REVISION
1a. [2 marks]
The following diagram shows part of the graph of 𝑓(𝑥) = (6 − 3𝑥)(4 + 𝑥), 𝑥 ∈ ℝ. The
shaded region R is bounded by the 𝑥-axis, 𝑦-axis and the graph of 𝑓.
Write down an integral for the area of region R.
1b. [1 mark]
Find the area of region R.
1c. [2 marks]
The three points A(0, 0) , B(3, 10) and C(𝑎, 0) define the vertices of a triangle.
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Find the value of 𝑎, the 𝑥-coordinate of C, such that the area of the triangle is equal to the
area of region R.
2a. [1 mark]
Jae Hee plays a game involving a biased six-sided die.
The faces of the die are labelled −3, −1, 0, 1, 2 and 5.
The score for the game, X, is the number which lands face up after the die is rolled.
The following table shows the probability distribution for X.
Find the exact value of 𝑝.
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2b. [2 marks]
Jae Hee plays the game once.
Calculate the expected score.
2c. [3 marks]
Jae Hee plays the game twice and adds the two scores together.
Find the probability Jae Hee has a total score of −3.
3a. [1 mark]
Mr Burke teaches a mathematics class with 15 students. In this class there are 6
female students and 9 male students.
Each day Mr Burke randomly chooses one student to answer a homework question.
Find the probability that on any given day Mr Burke chooses a female student to answer a
question.
3b. [2 marks]
In the first month, Mr Burke will teach his class 20 times.
Find the probability he will choose a female student 8 times.
3c. [3 marks]
Find the probability he will choose a male student at most 9 times.
4a. [1 mark]
At the end of a school day, the Headmaster conducted a survey asking students in how
many classes they had used the internet.
The data is shown in the following table.
State whether the data is discrete or continuous.
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4b. [4 marks]
The mean number of classes in which a student used the internet is 2.
Find the value of 𝑘
4c. [1 mark]
It was not possible to ask every person in the school, so the Headmaster arranged
the student names in alphabetical order and then asked every 10th person on the list.
Identify the sampling technique used in the survey.
5a. [1 mark]
As part of a study into healthy lifestyles, Jing visited Surrey Hills University. Jing recorded
a person’s position in the university and how frequently they ate a salad. Results are
shown in the table.
Jing conducted a 𝜒2 test for independence at a 5 % level of significance.
State the null hypothesis.
5b. [2 marks]
Calculate the 𝑝-value for this test.
5c. [2 marks]
State, giving a reason, whether the null hypothesis should be accepted.
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6a. [1 mark]
Ms Calhoun measures the heights of students in her mathematics class. She is interested to
see if the mean height of male students, 𝜇1 , is the same as the mean height of
female students, 𝜇2 . The information is recorded in the table.
At the 10 % level of significance, a 𝑡-test was used to compare the means of the two
groups. The data is assumed to be normally distributed and the standard deviations are
equal between the two groups.
State the null hypothesis.
6b. [1 mark]
State the alternative hypothesis.
6c. [2 marks]
Calculate the 𝑝-value for this test.
6d. [2 marks]
State, giving a reason, whether Ms Calhoun should accept the null hypothesis.
7a. [7 marks]
Give your answers to four significant figures.
A die is thrown 120 times with the following results.
Showing all steps clearly, test whether the die is fair
(i) at the 5% level of significance;
(ii) at the 1% level of significance.
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7b. [3 marks] Explain what is meant by “level of significance” in part (a)
8a. [7 marks]
A calculator generates a random sequence of digits. A sample of 200 digits is randomly
selected from the first 100 000 digits of the sequence. The following table gives the number
of times each digit occurs in this sample.
It is claimed that all digits have the same probability of appearing in the sequence.
Test this claim at the 5% level of significance.
8b. [2 marks]
Explain what is meant by the 5% level of significance.
9a. [5 marks]
Kayla wants to measure the extent to which two judges in a gymnastics competition are in
agreement. Each judge has ranked the seven competitors, as shown in the table, where 1 is
the highest ranking and 7 is the lowest.
Calculate Spearman’s rank correlation coefficient for this data.
9b. [1 mark]
State what conclusion Kayla can make from the answer in part (a).
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10a. [1 mark]
Charles wants to measure the strength of the relationship between the price of a house and
its distance from the city centre where he lives. He chooses houses of a similar size and
plots a graph of price, 𝑃 (in thousands of dollars) against distance from the city centre,
𝑑 (km).
Explain why it is not appropriate to use Pearson’s product moment correlation coefficient
to measure the strength of the relationship between 𝑃 and 𝑑.
10b. [1 mark]
Explain why it is appropriate to use Spearman’s rank correlation coefficient to measure the
strength of the relationship between 𝑃 and 𝑑.
10c. [6 marks]
The data from the graph is shown in the table.
Calculate Spearman’s rank correlation coefficient for this data.
10d. [1 mark] State what conclusion Charles can make from the answer in part (c).
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11a. [2 marks]
A set of data comprises of five numbers 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 , 𝑥5 which have been placed in
ascending order.
𝑛+1
Recalling definitions, such as the Lower Quartile is the 4 𝑡ℎ piece of data with the data
placed in order, find an expression for the Interquartile Range.
11b. [5 marks]
Hence, show that a data set with only 5 numbers in it cannot have any outliers.
11c. [2 marks]
Give an example of a set of data with 7 numbers in it that does have an outlier, justify this
fact by stating the Interquartile Range.
12a. [1 mark]
Anita is concerned that the construction of a new factory will have an adverse affect on the
fish in a nearby lake. Before construction begins she catches fish at random, records their
weight and returns them to the lake. After the construction is finished she collects a second,
random sample of weights of fish from the lake. Her data is shown in the table.
Anita decides to use a t-test, at the 5% significance level, to determine if the mean weight of
the fish changed after construction of the factory.
State an assumption that Anita is making, in order to use a t-test.
12b. [1 mark]
State the hypotheses for this t-test.
12c. [3 marks]
Find the p-value for this t-test.
12d. [2 marks] State the conclusion of this test, in context, giving a reason.
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13a. [5 marks]
In an effort to study the level of intelligence of students entering college, a psychologist
collected data from 4000 students who were given a standard test. The predictive norms
for this particular test were computed from a very large population of scores having a
normal distribution with mean 100 and standard deviation of 10. The psychologist wishes
to determine whether the 4000 test scores he obtained also came from a normal
distribution with mean 100 and standard deviation 10. He prepared the following table
(expected frequencies are rounded to the nearest integer):
Copy and complete the table, showing how you arrived at your answers.
13b. [6 marks]
Test the hypothesis at the 5% level of significance.
14. [9 marks]
Six coins are tossed simultaneously 320 times, with the following results.
At the 5% level of significance, test the hypothesis that all the coins are fair.
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15a. [1 mark]
The Malthouse Charity Run is a 5 kilometre race. The time taken for each runner to
complete the race was recorded. The data was found to be normally distributed with a
mean time of 28 minutes and a standard deviation of 5 minutes.
A runner who completed the race is chosen at random.
Write down the probability that the runner completed the race in more than 28 minutes.
15b. [2 marks]
Calculate the probability that the runner completed the race in less than 26 minutes.
15c. [3 marks]
It is known that 20% of the runners took more than 28 minutes and less than 𝑘 minutes
to complete the race.
Find the value of 𝑘.
16a. [1 mark]
Chicken eggs are classified by grade (4, 5, 6, 7 or 8), based on weight. A mixed
carton contains 12 eggs and could include eggs from any grade. As part of the science
project, Rocky buys 9 mixed cartons and sorts the eggs according to their weight.
State whether the weight of the eggs is a continuous or discrete variable.
16b. [1 mark]
Write down the modal grade of the eggs.
16c. [2 marks]
Use your graphic display calculator to find an estimate for the standard deviation of
the weight of the eggs.
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16d. [2 marks]
The mean weight of these eggs is 64.9 grams, correct to three significant figures.
Use the table and your answer to part (c) to find the smallest possible number of eggs that
could be within one standard deviation of the mean.
17a. [1 mark]
The diagram shows the curve 𝑦 =
𝑥2
2
+
2𝑎
𝑥
, 𝑥 ≠ 0.
The equation of the vertical asymptote of the curve is 𝑥 = 𝑘.
Write down the value of 𝑘.
17b. [3 marks]
d𝑦
Find d𝑥 .
17c. [2 marks]
At the point where 𝑥 = 2, the gradient of the tangent to the curve is 0.5.
Find the value of 𝑎.
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18a. [1 mark]
Stephen was invited to perform a piano recital. In preparation for the event, Stephen
recorded the amount of time, in minutes, that he rehearsed each day for the piano recital.
Stephen rehearsed for 32 days and data for all these days is displayed in the following boxand-whisker diagram.
Write down the median rehearsal time.
18b. [2 marks]
Stephen states that he rehearsed on each of the 32 days.
State whether Stephen is correct. Give a reason for your answer.
18c. [3 marks]
On 𝑘 days, Stephen practiced exactly 24 minutes.
Find the possible values of 𝑘.
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19a. [2 marks]
Galois Airways has flights from Hong Kong International Airport to different
destinations. The following table shows the distance, 𝑥 kilometres, between Hong Kong and
the different destinations and the corresponding airfare, 𝑦, in Hong Kong dollars (HKD).
The Pearson’s product–moment correlation coefficient for this data is 0.948, correct to
three significant figures.
Use your graphic display calculator to find the equation of the regression line 𝑦 on 𝑥.
19b. [2 marks]
The distance from Hong Kong to Tokyo is 2900 km.
Use your regression equation to estimate the cost of a flight from Hong Kong to Tokyo with
Galois Airways.
19c. [2 marks]
Explain why it is valid to use the regression equation to estimate the airfare between Hong
Kong and Tokyo.
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20a. [2 marks]
Sungwon plays a game where she rolls a fair 6-sided die and spins a fair spinner with
4 equal sectors. During each turn in the game, the die is rolled once and the spinner is
spun once. The score for each turn is the sum of the two results. For example, 1 on the die
and 2 on the spinner would receive a score of 3.
The following diagram represents the sample space.
Find the probability that Sungwon’s score on her first turn is greater than 4.
20b. [2 marks]
Sungwon takes a second turn.
Find the probability that Sungwon scores greater than 4 on both of her first two turns.
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20c. [2 marks]
Sungwon will play the game for 11 turns.
Find the expected number of times the score on a turn is greater than 4.
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