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Date	November Example question	Marks available	2	Reference code	EXN.2.AHL.TZ0.3
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Justify and State	Question number	3	Adapted from	N/A

Question

Dana has collected some data regarding the heights $h$ (metres) of waves against a pier at $50$ randomly chosen times in a single day. This data is shown in the table below.

She wishes to perform a $χ^{2}$ -test at the $5 %$ significance level to see if the height of waves could be modelled by a normal distribution. Her null hypothesis is

$H_{0} :$ The data can be modelled by a normal distribution.

From the table she calculates the mean of the heights in her sample to be $0.828 m$ and the standard deviation of the heights $s_{n}$ to be $0.257 m$ .

She calculates the expected values for each interval under this null hypothesis, and some of these values are shown in the table below.

Use the given value of $s_{n}$ to find the value of $s_{n - 1}$ .

[2]

a.

Find the value of $a$ and the value of $b$ , giving your answers correct to one decimal place.

[3]

b.

Find the value of the $χ^{2}$ test statistic $(χ_{c a l c}^{2})$ for this test.

[2]

c.

Determine the degrees of freedom for Dana’s test.

[2]

d.

It is given that the critical value for this test is $9.49$ .

State the conclusion of the test in context. Use your answer to part (c) to justify your conclusion.

[2]

e.

Markscheme

* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.

$s_{n - 1} = \sqrt{\frac{50}{49}} \times 0.257$ (M1)

Note: M1 is for the use of the correct formula

$= 0.260$ A1

[2 marks]

a.

Using $\bar{x} = 0.828$ and $s_{n - 1} = 0.260$ (M1)

$a = 7.3, b = 7.6$ A1A1

[3 marks]

b.

$χ_{c a l c}^{2} = 3.35$ (M1)A1

[2 marks]

c.

Combining columns with expected values less than $5$ leaves $7$ cells (M1)

$7 - 1 - 2 = 4$ A1

[2 marks]

d.

$3.35 < 9.49$ R1

hence insufficient evidence to reject $H_{0}$ that the heights of the waves are normally distributed. A1

Note: The A1 can be awarded independently of the R1.

[2 marks]

e.

Examiners report

[N/A]

a.

[N/A]

b.

[N/A]

c.

[N/A]

d.

[N/A]

e.

Syllabus sections

Topic 4—Statistics and probability » SL 4.10—Spearman’s rank correlation coefficient

Show 30 related questions

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EXM.1.SL.TZ0.2a:
Calculate Spearman’s rank correlation coefficient for this data.
22M.2.SL.TZ1.3d.i:
$a$ .
22M.2.SL.TZ1.3d.ii:
$b$ .
22M.2.SL.TZ1.3d.iii:
$c$ .
22M.2.SL.TZ1.3e.ii:
Interpret the value obtained for $r_{s}$ .
22M.2.SL.TZ1.3f:
When calculating the ranks, Chester incorrectly read the Netherlands’ score as $478$ . Explain why the value of the Spearman’s rank correlation $r_{s}$ does not change despite this error.
EXN.2.AHL.TZ0.3c:
Find the value of the $χ^{2}$ test statistic $(χ_{c a l c}^{2})$ for this test.
21M.2.SL.TZ1.1i.i:
Calculate the value of $r_{s}$ .
EXM.1.SL.TZ0.2b:
State what conclusion Kayla can make from the answer in part (a).
EXM.1.SL.TZ0.3d:
State what conclusion Charles can make from the answer in part (c).
EXM.1.SL.TZ0.3a:
Explain why it is not appropriate to use Pearson’s product moment correlation coefficient to measure the strength of the relationship between $P$ and $d$ .
SPM.2.SL.TZ0.3f:
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.
SPM.2.SL.TZ0.3a.ii:
Using the value of $r$ , interpret the relationship between Stan’s score and Minsun’s score.
SPM.2.SL.TZ0.3d:
Copy and complete the information in the following table.
SPM.2.SL.TZ0.3c.ii:
State whether this estimate is reliable. Justify your answer.
SPM.2.SL.TZ0.3e.ii:
Comment on the result obtained for ${r_s}$ .
SPM.2.SL.TZ0.3c.i:
Use your regression equation from part (b) to estimate Minsun’s score when Stan awards a perfect 10.
SPM.2.SL.TZ0.3e.i:
Find the value of the Spearman’s rank correlation coefficient, ${r_s}$ .
EXM.1.SL.TZ0.3c:
Calculate Spearman’s rank correlation coefficient for this data.
SPM.2.SL.TZ0.3b:
Write down the equation of the regression line $y$ on $x$ .
21M.2.SL.TZ1.1i.ii:
Interpret your result.
SPM.2.SL.TZ0.3a.i:
Write down the value of the Pearson’s product–moment correlation coefficient, $r$ .
EXM.1.SL.TZ0.3b:
Explain why it is appropriate to use Spearman’s rank correlation coefficient to measure the strength of the relationship between $P$ and $d$ .
EXN.2.SL.TZ0.1c:
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.
21M.2.SL.TZ1.1h:
Copy and complete the information in the following table.
21M.3.AHL.TZ1.2c.i:
Without calculation, explain why it might not be appropriate to calculate a correlation coefficient for the whole sample of $18$ employees.
21M.3.AHL.TZ1.2c.ii:
Find $r_{s}$ for the seven employees working in the international department.
21M.3.AHL.TZ1.2c.iii:
Hence comment on the validity of the written assessment as a measure of the level of performance of employees in this department. Justify your answer.
EXN.2.AHL.TZ0.3d:
Determine the degrees of freedom for Dana’s test.

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Topic 4—Statistics and probability