IB DP Maths Topic 7.7 Topics based on the assumption of bivariate normality: use of the t-statistic to test HL Paper 3

METHOD 1

\({\text{Cov}}(X,{\text{ }}Y) = {\text{E}}\left( {(X – {\mu _X})(Y – {\mu _Y})} \right)\)

\( = {\text{E}}(XY – X{\mu _Y} – Y{\mu _X} + {\mu _X}{\mu _Y})\)     (M1)

\( = {\text{E}}(XY) – {\mu _Y}{\text{E}}(X) – {\mu _X}{\text{E}}(Y) + {\mu _X}{\mu _Y}\)

\( = {\text{E}}(XY) – {\mu _X}{\mu _Y}\)     A1

as \(X\) and \(Y\) are independent \({\text{E}}(XY) = {\mu _X}{\mu _Y}\)     R1

\({\text{Cov}}(X,{\text{ }}Y) = 0\)     AG

METHOD 2

\({\text{Cov}}(X,{\text{ }}Y) = {\text{E}}\left( {(X – {\mu _x})(Y – {\mu _y})} \right)\)

\( = {\text{E}}(X – {\mu _x}){\text{E}}(Y – {\mu _y})\)     (M1)

since \(X,Y\) are independent     R1

\( = ({\mu _x} – {\mu _x})({\mu _y} – {\mu _y})\)     A1

\( = 0\)     AG

[3 marks]

\({H_0}:\rho  = 0\;\;\;{H_1}:\rho  < 0\)     A1

Note:     The hypotheses must be expressed in terms of \(\rho \).

test statistic \({t_{test}} =  – 0.35\sqrt {\frac{{20 – 2}}{{1 – {{( – 0.35)}^2}}}} \)     (M1)(A1)

\( =  – 1.585 \ldots \)     (A1)

\({\text{degrees of freedom}} = 18\)     (A1)

EITHER

\(p{\text{ – value}} = 0.0652\)     A1

this is greater than \(0.05\)     M1

OR

\({t_{5\% }}(18) =  – 1.73\)     A1

this is less than \( – {\text{1.59}}\)     M1

THEN

hence accept \({H_0}\) or reject \({H_1}\) or equivalent or contextual equivalent     R1

Note:     Allow follow through for the final R1 mark.

[8 marks]

Total [11 marks]

Solutions to (a) were often disappointing with few candidates gaining full marks, a common error being failure to state that

\(E(XY) = E(X)E(Y)\) or \({\text{E}}\left( {(X – {\mu _x})(Y – {\mu _y})} \right) = {\text{E}}(X – {\mu _x}){\text{E}}(Y – {\mu _y})\) in the case of independence.

In (b), the hypotheses were sometimes given incorrectly. Some candidates gave \({H_1}\) as \(\rho  \ne 0\), not seeing that a one-tailed test was required. A more serious error was giving the hypotheses as \({H_0}:r = 0,{\text{ }}{H_1}:r < 0\) which shows a complete misunderstanding of the situation. Subsequent parts of the question were well answered in general.