• Syllabus

7.7

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Sub sections and their related questions

Introduction to bivariate distributions.

• SPNone.3sp.hl.TZ0.4a: State suitable hypotheses.
• SPNone.3sp.hl.TZ0.4b: The marks obtained by the 12 students who sat both papers are given in the following...
• SPNone.3sp.hl.TZ0.4c: George obtained a mark of 63 on Paper 1 but was unable to sit Paper 2 because of illness. Predict...
• SPNone.3sp.hl.TZ0.4d: Another class of 16 students sat examinations in Physics and Chemistry and the product moment...
• 14M.3sp.hl.TZ0.2: The following table gives the average yield of olives per tree, in kg, and the rainfall, in cm,...
• 17N.3sp.hl.TZ0.4a: State suitable hypotheses to investigate whether or not $U$, $V$ are independent.
• 17N.3sp.hl.TZ0.4b: Find the least value of $|r|$ for which the test concludes that $\rho  \ne 0$.
• 18M.3sp.hl.TZ0.4a: State suitable hypotheses to investigate whether or not a negative linear association exists...
• 18M.3sp.hl.TZ0.4b.i: Determine the p-value.
• 18M.3sp.hl.TZ0.4b.ii: State your conclusion at the 1 % significance level.
• 18M.3sp.hl.TZ0.4c.i: Show that Cov(U, V) = E(UV) − E(U)E(V).
• 18M.3sp.hl.TZ0.4c.ii: Hence show that if U, V are independent random variables then the population product moment...

Covariance and (population) product moment correlation coefficient $\rho$.

• 14N.3sp.hl.TZ0.3a: If $X$ and $Y$ are two random variables such that ${\text{E}}(X) = {\mu _X}$ and...
• 18M.3sp.hl.TZ0.4a: State suitable hypotheses to investigate whether or not a negative linear association exists...
• 18M.3sp.hl.TZ0.4b.i: Determine the p-value.
• 18M.3sp.hl.TZ0.4b.ii: State your conclusion at the 1 % significance level.
• 18M.3sp.hl.TZ0.4c.i: Show that Cov(U, V) = E(UV) − E(U)E(V).
• 18M.3sp.hl.TZ0.4c.ii: Hence show that if U, V are independent random variables then the population product moment...

Proof that $\rho = 0$ in the case of independence and $\pm 1$ in the case of a linear relationship between $X$ and $Y$.

• 14N.3sp.hl.TZ0.3a: If $X$ and $Y$ are two random variables such that ${\text{E}}(X) = {\mu _X}$ and...
• 18M.3sp.hl.TZ0.4a: State suitable hypotheses to investigate whether or not a negative linear association exists...
• 18M.3sp.hl.TZ0.4b.i: Determine the p-value.
• 18M.3sp.hl.TZ0.4b.ii: State your conclusion at the 1 % significance level.
• 18M.3sp.hl.TZ0.4c.i: Show that Cov(U, V) = E(UV) − E(U)E(V).
• 18M.3sp.hl.TZ0.4c.ii: Hence show that if U, V are independent random variables then the population product moment...

Definition of the (sample) product moment correlation coefficient $R$ in terms of n paired observations on $X$ and $Y$. Its application to the estimation of $\rho$.

• 15N.3sp.hl.TZ0.2a: Determine the product moment correlation coefficient for these data.
• 18M.3sp.hl.TZ0.4a: State suitable hypotheses to investigate whether or not a negative linear association exists...
• 18M.3sp.hl.TZ0.4b.i: Determine the p-value.
• 18M.3sp.hl.TZ0.4b.ii: State your conclusion at the 1 % significance level.
• 18M.3sp.hl.TZ0.4c.i: Show that Cov(U, V) = E(UV) − E(U)E(V).
• 18M.3sp.hl.TZ0.4c.ii: Hence show that if U, V are independent random variables then the population product moment...

Informal interpretation of $r$, the observed value of $R$. Scatter diagrams.

• 18M.3sp.hl.TZ0.4a: State suitable hypotheses to investigate whether or not a negative linear association exists...
• 18M.3sp.hl.TZ0.4b.i: Determine the p-value.
• 18M.3sp.hl.TZ0.4b.ii: State your conclusion at the 1 % significance level.
• 18M.3sp.hl.TZ0.4c.i: Show that Cov(U, V) = E(UV) − E(U)E(V).
• 18M.3sp.hl.TZ0.4c.ii: Hence show that if U, V are independent random variables then the population product moment...

Topics based on the assumption of bivariate normality: use of the $t$-statistic to test the null hypothesis $\rho = 0$ .

• 14N.3sp.hl.TZ0.3b: In a particular company, it is claimed that the distance travelled by employees to work is...
• 17N.3sp.hl.TZ0.4a: State suitable hypotheses to investigate whether or not $U$, $V$ are independent.
• 17N.3sp.hl.TZ0.4b: Find the least value of $|r|$ for which the test concludes that $\rho  \ne 0$.
• 18M.3sp.hl.TZ0.4a: State suitable hypotheses to investigate whether or not a negative linear association exists...
• 18M.3sp.hl.TZ0.4b.i: Determine the p-value.
• 18M.3sp.hl.TZ0.4b.ii: State your conclusion at the 1 % significance level.
• 18M.3sp.hl.TZ0.4c.i: Show that Cov(U, V) = E(UV) − E(U)E(V).
• 18M.3sp.hl.TZ0.4c.ii: Hence show that if U, V are independent random variables then the population product moment...

Topics based on the assumption of bivariate normality: knowledge of the facts that the regression of $X$ on $Y$ (${E\left. {\left( X \right)} \right|Y = y}$) and $Y$ on $X$ (${E\left. {\left( Y \right)} \right|X = x}$) are linear.

• 17N.3sp.hl.TZ0.4a: State suitable hypotheses to investigate whether or not $U$, $V$ are independent.
• 17N.3sp.hl.TZ0.4b: Find the least value of $|r|$ for which the test concludes that $\rho  \ne 0$.
• 18M.3sp.hl.TZ0.4a: State suitable hypotheses to investigate whether or not a negative linear association exists...
• 18M.3sp.hl.TZ0.4b.i: Determine the p-value.
• 18M.3sp.hl.TZ0.4b.ii: State your conclusion at the 1 % significance level.
• 18M.3sp.hl.TZ0.4c.i: Show that Cov(U, V) = E(UV) − E(U)E(V).
• 18M.3sp.hl.TZ0.4c.ii: Hence show that if U, V are independent random variables then the population product moment...

Topics based on the assumption of bivariate normality: least-squares estimates of these regression lines (proof not required).

• 15N.3sp.hl.TZ0.2c: If the moisture content of a beam is found to be $9.5$, use the appropriate regression line to...
• 17N.3sp.hl.TZ0.4a: State suitable hypotheses to investigate whether or not $U$, $V$ are independent.
• 17N.3sp.hl.TZ0.4b: Find the least value of $|r|$ for which the test concludes that $\rho  \ne 0$.
• 18M.3sp.hl.TZ0.4a: State suitable hypotheses to investigate whether or not a negative linear association exists...
• 18M.3sp.hl.TZ0.4b.i: Determine the p-value.
• 18M.3sp.hl.TZ0.4b.ii: State your conclusion at the 1 % significance level.
• 18M.3sp.hl.TZ0.4c.i: Show that Cov(U, V) = E(UV) − E(U)E(V).
• 18M.3sp.hl.TZ0.4c.ii: Hence show that if U, V are independent random variables then the population product moment...

Topics based on the assumption of bivariate normality: the use of these regression lines to predict the value of one of the variables given the value of the other.

• 15N.3sp.hl.TZ0.2c: If the moisture content of a beam is found to be $9.5$, use the appropriate regression line to...
• 17N.3sp.hl.TZ0.4a: State suitable hypotheses to investigate whether or not $U$, $V$ are independent.
• 17N.3sp.hl.TZ0.4b: Find the least value of $|r|$ for which the test concludes that $\rho  \ne 0$.
• 18M.3sp.hl.TZ0.4a: State suitable hypotheses to investigate whether or not a negative linear association exists...
• 18M.3sp.hl.TZ0.4b.i: Determine the p-value.
• 18M.3sp.hl.TZ0.4b.ii: State your conclusion at the 1 % significance level.
• 18M.3sp.hl.TZ0.4c.i: Show that Cov(U, V) = E(UV) − E(U)E(V).
• 18M.3sp.hl.TZ0.4c.ii: Hence show that if U, V are independent random variables then the population product moment...