Edexcel S1 2015 June — Question 4 14 marks

Exam BoardEdexcel
ModuleS1 (Statistics 1)
Year2015
SessionJune
Marks14
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLinear regression
TypeIdentify response/explanatory variables
DifficultyEasy -1.2 This is a standard S1 linear regression question with routine calculations (finding Sxy, regression equation) using given summary statistics, plus straightforward interpretation questions. All steps follow textbook procedures with no problem-solving or novel insight required, making it easier than average A-level material.
Spec5.09a Dependent/independent variables5.09b Least squares regression: concepts5.09c Calculate regression line5.09d Linear coding: effect on regression5.09e Use regression: for estimation in context
  1. Statistical models can provide a cheap and quick way to describe a real world situation.
    1. Give two other reasons why statistical models are used.
    A scientist wants to develop a model to describe the relationship between the average daily temperature, \(x ^ { \circ } \mathrm { C }\), and her household's daily energy consumption, \(y \mathrm { kWh }\), in winter. A random sample of the average daily temperature and her household's daily energy consumption are taken from 10 winter days and shown in the table.
    \(x\)- 0.4- 0.20.30.81.11.41.82.12.52.6
    \(y\)28302625262726242221
    $$\text { [You may use } \sum x ^ { 2 } = 24.76 \quad \sum y = 255 \quad \sum x y = 283.8 \quad \mathrm {~S} _ { x x } = 10.36 \text { ] }$$
  2. Find \(\mathrm { S } _ { x y }\) for these data.
  3. Find the equation of the regression line of \(y\) on \(x\) in the form \(y = a + b x\) Give the value of \(a\) and the value of \(b\) to 3 significant figures.
  4. Give an interpretation of the value of \(a\)
  5. Estimate her household's daily energy consumption when the average daily temperature is \(2 ^ { \circ } \mathrm { C }\) The scientist wants to use the linear regression model to predict her household's energy consumption in the summer.
  6. Discuss the reliability of using this model to predict her household's energy consumption in the summer.
Question 4:
Part (a)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
To simplify (or represent) a real world problemB1g Be fairly generous for a sensible reason; not "quick", "cheap" or "describe"
To improve understanding / To analyse a real world problem or can change variables/replicate easily / To make predictions or find estimatesB1h Slightly harder; both reasons must be from the list; use professional judgement; do not use B0B1
Part (b)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\sum x=12\)B1 May be implied by 3060 seen or next line
\(S_{xy}=283.8-\frac{12\times255}{10}=-\mathbf{22.2}\)M1, A1cao M1 for attempt at correct formula (ft their \(\sum x\) where \(10<\sum x<14\)); A1 for \(-22.2\) only
Part (c)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(b=\frac{'-22.2'}{10.36}=,-2.142857...\) (A1 for awrt \(-2.1\))M1A1 A1 for awrt \(-2.1\) (allow \(-15/7\))
\(\left[a=\bar{y}-b\bar{x}\Rightarrow\right]\ a=\frac{255}{10}-'b'\times\frac{"12"}{10}=28.07143\)M1 M1 for correct expression for \(a\) and ft their 12 (allow use of a letter \(b\))
\(y=28.1-2.14x\) [Condone: \(y=28.1+-2.14x\)]A1 A1 for \(y=28.1-2.14x\) (awrt 28.1 and awrt \(-2.14\)); must be \(y\) and \(x\) and no fractions
Part (d)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
(28.1 kWh) of energy are used when the temperature is \(0[°C]\)B1 B1 for contextualised interpretation; need temp or °sign; [B0 for "value of \(y\) when \(x=0\)"]
Part (e)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(y=28.1-2.14(2)=\) awrt \(\mathbf{23.8}\)M1, A1 M1 for substituting \(x=2\) into their equation
Part (f)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
The regression model is based on temperatures from the winter, so not reliable in the summerB1, dB1 B1 for reasoning that temperatures are different in summer or model based only on winter data; allow mention of extrapolation; dB1 so not reliable
Stating it is reliable (whatever the reason) is B0B0 
# Question 4:

## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| To simplify (or represent) a real world problem | B1g | Be fairly generous for a sensible reason; not "quick", "cheap" or "describe" |
| To improve understanding / To analyse a real world problem or can change variables/replicate easily / To make predictions or find estimates | B1h | Slightly harder; both reasons must be from the list; use professional judgement; do **not** use B0B1 |

## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\sum x=12$ | B1 | May be implied by 3060 seen or next line |
| $S_{xy}=283.8-\frac{12\times255}{10}=-\mathbf{22.2}$ | M1, A1cao | M1 for attempt at correct formula (ft their $\sum x$ where $10<\sum x<14$); A1 for $-22.2$ only |

## Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $b=\frac{'-22.2'}{10.36}=,-2.142857...$ (A1 for awrt $-2.1$) | M1A1 | A1 for awrt $-2.1$ (allow $-15/7$) |
| $\left[a=\bar{y}-b\bar{x}\Rightarrow\right]\ a=\frac{255}{10}-'b'\times\frac{"12"}{10}=28.07143$ | M1 | M1 for correct expression for $a$ and ft their 12 (allow use of a letter $b$) |
| $y=28.1-2.14x$ [Condone: $y=28.1+-2.14x$] | A1 | A1 for $y=28.1-2.14x$ (awrt 28.1 and awrt $-2.14$); must be $y$ and $x$ and no fractions |

## Part (d)
| Answer/Working | Marks | Guidance |
|---|---|---|
| (28.1 kWh) of energy are used when the temperature is $0[°C]$ | B1 | B1 for contextualised interpretation; need temp or °sign; [B0 for "value of $y$ when $x=0$"] |

## Part (e)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y=28.1-2.14(2)=$ awrt $\mathbf{23.8}$ | M1, A1 | M1 for substituting $x=2$ into their equation |

## Part (f)
| Answer/Working | Marks | Guidance |
|---|---|---|
| The regression model is based on temperatures from the winter, so not reliable in the summer | B1, dB1 | B1 for reasoning that temperatures are different in summer or model based only on winter data; allow mention of extrapolation; dB1 so not reliable |
| Stating it **is** reliable (whatever the reason) is B0B0 | | |

---
\begin{enumerate}
  \item Statistical models can provide a cheap and quick way to describe a real world situation.\\
(a) Give two other reasons why statistical models are used.
\end{enumerate}

A scientist wants to develop a model to describe the relationship between the average daily temperature, $x ^ { \circ } \mathrm { C }$, and her household's daily energy consumption, $y \mathrm { kWh }$, in winter.

A random sample of the average daily temperature and her household's daily energy consumption are taken from 10 winter days and shown in the table.

\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | c | }
\hline
$x$ & - 0.4 & - 0.2 & 0.3 & 0.8 & 1.1 & 1.4 & 1.8 & 2.1 & 2.5 & 2.6 \\
\hline
$y$ & 28 & 30 & 26 & 25 & 26 & 27 & 26 & 24 & 22 & 21 \\
\hline
\end{tabular}
\end{center}

$$\text { [You may use } \sum x ^ { 2 } = 24.76 \quad \sum y = 255 \quad \sum x y = 283.8 \quad \mathrm {~S} _ { x x } = 10.36 \text { ] }$$

(b) Find $\mathrm { S } _ { x y }$ for these data.\\
(c) Find the equation of the regression line of $y$ on $x$ in the form $y = a + b x$

Give the value of $a$ and the value of $b$ to 3 significant figures.\\
(d) Give an interpretation of the value of $a$\\
(e) Estimate her household's daily energy consumption when the average daily temperature is $2 ^ { \circ } \mathrm { C }$

The scientist wants to use the linear regression model to predict her household's energy consumption in the summer.\\
(f) Discuss the reliability of using this model to predict her household's energy consumption in the summer.

\hfill \mbox{\textit{Edexcel S1 2015 Q4 [14]}}
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