| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2018 |
| Session | December |
| Marks | 10 |
| Topic | Bivariate data |
| Type | Calculate r from raw bivariate data |
| Difficulty | Moderate -0.3 This is a straightforward Further Maths Statistics question requiring standard application of the correlation coefficient formula with all summations provided. Parts (b)-(e) test basic conceptual understanding of correlation properties. While it's a multi-part question worth several marks, it involves routine calculations and standard interpretations with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation5.09a Dependent/independent variables5.09c Calculate regression line |
| Country | \(x\) | \(y\) | Country | \(x\) | \(y\) |
| Benin | 4.8 | 59.2 | Mozambique | 5.4 | 54.63 |
| Cameroon | 4.7 | 54.87 | Nigeria | 5.7 | 52.29 |
| Congo | 4.9 | 61.42 | Senegal | 5.1 | 65.81 |
| Gambia | 5.7 | 59.83 | Somalia | 6.5 | 54.88 |
| Liberia | 4.7 | 60.25 | Sudan | 4.4 | 63.08 |
| Malawi | 5.1 | 60.97 | Uganda | 5.8 | 57.25 |
| Mauretania | 4.6 | 62.77 | Zambia | 5.4 | 58.75 |
| Answer | Marks |
|---|---|
| B2 | art -0.554; SC: if B0, allow M1 for correct method with \(S_{xx}, S_{yy}, S_{xy}\) etc |
| Answer | Marks |
|---|---|
| B1 | "Unchanged", same value written out, etc |
| Answer | Marks |
|---|---|
| B1 | oe |
| Answer | Marks |
|---|---|
| B2, B1, M1ft, A1ft, A1ft | One error or omission, e.g. no symbol, <, ρ not defined: B1; Ignore sign for this mark; Correct first conclusion, needs consistent sign; In context, not too definite [not: there is association...]; Or Zimbabwe different, etc |
| Answer | Marks |
|---|---|
| B1 | Or Zimbabwe different, etc |
## (a)
$r = -0.55397$
| B2 | art -0.554; SC: if B0, allow M1 for correct method with $S_{xx}, S_{yy}, S_{xy}$ etc |
## (b)
None
| B1 | "Unchanged", same value written out, etc |
## (c)
Increasing birth rate is associated with lower life expectancy
| B1 | oe |
## (d)
$H_0: \rho = 0$
$H_1: \rho \neq 0$ where $\rho$ is the population pmcc
CV ± 0.5324
$-0.55397 < -0.5324$ so reject $H_0$
There is significant evidence of association between birth rate and life expectancy
No as 3.9 is outside the data range
| B2, B1, M1ft, A1ft, A1ft | One error or omission, e.g. no symbol, <, ρ not defined: B1; Ignore sign for this mark; Correct first conclusion, needs consistent sign; In context, not too definite [not: there is association...]; Or Zimbabwe different, etc |
## (e)
| B1 | Or Zimbabwe different, etc |
---5 The birth rate, $x$ per thousand members of the population, and the life expectancy at birth, $y$ years, in 14 randomly selected African countries are given in the table.
\begin{center}
\begin{tabular}{ | l l l | l l l | }
\hline
Country & $x$ & $y$ & Country & $x$ & $y$ \\
\hline
Benin & 4.8 & 59.2 & Mozambique & 5.4 & 54.63 \\
Cameroon & 4.7 & 54.87 & Nigeria & 5.7 & 52.29 \\
Congo & 4.9 & 61.42 & Senegal & 5.1 & 65.81 \\
Gambia & 5.7 & 59.83 & Somalia & 6.5 & 54.88 \\
Liberia & 4.7 & 60.25 & Sudan & 4.4 & 63.08 \\
Malawi & 5.1 & 60.97 & Uganda & 5.8 & 57.25 \\
Mauretania & 4.6 & 62.77 & Zambia & 5.4 & 58.75 \\
\hline
\end{tabular}
\end{center}
$n = 14 , \sum x = 72.8 , \sum y = 826 , \sum x ^ { 2 } = 392.96 , \sum y ^ { 2 } = 48924.54 , \sum x y = 4279.16$
\begin{enumerate}[label=(\alph*)]
\item Calculate Pearson's product-moment correlation coefficient $r$ for the data.
\item State what would be the effect on the value of $r$ if the birth rate were given per hundred and not per thousand.
\item Explain what the sign of $r$ tells you about the relationship between life expectancy and birth rate for these countries.
\item Test at the $5 \%$ significance level whether there is correlation between birth rate and life expectancy at birth in African countries.
\item A researcher wants to estimate the life expectancy at birth in Zimbabwe, where the birth rate is 3.9 per thousand. Explain whether a reliable estimate could be obtained using the regression line of $y$ on $x$ for the given data.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics 2018 Q5 [10]}}