| 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 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.
Find the least value of \(|r|\) for which the test concludes that \(\rho \ne 0\).
Markscheme
\({{\text{H}}_0}:\rho = 0;{\text{ }}{{\text{H}}_1}:\rho \ne 0\) 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]
