HL Paper 3
The random variables X , Y follow a bivariate normal distribution with product moment correlation coefficient ρ.
A random sample of 11 observations on X, Y was obtained and the value of the sample product moment correlation coefficient, r, was calculated to be −0.708.
The covariance of the random variables U, V is defined by
Cov(U, V) = E((U − E(U))(V − E(V))).
State suitable hypotheses to investigate whether or not a negative linear association exists between X and Y.
Determine the p-value.
State your conclusion at the 1 % significance level.
Show that Cov(U, V) = E(UV) − E(U)E(V).
Hence show that if U, V are independent random variables then the population product moment correlation coefficient, ρ, is zero.
Peter, the Principal of a college, believes that there is an association between the score in a Mathematics test, , and the time taken to run 500 m, seconds, of his students. The following paired data are collected.
It can be assumed that follow a bivariate normal distribution with product moment correlation coefficient .
State suitable hypotheses and to test Peter’s claim, using a two-tailed test.
Carry out a suitable test at the 5 % significance level. With reference to the -value, state your conclusion in the context of Peter’s claim.
Peter uses the regression line of on as and calculates that a student with a Mathematics test score of 73 will have a running time of 101 seconds. Comment on the validity of his calculation.