Edexcel AS Paper 2 2020 June — Question 4 7 marks

Exam BoardEdexcel
ModuleAS Paper 2 (AS Paper 2)
Year2020
SessionJune
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicMeasures of Location and Spread
TypeDescribing sampling methods
DifficultyEasy -1.2 This is a straightforward AS-level statistics question testing basic understanding of sampling methods and standard deviation calculations. Part (a) requires recognizing that stratified sampling needs whole numbers, part (b) is naming quota sampling, part (c) is a standard formula application with given summations, and part (d) tests conceptual understanding of how outliers affect spread. All parts are routine recall or simple reasoning with no novel problem-solving required.
Spec2.01d Select/critique sampling: in context2.02g Calculate mean and standard deviation
  1. A lake contains three different types of carp.
There are an estimated 450 mirror carp, 300 leather carp and 850 common carp.
Tim wishes to investigate the health of the fish in the lake.
He decides to take a sample of 160 fish.
  1. Give a reason why stratified random sampling cannot be used.
  2. Explain how a sample of size 160 could be taken to ensure that the estimated populations of each type of carp are fairly represented. You should state the name of the sampling method used. As part of the health check, Tim weighed the fish.
    His results are given in the table below.
    Weight (wkg)Frequency (f)Midpoint (m kg)
    \(2 \leqslant w < 3.5\)82.75
    \(3.5 \leqslant w < 4\)323.75
    \(4 \leqslant w < 4.5\)644.25
    \(4.5 \leqslant w < 5\)404.75
    \(5 \leqslant w < 6\)165.5
    $$\left( \text { You may use } \sum \mathrm { f } m = 692 \quad \text { and } \quad \sum \mathrm { f } m ^ { 2 } = 3053 \right)$$
  3. Calculate an estimate for the standard deviation of the weight of the carp. Tim realised that he had transposed the figures for 2 of the weights of the fish.
    He had recorded in the table 2.3 instead of 3.2 and 4.6 instead of 6.4
  4. Without calculating a new estimate for the standard deviation, state what effect
    1. using the correct figure of 3.2 instead of 2.3
    2. using the correct figure of 6.4 instead of 4.6
      would have on your estimated standard deviation.
      Give a reason for each of your answers.
Question 4:
Part 4(a):
AnswerMarks Guidance
AnswerMark Guidance
It is not possible to have a sampling frame/listB1 For the idea there cannot be a sampling frame/list
Part 4(b):
AnswerMarks Guidance
AnswerMark Guidance
Quota sampling and (catch 85 common carp, 45 mirror carp and 30 leather carp) or (ignore any fish caught of a type where the quota is full)M1 Quota sampling and either correct numbers of each type or idea that if quota full ignore the fish
Quota sampling and catch 85 common carp, 45 mirror carp and 30 leather carp and ignore any fish caught of a type where the quota is fullA1 Quota sampling and both correct numbers and idea that if quota full ignore the fish, or sample until all quotas are full
Part 4(c):
AnswerMarks Guidance
AnswerMark Guidance
\(\sigma = \sqrt{\dfrac{3053}{160} - \left(\dfrac{692}{160}\right)^2}\)M1 A correct expression for \(\sigma\)
\(= 0.6129\ldots\) awrt \(0.613\)A1 awrt \(0.613\); allow \(s =\) awrt \(0.615\)
Part 4(d)(i):
AnswerMarks Guidance
AnswerMark Guidance
This would have no effect as the piece of data would remain in the same classB1 Correct deduction with suitable explanation. Allow range for class. Do not allow "there is no differences"
Part 4(d)(ii):
AnswerMarks Guidance
AnswerMark Guidance
This would increase the standard deviation as change in mean is small and \(6.4 - 4.6 \approx 3\sigma\), therefore estimate of standard deviation will increaseB1 Correct deduction with suitable explanation; so would increase the standard deviation and a suitable reason. Allow "the value is bigger than any others in the table" oe 
# Question 4:

## Part 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| It is not possible to have a sampling frame/list | B1 | For the idea there cannot be a sampling frame/list |

## Part 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Quota sampling **and** (catch 85 common carp, 45 mirror carp and 30 leather carp) **or** (ignore any fish caught of a type where the quota is full) | M1 | Quota sampling **and** either correct numbers of each type **or** idea that if quota full ignore the fish |
| Quota sampling **and** catch 85 common carp, 45 mirror carp and 30 leather carp **and** ignore any fish caught of a type where the quota is full | A1 | Quota sampling **and** both correct numbers **and** idea that if quota full ignore the fish, or sample until all quotas are full |

## Part 4(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\sigma = \sqrt{\dfrac{3053}{160} - \left(\dfrac{692}{160}\right)^2}$ | M1 | A correct expression for $\sigma$ |
| $= 0.6129\ldots$ awrt $0.613$ | A1 | awrt $0.613$; allow $s =$ awrt $0.615$ |

## Part 4(d)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| This would have no effect as the piece of data would remain in the same class | B1 | Correct deduction with suitable explanation. Allow range for class. Do not allow "there is no differences" |

## Part 4(d)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| This would increase the standard deviation as change in mean is small and $6.4 - 4.6 \approx 3\sigma$, therefore estimate of standard deviation will increase | B1 | Correct deduction with suitable explanation; so would increase the standard deviation and a suitable reason. Allow "the value is bigger than any others in the table" **oe** |

---
\begin{enumerate}
  \item A lake contains three different types of carp.
\end{enumerate}

There are an estimated 450 mirror carp, 300 leather carp and 850 common carp.\\
Tim wishes to investigate the health of the fish in the lake.\\
He decides to take a sample of 160 fish.\\
(a) Give a reason why stratified random sampling cannot be used.\\
(b) Explain how a sample of size 160 could be taken to ensure that the estimated populations of each type of carp are fairly represented.

You should state the name of the sampling method used.

As part of the health check, Tim weighed the fish.\\
His results are given in the table below.

\begin{center}
\begin{tabular}{|l|l|l|}
\hline
Weight (wkg) & Frequency (f) & Midpoint (m kg) \\
\hline
$2 \leqslant w < 3.5$ & 8 & 2.75 \\
\hline
$3.5 \leqslant w < 4$ & 32 & 3.75 \\
\hline
$4 \leqslant w < 4.5$ & 64 & 4.25 \\
\hline
$4.5 \leqslant w < 5$ & 40 & 4.75 \\
\hline
$5 \leqslant w < 6$ & 16 & 5.5 \\
\hline
\end{tabular}
\end{center}

$$\left( \text { You may use } \sum \mathrm { f } m = 692 \quad \text { and } \quad \sum \mathrm { f } m ^ { 2 } = 3053 \right)$$

(c) Calculate an estimate for the standard deviation of the weight of the carp.

Tim realised that he had transposed the figures for 2 of the weights of the fish.\\
He had recorded in the table 2.3 instead of 3.2 and 4.6 instead of 6.4\\
(d) Without calculating a new estimate for the standard deviation, state what effect\\
(i) using the correct figure of 3.2 instead of 2.3\\
(ii) using the correct figure of 6.4 instead of 4.6\\
would have on your estimated standard deviation.\\
Give a reason for each of your answers.\\

\hfill \mbox{\textit{Edexcel AS Paper 2 2020 Q4 [7]}}
This paper (5 questions)
View full paper
Q1 4 Q2 5 Q3 6 Q4 7 Q5 8