17N.3.AHL.TZ0.HSP_4

pestleMathematicsAIHLPaper 317N· ahl-4-18-t-and-z-test-type-i-and-ii-errorssource ↗

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 U, V are independent is carried out at the 1% level of significance.

State suitable hypotheses to investigate whether or not $U$ , $V$ are independent.

[2]

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

[6]

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

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

[2 marks]

$\nu = 10$ (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]

Examiners’ report

[N/A]