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Date November 2016 Marks available 6 Reference code 16N.3sp.hl.TZ0.1
Level HL only Paper Paper 3 Statistics and probability Time zone TZ0
Command term Find and State Question number 1 Adapted from N/A

Question

In this question you may assume that these data are a random sample from a bivariate normal distribution, with population product moment correlation coefficient \(\rho \).

Richard wishes to do some research on two types of exams which are taken by a large number of students. He takes a random sample of the results of 10 students, which are shown in the following table.

N16/5/MATHL/HP3/ENG/TZ0/SP/01

Using these data, it is decided to test, at the 1% level, the null hypothesis \({H_0}:\rho  = 0\) against the alternative hypothesis \({H_1}:\rho  > 0\).

Richard decides to take the exams himself. He scored 11 on Exam 1 but his result on Exam 2 was lost.

Caroline believes that the population mean mark on Exam 2 is 6 marks higher than the population mean mark on Exam 1. Using the original data from the 10 students, it is decided to test, at the 5% level, this hypothesis against the alternative hypothesis that the mean of the differences, \({\text{d}} = {\text{exam 2 mark }} - {\text{ exam 1 mark}}\), is less than 6 marks.

For these data find the product moment correlation coefficient, \(r\).

[2]
a.

(i)     State the distribution of the test statistic (including any parameters).

(ii)     Find the \(p\)-value for the test.

(iii)     State the conclusion, in the context of the question, with the word “correlation” in your answer. Justify your answer.

[6]
b.

Using a suitable regression line, find an estimate for his score on Exam 2, giving your answer to the nearest integer.

[3]
c.

(i)     State the distribution of your test statistic (including any parameters).

(ii)     Find the \(p\)-value.

(iii)     State the conclusion, justifying the answer.

[6]
d.

Markscheme

\(r = 0.804\)    A2

 

Note: Accept any number that rounds to 0.80.

 

[2 marks]

a.

(i)     \(t\) distribution with 8 degrees of freedom     A1A1

(ii)     \(p{\text{ - value}} = 0.00254\)     A2

 

Notes: Accept any number that rounds to 0.0025.

Award A1 for 2-tail test giving an answer that rounds to 0.0051.

 

(iii)     \(p{\text{ - value}} < 0.01\), so conclude that there is positive correlation     R1A1

 

Notes: Only award the A1 if the R1 is awarded.

Do not accept just “reject \({H_0}\)” or “accept \({H_1}\)”.

The words “positive correlation” must be seen.

 

[6 marks]

b.

regression line of \(y\) (Exam 2 mark) on \(x\) (Exam 1 mark) is     (M1)

\(y = 0.59407 \ldots x + 21.387 \ldots \)    (A1)

\(x = 11\) gives \(y = 28\) (to nearest integer)     A1

[3 marks]

c.

(i)     applying the \(t\) test to the differences

\(t\) distribution with 9 degrees of freedom     A1A1

(ii)     \(p{\text{ - value}} = 0.239\)     A2

 

Notes: Accept any number that rounds to 0.24.

Award A1 if subtraction done the wrong way round giving \(p{\text{ - value}} = 0.109\).

 

(iii)     \(p{\text{ - value}} > 0.05\), so accept \({H_0}\) or \({u_d} = 6\)     R1A1

[6 marks]

d.

Examiners report

[N/A]
a.
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b.
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c.
[N/A]
d.

Syllabus sections

Topic 7 - Option: Statistics and probability » 7.6 » Testing hypotheses for the mean of a normal population.

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