DP1 Practise for TEA-2 [119 marks]
1.
[Maximum mark: 6]
SPM.1.SL.TZ0.13
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.
(a)
Find the probability that on any given day Mr Burke chooses a
female student to answer a question.
[1]
In the first month, Mr Burke will teach his class 20 times.
(b)
Find the probability he will choose a female student 8 times.
[2]
(c)
Find the probability he will choose a male student at most 9
times.
[3]
2.
[Maximum mark: 5]
SPM.1.SL.TZ0.6
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.
(a)
State the null hypothesis.
[1]
(b)
Calculate the p-value for this test.
[2]
(c)
State, giving a reason, whether the null hypothesis should be
accepted.
[2]
3.
[Maximum mark: 6]
SPM.1.SL.TZ0.9
Ms Calhoun measures the heights of students in her mathematics class. She is
interested to see if the mean height of male students, μ 1 , is the same as the
mean height of female students, μ 2 . 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.
(a.i)
State the null hypothesis.
[1]
(a.ii)
State the alternative hypothesis.
[1]
(b)
Calculate the p-value for this test.
[2]
(c)
State, giving a reason, whether Ms Calhoun should accept the
null hypothesis.
[2]
4.
[Maximum mark: 17]
SPM.2.SL.TZ0.3
The Malvern Aquatic Center hosted a 3 metre spring board diving event. The
judges, Stan and Minsun awarded 8 competitors a score out of 10. The raw data is
collated in the following table.
(a.i)
Write down the value of the Pearson’s product–moment
correlation coefficient, r.
[2]
Using the value of r, interpret the relationship between Stan’s
score and Minsun’s score.
[2]
(b)
Write down the equation of the regression line y on x.
[2]
(c.i)
Use your regression equation from part (b) to estimate Minsun’s
score when Stan awards a perfect 10.
[2]
State whether this estimate is reliable. Justify your answer.
[2]
(a.ii)
(c.ii)
The Commissioner for the event would like to find the Spearman’s rank
correlation coefficient.
(d)
Copy and complete the information in the following table.
[2]
(e.i)
Find the value of the Spearman’s rank correlation coefficient, r s
.
[2]
(e.ii)
Comment on the result obtained for r s .
[2]
(f )
The Commissioner believes Minsun’s score for competitor G is
too high and so decreases the score from 9.5 to 9.1.
Explain why the value of the Spearman’s rank correlation
coefficient r s does not change.
5.
[1]
[Maximum mark: 6]
EXN.1.SL.TZ0.3
The weights of apples on a tree can be modelled by a normal distribution with a
mean of 85 grams and a standard deviation of 7. 5 grams.
(a)
Find the probability that an apple from the tree has a weight
greater than 90 grams.
[2]
A sample of apples are taken from 2 trees, A and B, in different parts of the
orchard.
The data is shown in the table below.
The owner of the orchard wants to know whether the mean weight of the apples
from tree A(μ A ) is greater than the mean weight of the apples from tree
B(μ B ) so sets up the following test:
H 0 : μ A = μ B and H 1 : μ A > μ B
(b.i)
Find the p-value for the owner’s test.
(b.ii)
The test is performed at the 5% significance level.
State the conclusion of the test, giving a reason for your answer.
[2]
[2]
6.
[Maximum mark: 16]
EXN.2.SL.TZ0.4
A dice manufacturer claims that for a novelty die he produces the probability of
scoring the numbers 1 to 5 are all equal, and the probability of a 6 is two times
the probability of scoring any of the other numbers.
(a)
(b)
Find the probability of scoring a six when rolling the novelty
die.
[3]
Find the probability of scoring more than 2 sixes when this die
is rolled 5 times.
[4]
To test the manufacture’s claim one of the novelty dice is rolled 350 times and
the numbers scored on the die are shown in the table below.
(c.i)
Find the expected frequency for each of the numbers if the
manufacturer’s claim is true.
[2]
A χ 2 goodness of fit test is to be used with a 5% significance level.
(c.ii)
Write down the null and alternative hypotheses.
[2]
(c.iii)
State the degrees of freedom for the test.
[1]
(c.iv)
Determine the conclusion of the test, clearly justifying your
answer.
[4]
7.
8.
[Maximum mark: 2]
Which of the correlation coefficients would you recommend is used to
assess whether or not there is an association between total number of
minutes late and distance from school? Fully justify your answer.
EXN.2.SL.TZ0.c
[Maximum mark: 9]
Six coins are tossed simultaneously 320 times, with the following
results.
EXM.1.SL.TZ0.9
At the 5% level of significance, test the hypothesis that all the coins are
fair.
[2]
[9]
9.
[Maximum mark: 11]
EXM.1.SL.TZ0.8
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):
(a)
(b)
Copy and complete the table, showing how you arrived at your
answers.
[5]
Test the hypothesis at the 5% level of significance.
[6]
10.
[Maximum mark: 9]
EXM.1.SL.TZ0.3
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).
(a)
(b)
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.
[1]
Explain why it is appropriate to use Spearman’s rank correlation
coefficient to measure the strength of the relationship between
P and d.
[1]
The data from the graph is shown in the table.
(c)
Calculate Spearman’s rank correlation coefficient for this data.
(d)
State what conclusion Charles can make from the answer in
part (c).
[6]
[1]
11.
[Maximum mark: 6]
EXM.1.SL.TZ0.2
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.
(a)
Calculate Spearman’s rank correlation coefficient for this data.
[5]
(b)
State what conclusion Kayla can make from the answer in part
(a).
[1]
12.
[Maximum mark: 7]
EXM.1.SL.TZ0.7
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.
(a)
State an assumption that Anita is making, in order to use a ttest.
[1]
(b)
State the hypotheses for this t-test.
[1]
(c)
Find the p-value for this t-test.
[3]
(d)
State the conclusion of this test, in context, giving a reason.
[2]
13.
[Maximum mark: 9]
EXM.1.SL.TZ0.11
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.
14.
(a)
Test this claim at the 5% level of significance.
[7]
(b)
Explain what is meant by the 5% level of significance.
[2]
[Maximum mark: 10]
Give your answers to four significant figures.
EXM.1.SL.TZ0.10
A die is thrown 120 times with the following results.
(a)
Showing all steps clearly, test whether the die is fair
(i) at the 5% level of significance;
(b)
(ii) at the 1% level of significance.
[7]
Explain what is meant by “level of significance” in part (a).
[3]
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