted due to a
mechanical failure in the apparatus used. After the adjustment, the mean and standard deviation will increase by 8% and 60% respectively, while the standard scores of the speeds of the cars remain unchanged. Find the speed of Vivian’s car after the adjustment. (6 marks)
3. The heights of a group of students are normally distributed with a mean of 158 cm and a
standard deviation of 6 cm.
(a) Find the percentage of students with heights between 146 cm and 164 cm. (6 marks) (b) If there are 200 students in the group, find the number of students with heights less
4. The lifetimes of the light bulbs produced by a machine are normally distributed, and the mean of the lifetimes is 1 800 hours. If the lower limit of the standard lifetime of a light bulb is 1 500 hours, 2.5% of the light bulbs failed because their lifetimes are too short. Find the standard deviation of the lifetimes of the light bulbs. (8 marks)
5. Find the mode, median, range and inter-quartile range of the following set of data.
(b) (i) Complete the following frequency distribution table. (4 marks)
Mark (x) Class mark Frequency
40 ≤ x < 50
50 ≤ x < 60
60 ≤ x < 70
70 ≤ x < 80
(ii) Hence find the mean and standard deviation. (Give your answers correct to 3 significant figures if necessary.) (4 marks)
(c) The full marks of the Mathematics test are 80. If the full marks are adjusted to 100
by multiplication, find the range, inter-quartile range, mean and standard deviation of the test marks after the adjustment. (Give your answers correct to 3 significant figures if necessary.) (4 marks)
7. The following stem-and-leaf diagram shows the Geography test marks of 20 students.
1 4
4 0 6 8
5 0 0 0 2 2 2 2 26 2 6 8
Leaf (units)
Geography test marks of 20 studentsStem (tens)
7 6 6 8 88 2
(a) Find the median, range, inter-quartile range, mean and standard deviation of the Geography test marks of these 20 students. (Give your answers correct to 3 significant figures if necessary.) (9 marks)
(b) Find the new median, range, inter-quartile range, mean and standard deviation for each of the following cases. (Give your answers correct to 3 significant figures if necessary.)
(i) Remove the largest datum. (9 marks) (ii) Add a datum ‘0’. (9 marks)
(b) cm 152 =σ−x 1M ∴ Percentage of students with heights less than 152 cm = Percentage of data below σ−x
%16
%34%50=
−=
∴ Number of students with heights less than 152 cm
32%16200
=
×=
4. Let σ hours be the standard deviation of the lifetimes of the light bulbs. 2.5% of the light bulbs have lifetimes less than 1 500 hours, while the percentage of data below σ− 2x is 2.5%.
1503002
500 12800 1
=σ
=σ
=σ−∴
∴ The standard deviation of the lifetimes of the light bulbs is 150 hours.
5. (a) Arrange the data in ascending order as follows : 7, 7, 8, 9, 10, 10, 10, 12, 14, 36 Mode 10= 1A
Median
1021010
=
+=
1A
Range29
736=
−=
128
3
1
=
=
QQ
Inter-quartile range4
812=
−=
(b) (i) Every datum is obtained by adding 5 to each datum in (a) . Mode
