Edexcel S1 2005 January — Question 3 15 marks

Exam BoardEdexcel
ModuleS1 (Statistics 1)
Year2005
SessionJanuary
Marks15
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicBivariate data
TypeDraw scatter diagram from data
DifficultyEasy -1.3 This is a routine S1 statistics question testing standard procedures: plotting a scatter diagram, identifying correlation direction by inspection, and applying memorized formulas for Sxy, regression coefficient b, mean, and standard deviation. All calculations are straightforward substitutions into given formulas with provided summary statistics. No problem-solving insight or novel reasoning required—purely procedural recall and arithmetic.
Spec2.02c Scatter diagrams and regression lines2.02d Informal interpretation of correlation2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.08a Pearson correlation: calculate pmcc5.09b Least squares regression: concepts5.09c Calculate regression line
3. The following table shows the height \(x\), to the nearest cm , and the weight \(y\), to the nearest kg , of a random sample of 12 students.
\(x\)148164156172147184162155182165175152
\(y\)395956774477654980727052
  1. On graph paper, draw a scatter diagram to represent these data.
  2. Write down, with a reason, whether the correlation coefficient between \(x\) and \(y\) is positive or negative. The data in the table can be summarised as follows. $$\Sigma x = 1962 , \quad \Sigma y = 740 , \quad \Sigma y ^ { 2 } = 47746 , \quad \Sigma x y = 122783 , \quad S _ { x x } = 1745 .$$
  3. Find \(S _ { x y }\). The equation of the regression line of \(y\) on \(x\) is \(y = - 106.331 + b x\).
  4. Find, to 3 decimal places, the value of \(b\).
  5. Find, to 3 significant figures, the mean \(\bar { y }\) and the standard deviation \(s\) of the weights of this sample of students.
  6. Find the values of \(\bar { y } \pm 1.96 s\).
  7. Comment on whether or not you think that the weights of these students could be modelled by a normal distribution.
Question 3:
Part (a)
AnswerMarks Guidance
AnswerMarks Guidance
Scales and labels correctB1
10 or 11 points correct, all correctB1, B1
Part (b)
AnswerMarks Guidance
AnswerMarks Guidance
Positive; as \(x\) increases, \(y\) increasesB1; B1
Part (c)
AnswerMarks Guidance
AnswerMarks Guidance
\(S_{xy} = 122783 - \dfrac{1962 \times 740}{12} = 1793\)M1, A1 Use of formula, cao
Part (d)
AnswerMarks Guidance
AnswerMarks Guidance
\(b = \dfrac{S_{xy}}{S_{xx}} = \dfrac{1793}{1745} = 1.027507\ldots\)M1, A1 Division, 1.028
Part (e)
AnswerMarks Guidance
AnswerMarks Guidance
\(\bar{y} = \dfrac{740}{12} = 61\dfrac{2}{3}\)B1 \(61\frac{2}{3}\) or \(61.\dot{6}\) or 61.7
\(s = \sqrt{\dfrac{47746}{12} - \left(\dfrac{740}{12}\right)^2} = 13.26859\ldots\)M1, A1 Use of formula including root, 13.3
Part (f)
AnswerMarks Guidance
AnswerMarks Guidance
35.7, 87.7B1, B1
Part (g)
AnswerMarks Guidance
AnswerMarks Guidance
All values between 35.7 and 87.7 so could be normalB1 Reason required 
# Question 3:

## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Scales and labels correct | B1 | |
| 10 or 11 points correct, all correct | B1, B1 | |

## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Positive; as $x$ increases, $y$ increases | B1; B1 | |

## Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $S_{xy} = 122783 - \dfrac{1962 \times 740}{12} = 1793$ | M1, A1 | Use of formula, cao |

## Part (d)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $b = \dfrac{S_{xy}}{S_{xx}} = \dfrac{1793}{1745} = 1.027507\ldots$ | M1, A1 | Division, 1.028 |

## Part (e)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\bar{y} = \dfrac{740}{12} = 61\dfrac{2}{3}$ | B1 | $61\frac{2}{3}$ or $61.\dot{6}$ or 61.7 |
| $s = \sqrt{\dfrac{47746}{12} - \left(\dfrac{740}{12}\right)^2} = 13.26859\ldots$ | M1, A1 | Use of formula including root, 13.3 |

## Part (f)
| Answer | Marks | Guidance |
|--------|-------|----------|
| 35.7, 87.7 | B1, B1 | |

## Part (g)
| Answer | Marks | Guidance |
|--------|-------|----------|
| All values between 35.7 and 87.7 so could be normal | B1 | Reason required |
3. The following table shows the height $x$, to the nearest cm , and the weight $y$, to the nearest kg , of a random sample of 12 students.

\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | c | c | c | }
\hline
$x$ & 148 & 164 & 156 & 172 & 147 & 184 & 162 & 155 & 182 & 165 & 175 & 152 \\
\hline
$y$ & 39 & 59 & 56 & 77 & 44 & 77 & 65 & 49 & 80 & 72 & 70 & 52 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item On graph paper, draw a scatter diagram to represent these data.
\item Write down, with a reason, whether the correlation coefficient between $x$ and $y$ is positive or negative.

The data in the table can be summarised as follows.

$$\Sigma x = 1962 , \quad \Sigma y = 740 , \quad \Sigma y ^ { 2 } = 47746 , \quad \Sigma x y = 122783 , \quad S _ { x x } = 1745 .$$
\item Find $S _ { x y }$.

The equation of the regression line of $y$ on $x$ is $y = - 106.331 + b x$.
\item Find, to 3 decimal places, the value of $b$.
\item Find, to 3 significant figures, the mean $\bar { y }$ and the standard deviation $s$ of the weights of this sample of students.
\item Find the values of $\bar { y } \pm 1.96 s$.
\item Comment on whether or not you think that the weights of these students could be modelled by a normal distribution.
\end{enumerate}

\hfill \mbox{\textit{Edexcel S1 2005 Q3 [15]}}
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