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

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

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