Math AI Class of 2025
Statistics
Unit Test Review
SL Questions
1.
2.
–6–
240 cars were tested to see how far they travelled on 10 litres of fuel. The graph shows the
cumulative frequency distribution of the results.
240
200
Cumulative frequency
3.4.
N13/5/MATSD/SP1/ENG/TZ0/XX
160
120
80
40
0
100
110
120
130
140
150
Distance (km)
(a)
Find the median distance travelled by the cars.
[ 2]
(b)
Calculate the interquartile range of the distance travelled by the cars.
[ 2]
(c)
Find the number of cars that travelled more than 130 km.
[ 2]
(This question continues on the following page)
24EP06
4.
5.
written working. Solutions found from a graphic display calculator should be supported by suitable working,
for example, d
if graphs are used to fin a sol ut ion, you shoul d sket ch these as par t of your answe r .
6. 1.
[ Maximum mark: 18]
The table shows the distance, in km, of eight regional railway stations from a city centre
terminus and the price, in $, of a return ticket from each regional station to the terminus.
(a)
(b)
Distance in km (x)
3
15
23
42
56
62
74
93
Price in $ ( y)
5
24
43
56
68
74
86
100
Draw a scatter diagram for the above data. Use a scale of 1 cm to represent 10 km on
the x-axis and 1 cm to represent $10 on the y-axis.
Use your graphic display calculator
d
to fin
(i)
x , the mean of the distances;
(ii)
y , the mean of the prices.
(c)
Plot and label the point M (x , y ) on your scatter diagram.
(d)
Use your graphic display calculator
d
to fin
(e)
[ 4]
[ 2]
[ 1]
(i)
the product–moment correlation
i
coefficent , r ;
(ii)
the equation of the regression line y on x .
[ 3]
Draw the regression line y on x on your scatter diagram.
[ 2]
A ninth regional station is 76 km from the city centre terminus.
(f)
(g)
Use the equation of the regression line to estimate the price of a return ticket to the city
centre terminus from this regional station. Give your answer correct to the nearest $.
[ 3]
Give a reason why it is valid to use your regression line to estimate the price of this
return ticket.
[ 1]
The actual price of the return ticket is $80.
(h)
8813-7402
Using your answer to part (f), calculate the percentage error in the estimated price of
the ticket.
[ 2]
–3–
7.2.
M13/5/MATSD/SP2/ENG/TZ2/XX
[ Maximum mark: 17]
Francesca is a chef in a restaurant. She cooks eight chickens and records their
masses and cooking times. The mass m of each chicken, in kg, and its cooking
time t , in minutes, are shown in the following table.
M ass m (kg)
1.5
1.6
1.8
1.9
2.0
2.1
2.1
2.3
(a)
(b)
Cooking time t (minutes)
62
75
82
83
86
87
91
98
Draw a scatter diagram to show the relationship between the mass of a
chicken and its cooking time. Use 2 cm to represent 0.5 kg on the horizontal
axis and 1 cm to represent 10 minutes on the vertical axis.
[ 4 marks]
Write down for this set of data
(i)
the mean mass, m;
(ii)
the mean cooking time, t .
[ 2 marks]
(c)
Label the point M (m, t ) on the scatter diagram.
[ 1 mark]
(d)
Draw the line of best fit on the scatter diagram.
[ 2 marks]
(e)
Using your line of best, fit estimate the cooking time, in minutes, for a 1.7 kg
chicken.
[ 2 marks]
(f)
Write down the Pearson’s product–moment correlation
i
coefficent , r .
[ 2 marks]
(g)
Using your value for r , comment on the correlation.
[ 2 marks]
The cooking time of an additional 2.0 kg chicken is recorded. If the mass and
cooking time of this chicken is included in the data, the correlation is weak.
(h)
2213-7406
(i)
Explain how the cooking time of this additional chicken might differ from
that of the other eight chickens.
(ii)
Explain how a new line of best fit might differ from that drawn in part (d).
[ 2 marks]
Turn over
–4–
8.3.
M13/5/MATSD/SP2/ENG/TZ1/XX
[ Maximum mark: 23]
George leaves a cup of hot coffee to cool and measures its temperature every minute.
His results are shown in the table below.
Time, t (minutes)
0
1
2
3
4
5
6
Temperature, y (°C)
94
54
34
24
k
16.5
15.25
(a)
Write down the decrease in the temperature of the coffee
s
(i)
during the firt m
i nut e (bet we en t = 0 and t = 1 ) ;
(ii)
during the second minute;
(iii) during the third minute.
[ 3 marks]
(b)
Assuming the pattern in the answers to part (a) continues, show thats k = 19 .
[ 2 marks]
(c)
Use the seven results in the table to draw a graph that shows how the temperature
of the coffee changes during the firt si x m
i nut es.
Use a scale of 2 cm to represent 1 minute on the horizontal axis and 1 cm to
represent 10 °C on the vertical axis.
[ 4 marks]
The function that models the change in temperature of the coffee is y = p ( 2−t ) + q .
(d)
(e)
(i)
Use the values t = 0 and y = 94 to form an equation in p and q .
(ii)
Use the values t = 1 and y = 54 to form a second equation in p and q .
[ 2 marks]
Solve the equations found in part (d) to find the value of p and the value of q .
[ 2 marks]
The graph of this function has a horizontal asymptote.
–5–
M13/5/MATSD/SP2/ENG/TZ1/XX
(f) 3 Write
down the equation of this asymptote.
(Question
continued)
[ 2 marks]
George decides to model the change in temperature of the coffee with a linear function
using correlation and linear regression.
(This question continues on the following page)
(g) Use the seven results in the table ito write down
(h)
2213-7404
(i)
(i)
the correlation coefficent ;
(ii)
the equation of the regression line y on t .
[ 4 marks]
Use the equation of the regression line to estimate the temperature of the coffee
at t = 3 .
[ 2 marks]
Find the percentage error in this estimate of the temperature of the coffee
at t = 3 .
[ 2 marks]
9.
–4–
[ Maximum mark: 7]
The following is a cumulative frequency diagram for the time t , in minutes, taken by 80 students
to complete a task.
80
70
60
50
Number of students
3.
10.
N13/5/MATME/SP1/ENG/TZ0/XX
40
30
20
10
0
10
20
30
40
Time t (minutes)
50
60
(Question 3 continued)
(a)
Find the number of students who completed the task in less than 45 minutes.
[ 2]
(b)
Find the number of students who took between 35 and 45 minutes to complete the task.
[ 3]
(c)
Given that 50 students take less than k minutes to complete the task,
d fin the value of k .
[ 2]
HL Question
1.
.......................................................................
.......................................................................
.......................................................................
.......................................................................
.......................................................................
.......................................................................
.......................................................................
.......................................................................
.......................................................................
.......................................................................
.......................................................................
.......................................................................
t urn over
12EP05
ANSWERS – SL Questions
1.
– 15 –
M13/5/MATSD/SP1/ENG/TZ2/XX/M
QUESTI ON 4
2.
(a)
(i)
8
(A1)
(ii)
48
(A1)(ft)
(C2)
(A1)(A1)(A1)(A1)
(C4)
Note: Follow through from their t, even if no workings seen as long as w 50 .
(b)
Statement
True
False

Every household owns at least 1 bicycle.
The median number of bicycles per household is 3.


The 25th percentile is 1 bicycle per household.
There are 10 households with at most 1 bicycle.

[6 marks]
3.QUESTI ON 5
(a)
AC
sin100
10
sin 50
(M1)(A1)
Note: Award (M1) for substitution in the sine rule formula, (A1) for correct
substitutions.
(A1)
12.9 12.8557
(C3)
Note: Radian answer is 19.3, award (M1)(A1)(A0).
(b)
122
7 2 12.8557
2 12 7
2
(M1)(A1)(ft)
Note: Award (M1) for substitution in the cosine rule formula, (A1)(ft) for
correct substitutions.
80.5 80.4994
(A1)(ft)
(C3)
4.
5.
6.
7.
– 15 –
M13/5/MATSD/SP2/ENG/TZ2/XX/M
Question 2 continued
(e)
75
(M1)(A1)(ft)(G2)
[2 marks]
Notes: Accept 74.77 from the regression line equation.
Award (M1) for indication of the use of their graph to get an estimate
OR for correct substitution of 1.7 in the correct regression line
equation t 38.5m 9.32 .
(f)
0.960 (0.959614 )
(G2)
[2 marks]
(A1)(ft)(A1)(ft)
[2 marks]
Note: Award (G0)(G1)(ft) for 0. 95, 0.959
(g)
Strong and positive
Note: Follow through from their correlation coefficient in part (f).
(h)
(i)
Cooking time is much larger (or smaller) than the other eight
(A1)
(ii)
The gradient of the new line of best fit will be larger (or smaller)
(A1)
[2 marks]
Note: Some acceptable explanations may include but are not limited to:
The line of best fit may be further away from the plotted points
It may be steeper than the previous line (as the mean would change)
The t-intercept of the new line is smaller (larger)
Do not accept vague explanations, like:
The new line would vary
It would not go through all points
It would not fit the patterns
The line may be slightly tilted
Total [17 marks]
8.
Total [7 marks]
9.2.
(a)
valid approach
eg 35 − 26 , 26 + p = 35
(M1)
p=9
(b)
(i)
mean = 26.7
(ii)
recognizing that variance is (sd) 2
A1
N2
[2 marks]
A2
N2
(M1)
eg 11.021− 2 , − = var , 11.158− 2
− 2 = 121
A1
–9–
10.
3.
(a)
(b)
(c)
N2
[4 marks]
N13/5/MATME/SP1/ENG/TZ0/XX/M
Total [6 marks]
attempt to find number who took less than 45 minutes
eg line on graph (vertical at approx 45, or horizontal at approx 70)
(M1)
70 students (accept 69)
A1
55 students completed task in less than 35 minutes
(A1)
subtracting their values
eg 70 – 55
(M1)
15 students
A1
correct approach
eg line from y-axis on 50
(A1)
k = 33
A1
N2
[2 marks]
N2
[3 marks]
N2
[2 marks]
[Total 7 marks]
HL Questions
1.4. (a) appropriate approach
eg
(M1)
2 −f ( x) , 2(8)
−2 f ( x) dx = 16
6
1
A1
N2
[2 marks]
(b)
appropriate approach
eg
−f ( x) + −2 , 8 + −2
(M1)
−2dx = 2 x (seen anywhere)
(A1)
substituting limits into their integrated function and subtracting
(in any order)
eg
2(6) − 2(1) , 8 + 12 − 2
(M1)
−( f ( x) + 2 ) dx = 18
6
1
A1
N3
[4 marks]
[Total 6 marks]