OCR MEI Further Statistics A AS 2020 November — Question 2 12 marks

Exam BoardOCR MEI
ModuleFurther Statistics A AS (Further Statistics A AS)
Year2020
SessionNovember
Marks12
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Mark schemeDownload PDF ↗
TopicHypothesis test of Pearson’s product-moment correlation coefficient
TypeTwo-tailed test for any correlation
DifficultyStandard +0.3 This is a standard hypothesis testing question on Pearson's correlation coefficient with routine calculations and interpretation. While it's Further Maths content (making it slightly harder on an absolute scale), the question follows a textbook structure: justify the test, calculate r using given summary statistics with the standard formula, perform a two-tailed test at 5% level, and explain random sampling. No novel insight or complex problem-solving is requiredβ€”just methodical application of learned procedures.
Spec2.02c Scatter diagrams and regression lines2.02d Informal interpretation of correlation2.05e Hypothesis test for normal mean: known variance2.05f Pearson correlation coefficient2.05g Hypothesis test using Pearson's r5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation
2 A researcher is investigating the concentration of bacteria and fungi in the air in buildings. The researcher selects a random sample of 12 buildings and measures the concentrations of bacteria, \(x\), and fungi, \(y\), in the air in each building. Both concentrations are measured in the same standard units. Fig. 2 illustrates the data collected. The researcher wishes to test for a relationship between \(x\) and \(y\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ba3fcd3c-6834-4116-be0e-d5b27aed0a7e-3_595_844_513_255} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. Explain why a test based on the product moment correlation coefficient is likely to be appropriate for these data. Summary statistics for the data are as follows. \(n = 12 \quad \sum x = 18030 \quad \sum y = 15550 \quad \sum x ^ { 2 } = 31458700 \quad \sum y ^ { 2 } = 21980500 \quad \sum x y = 25626800\)
  2. In this question you must show detailed reasoning. Calculate the product moment correlation coefficient between \(x\) and \(y\).
  3. Carry out a test at the \(5 \%\) significance level based on the product moment correlation coefficient to investigate whether there is any correlation between concentrations of bacteria and fungi.
  4. Explain why, in order for proper inference to be undertaken, the sample should be chosen randomly.
Question 2:
AnswerMarks Guidance
2(a) Shape of scatter diagram is approx elliptical
so evidence to suggest (underlying) bivariate
AnswerMarks
Normality required for test using pmcc to be validE1
E1
AnswerMarks
[2]3.5a
2.4
AnswerMarks Guidance
2(b) DR
S =
xy
1
= 2562βˆ‘6π‘₯π‘₯8π‘₯π‘₯00βˆ’ β€“π‘›π‘›βˆ‘Γ—1π‘₯π‘₯8βˆ‘03π‘₯π‘₯0Γ—15550 = 2262925
1
S = 12
xx
2 1 2
= 314βˆ‘58π‘₯π‘₯700βˆ’ –𝑛𝑛(βˆ‘Γ— 1π‘₯π‘₯8)0302 = 4368625
1
S = 12
yy
2 1 2
= 219βˆ‘80π‘₯π‘₯500βˆ’ –𝑛𝑛(Γ—βˆ‘ 1π‘₯π‘₯5)5502 = 1830292
1
12 2262925
r = =
4368625Γ—1830292
𝑆𝑆π‘₯π‘₯π‘₯π‘₯
= 0.800
AnswerMarks
�𝑆𝑆π‘₯π‘₯π‘₯π‘₯𝑆𝑆π‘₯π‘₯π‘₯π‘₯M1
A1
M1
A1
AnswerMarks
[4]1.1a
1.1
1.1
AnswerMarks
1.1For attempt to find S xx , S xy or S yy
For any of S xx , S xy or S yy correct
AnswerMarks Guidance
For general form including √Allow 0.80 www
2(c) H : ρ = 0
0
H : ρ β‰  0 (two-tailed test)
1
where ρ is the (population) correlation coefficient
between x and y
For n = 12, 5% critical value (two tailed) = 0.5760
Since 0.800 > 0.5760 the result is significant. There is
sufficient evidence to reject H
0
There is sufficient evidence at the 5% level to suggest
that there is correlation between the concentrations of
AnswerMarks
bacteria and fungi.B1
B1
B1
M1
A1
AnswerMarks
[5]3.3
2.5
3.4
1.1
AnswerMarks
2.2bFor both hypotheses
For defining ρ in context
For critical value
For comparison leading to a
conclusion e.g. β€˜significant’ or β€˜reject
H ’
0
For non-assertive conclusion in
AnswerMarks
context. FT their r.Other symbols may be
used as long as they are
defined as population
correlation coefficient
No further marks if
cv incorrect
AnswerMarks Guidance
2(d) Because then the probability basis on which the sample
has been selected is known.E1
[1]2.2a 
Question 2:
2 | (a) | Shape of scatter diagram is approx elliptical
so evidence to suggest (underlying) bivariate
Normality required for test using pmcc to be valid | E1
E1
[2] | 3.5a
2.4
2 | (b) | DR
S =
xy
1
= 2562βˆ‘6π‘₯π‘₯8π‘₯π‘₯00βˆ’ β€“π‘›π‘›βˆ‘Γ—1π‘₯π‘₯8βˆ‘03π‘₯π‘₯0Γ—15550 = 2262925
1
S = 12
xx
2 1 2
= 314βˆ‘58π‘₯π‘₯700βˆ’ –𝑛𝑛(βˆ‘Γ— 1π‘₯π‘₯8)0302 = 4368625
1
S = 12
yy
2 1 2
= 219βˆ‘80π‘₯π‘₯500βˆ’ –𝑛𝑛(Γ—βˆ‘ 1π‘₯π‘₯5)5502 = 1830292
1
12 2262925
r = =
4368625Γ—1830292
𝑆𝑆π‘₯π‘₯π‘₯π‘₯
= 0.800
�𝑆𝑆π‘₯π‘₯π‘₯π‘₯𝑆𝑆π‘₯π‘₯π‘₯π‘₯ | M1
A1
M1
A1
[4] | 1.1a
1.1
1.1
1.1 | For attempt to find S xx , S xy or S yy
For any of S xx , S xy or S yy correct
For general form including √ | Allow 0.80 www
2 | (c) | H : ρ = 0
0
H : ρ β‰  0 (two-tailed test)
1
where ρ is the (population) correlation coefficient
between x and y
For n = 12, 5% critical value (two tailed) = 0.5760
Since 0.800 > 0.5760 the result is significant. There is
sufficient evidence to reject H
0
There is sufficient evidence at the 5% level to suggest
that there is correlation between the concentrations of
bacteria and fungi. | B1
B1
B1
M1
A1
[5] | 3.3
2.5
3.4
1.1
2.2b | For both hypotheses
For defining ρ in context
For critical value
For comparison leading to a
conclusion e.g. β€˜significant’ or β€˜reject
H ’
0
For non-assertive conclusion in
context. FT their r. | Other symbols may be
used as long as they are
defined as population
correlation coefficient
No further marks if
cv incorrect
2 | (d) | Because then the probability basis on which the sample
has been selected is known. | E1
[1] | 2.2a
2 A researcher is investigating the concentration of bacteria and fungi in the air in buildings. The researcher selects a random sample of 12 buildings and measures the concentrations of bacteria, $x$, and fungi, $y$, in the air in each building. Both concentrations are measured in the same standard units. Fig. 2 illustrates the data collected. The researcher wishes to test for a relationship between $x$ and $y$.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{ba3fcd3c-6834-4116-be0e-d5b27aed0a7e-3_595_844_513_255}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{center}
\end{figure}
\begin{enumerate}[label=(\alph*)]
\item Explain why a test based on the product moment correlation coefficient is likely to be appropriate for these data.

Summary statistics for the data are as follows.\\
$n = 12 \quad \sum x = 18030 \quad \sum y = 15550 \quad \sum x ^ { 2 } = 31458700 \quad \sum y ^ { 2 } = 21980500 \quad \sum x y = 25626800$
\item In this question you must show detailed reasoning.

Calculate the product moment correlation coefficient between $x$ and $y$.
\item Carry out a test at the $5 \%$ significance level based on the product moment correlation coefficient to investigate whether there is any correlation between concentrations of bacteria and fungi.
\item Explain why, in order for proper inference to be undertaken, the sample should be chosen randomly.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Statistics A AS 2020 Q2 [12]}}
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