| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2002 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Calculate y on x from raw data table |
| Difficulty | Moderate -0.8 This is a standard S1 linear regression question requiring routine application of formulae for correlation coefficient and regression line from given summary statistics. All calculations follow textbook procedures with no problem-solving insight needed, making it easier than average but not trivial due to the multi-step arithmetic involved. |
| Spec | 2.02c Scatter diagrams and regression lines2.02d Informal interpretation of correlation5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation5.09b Least squares regression: concepts5.09c Calculate regression line5.09e Use regression: for estimation in context |
| \(t\) | \(c\) |
| 13 | 24 |
| 22 | 55 |
| 17 | 35 |
| 20 | 45 |
| 10 | 20 |
| 15 | 30 |
| 19 | 39 |
| 12 | 19 |
| 18 | 36 |
| 23 | 54 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\Sigma t = 169;\ \Sigma c = 357\) | ||
| \(S_{cc} = 14245 - \dfrac{357^2}{10} = 1500.1\) | M1, A1 | |
| \(S_{tt} = 168.9,\ S_{ct} = 492.7\) | A1, A1 | |
| \(r = \dfrac{492.7}{\sqrt{1500.1 \times 168.9}} = 0.97883\ldots\) accept \(0.979\) | M1, A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Since \(r\) close to 1, value supports use of regression line | B1, B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(b = \dfrac{S_{ct}}{S_{tt}} = \dfrac{492.7}{168.9} = 2.91711\ldots\) | B1 | |
| \(a = \bar{c} - b\bar{t} = \dfrac{357}{10} - \dfrac{492.7}{168.9} \times \dfrac{169}{10} = -13.59917\ldots\) | B1 | |
| \(c = -13.6 + 2.92t\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| 3 extra ice-creams are sold for every \(1\ ^\circ\text{C}\) increase in temperature | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(c = -13.6 + 2.92 \times 16 = 33.12\), i.e. 33 ice-creams | M1, A1, A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Temperature likely to be outside range of validity | B1 |
## Question 7:
### Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\Sigma t = 169;\ \Sigma c = 357$ | | |
| $S_{cc} = 14245 - \dfrac{357^2}{10} = 1500.1$ | M1, A1 | |
| $S_{tt} = 168.9,\ S_{ct} = 492.7$ | A1, A1 | |
| $r = \dfrac{492.7}{\sqrt{1500.1 \times 168.9}} = 0.97883\ldots$ accept $0.979$ | M1, A1 | |
### Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Since $r$ close to 1, value supports use of regression line | B1, B1 | |
### Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $b = \dfrac{S_{ct}}{S_{tt}} = \dfrac{492.7}{168.9} = 2.91711\ldots$ | B1 | |
| $a = \bar{c} - b\bar{t} = \dfrac{357}{10} - \dfrac{492.7}{168.9} \times \dfrac{169}{10} = -13.59917\ldots$ | B1 | |
| $c = -13.6 + 2.92t$ | B1 | |
### Part (d)
| Answer | Marks | Guidance |
|--------|-------|----------|
| 3 extra ice-creams are sold for every $1\ ^\circ\text{C}$ increase in temperature | B1 | |
### Part (e)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $c = -13.6 + 2.92 \times 16 = 33.12$, i.e. 33 ice-creams | M1, A1, A1 | |
### Part (f)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Temperature likely to be outside range of validity | B1 | |7. An ice cream seller believes that there is a relationship between the temperature on a summer day and the number of ice creams sold. Over a period of 10 days he records the temperature at 1 p.m., $t ^ { \circ } \mathrm { C }$, and the number of ice creams sold, $c$, in the next hour. The data he collects is summarised in the table below.
\begin{center}
\begin{tabular}{|l|l|}
\hline
$t$ & $c$ \\
\hline
13 & 24 \\
\hline
22 & 55 \\
\hline
17 & 35 \\
\hline
20 & 45 \\
\hline
10 & 20 \\
\hline
15 & 30 \\
\hline
19 & 39 \\
\hline
12 & 19 \\
\hline
18 & 36 \\
\hline
23 & 54 \\
\hline
\end{tabular}
\end{center}
[Use $\left. \Sigma t ^ { 2 } = 3025 , \Sigma c ^ { 2 } = 14245 , \Sigma c t = 6526 .\right]$
\begin{enumerate}[label=(\alph*)]
\item Calculate the value of the product moment correlation coefficient between $t$ and $c$.
\item State whether or not your value supports the use of a regression equation to predict the number of ice creams sold. Give a reason for your answer.
\item Find the equation of the least squares regression line of $c$ on $t$ in the form $c = a + b t$.
\item Interpret the value of $b$.
\item Estimate the number of ice creams sold between 1 p.m. and 2 p.m. when the temperature at 1 p.m. is $16 ^ { \circ } \mathrm { C }$.\\
(3)
\item At 1 p.m. on a particular day, the highest temperature for 50 years was recorded. Give a reason why you should not use the regression equation to predict ice cream sales on that day.\\
(1)
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2002 Q7 [16]}}