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Date November 2017 Marks available 2 Reference code 17N.3sp.hl.TZ0.4
Level HL only Paper Paper 3 Statistics and probability Time zone TZ0
Command term State Question number 4 Adapted from N/A

Question

The random variables \(U,{\text{ }}V\) follow a bivariate normal distribution with product moment correlation coefficient \(\rho \).

A random sample of 12 observations on U, V is obtained to determine whether there is a correlation between U and V. The sample product moment correlation coefficient is denoted by r. A test to determine whether or not UV are independent is carried out at the 1% level of significance.

State suitable hypotheses to investigate whether or not \(U\), \(V\) are independent.

[2]
a.

Find the least value of \(|r|\) for which the test concludes that \(\rho  \ne 0\).

[6]
b.

Markscheme

\({{\text{H}}_0}:\rho  = 0;{\text{ }}{{\text{H}}_1}:\rho  \ne 0\)     A1A1

[2 marks]

a.

\(\nu  = 10\)     (A1)

\({t_{0.005}} = 3.16927 \ldots \)     (M1)(A1)

we reject \({{\text{H}}_0}:\rho  = 0\) if \(\left| t \right| > 3.16927 \ldots \)     (R1)

attempting to solve \(\left| r \right|\sqrt {\frac{{10}}{{1 - {r^2}}}}  > 3.16927 \ldots \) for \(\left| r \right|\)     M1

 

Note:     Allow = instead of >.

 

(least value of \(\left| r \right|\) is) 0.708 (3 sf)     A1

 

Note:     Award A1M1A0R1M1A0 to candidates who use a one-tailed test. Award A0M1A0R1M1A0 to candidates who use an incorrect number of degrees of freedom or both a one-tailed test and incorrect degrees of freedom.

 

Note: Possible errors are

10 DF 1-tail, \(t = 2.763 \ldots \), least value \( = \) 0.658

11 DF 2-tail, \(t = 3.105 \ldots \), least value \( = \) 0.684

11 DF 1-tail, \(t = 2.718 \ldots \), least value \( = \) 0.634.

[6 marks]

b.

Examiners report

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

Syllabus sections

Topic 7 - Option: Statistics and probability » 7.7 » Introduction to bivariate distributions.

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