Question 5 - A-Level Maths

OCR Further Statistics AS 2023 June — Question 5 9 marks

Exam Board	OCR
Module	Further Statistics AS (Further Statistics AS)
Year	2023
Session	June
Marks	9
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Hypothesis test of Pearson’s product-moment correlation coefficient
Type	One-tailed test for positive correlation
Difficulty	Standard +0.3 This is a standard textbook application of Pearson's correlation coefficient hypothesis test with all summary statistics provided. Part (a) requires routine calculation and comparison to critical values, part (b) tests basic interpretation, and part (c) requires understanding that standardisation doesn't affect correlation—all straightforward for Further Maths students with no novel problem-solving required.
Spec	5.08a Pearson correlation: calculate pmcc 5.08d Hypothesis test: Pearson correlation

5 A psychologist investigates the relationship between 'openness' and 'creativity' in adults. Each member of a random sample of 15 adults is given two tests, one on openness and one on creativity. Each test has a maximum score of 75 . The results are given in the table.

Adult	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O
Openness, \(x\)	39	34	29	20	40	35	20	36	55	31	41	43	33	30	33
Creativity, \(y\)	59	34	17	29	49	46	45	54	60	38	46	35	43	56	34

\(n = 15 \quad \sum x = 519 \quad \sum y = 645 \quad \sum x ^ { 2 } = 19033 \quad \sum y ^ { 2 } = 29751 \quad \sum x y = 23034\)

Use Pearson's product-moment correlation coefficient to test, at the \(5 \%\) significance level, whether there is positive association between openness and creativity.
State what the value of Pearson's product-moment correlation coefficient shows about a scatter diagram illustrating the data.
A student suggests that there is a way to obtain a more accurate measure of the correlation. Before carrying out the test it would be better to standardise the test scores so that they have the same mean and variance. Explain whether you agree with this suggestion.

Show mark scheme Show mark scheme source

Question 5:

Part (a):

Answer	Marks	Guidance
Answer	Marks	Guidance
\(H_0: \rho = 0\), \(H_1: \rho > 0\) where \(\rho\) is the population product-moment correlation coefficient between the test scores	B2	One error e.g. two-tailed, or \(\rho\) not defined: B1. Allow \(H_0: \rho \leq 0\). Symbols: definition of \(\rho\) needs "population" or context or both, and "correlation". Allow \(r\) in place of \(\rho\). Do not allow "association". Verbal: \(H_0\): no correlation between openness and creativity, \(H_1\): positive correlation: max B1 unless "population" explicit. Needs "positive". Allow "association" here.
\(\rho = 717 \div \sqrt{1075.6 \times 2016} = 0.487\) (0.48691)	M1 A1	Art 0.487 seen gets M1A1. Else allow M1 for correct subs into formula, or any two of 1075.6, 2016 and 717, or any two of 71.7, 134.3 and 47.8
\(0.487 > 0.4409\) so reject \(H_0\)	M1ft	Compare their \(r\) with 0.4409 or 0.441 and reject (ft on TS)
There is significant evidence of positive association between openness and creativity	A1ft [6]	Correct contextualised conclusion, not too assertive, allow omission of "positive", FT on their \(r\), no FT for hypotheses wrong way round

Part (b):

Answer	Marks	Guidance
Answer	Marks	Guidance
Points lie fairly (but not very) close to straight line	B1	Must refer to diagram of points, not just to correlation. Not "points lie close to a line" — some level of nuance needed. Allow general statement, e.g. "it shows how close to a straight line the points are"
… with positive gradient	B1 [2]	Any wrong statement: max B1B0

Part (c):

Answer	Marks	Guidance
Answer	Marks	Guidance
Disagree as \(\rho\) is unchanged by linear scaling	B1 [1]	"Disagree" oe and correct reason, allow omission of "linear". Allow "It wouldn't affect the value"

# Question 5:

## Part (a):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: \rho = 0$, $H_1: \rho > 0$ where $\rho$ is the population product-moment correlation coefficient between the test scores | **B2** | One error e.g. two-tailed, or $\rho$ not defined: B1. Allow $H_0: \rho \leq 0$. Symbols: definition of $\rho$ needs "population" or context or both, and "correlation". Allow $r$ in place of $\rho$. Do *not* allow "association". Verbal: $H_0$: no correlation between openness and creativity, $H_1$: positive correlation: max B1 unless "population" explicit. Needs "positive". *Allow "association" here.* |
| $\rho = 717 \div \sqrt{1075.6 \times 2016} = 0.487$ (0.48691) | **M1 A1** | Art 0.487 seen gets M1A1. Else allow M1 for correct subs into formula, or any two of 1075.6, 2016 and 717, or any two of 71.7, 134.3 and 47.8 |
| $0.487 > 0.4409$ so reject $H_0$ | **M1ft** | Compare their $r$ with 0.4409 or 0.441 and reject (ft on TS) |
| There is significant evidence of positive association between openness and creativity | **A1ft [6]** | Correct contextualised conclusion, not too assertive, allow omission of "positive", FT on their $r$, no FT for hypotheses wrong way round |

## Part (b):

| Answer | Marks | Guidance |
|--------|-------|----------|
| Points lie fairly (but not very) close to straight line | **B1** | Must refer to diagram of points, not just to correlation. Not "points lie close to a line" — some level of nuance needed. Allow general statement, e.g. "it shows how close to a straight line the points are" |
| … with positive gradient | **B1 [2]** | Any wrong statement: max B1B0 |

## Part (c):

| Answer | Marks | Guidance |
|--------|-------|----------|
| Disagree as $\rho$ is unchanged by linear scaling | **B1 [1]** | "Disagree" oe and correct reason, allow omission of "linear". Allow "It wouldn't affect the value" |

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Show LaTeX source

5 A psychologist investigates the relationship between 'openness' and 'creativity' in adults. Each member of a random sample of 15 adults is given two tests, one on openness and one on creativity. Each test has a maximum score of 75 . The results are given in the table.

\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | c | c | c | c | c | }
\hline
Adult & A & B & C & D & E & F & G & H & I & J & K & L & M & N & O \\
\hline
Openness, $x$ & 39 & 34 & 29 & 20 & 40 & 35 & 20 & 36 & 55 & 31 & 41 & 43 & 33 & 30 & 33 \\
\hline
Creativity, $y$ & 59 & 34 & 17 & 29 & 49 & 46 & 45 & 54 & 60 & 38 & 46 & 35 & 43 & 56 & 34 \\
\hline
\end{tabular}
\end{center}

$n = 15 \quad \sum x = 519 \quad \sum y = 645 \quad \sum x ^ { 2 } = 19033 \quad \sum y ^ { 2 } = 29751 \quad \sum x y = 23034$
\begin{enumerate}[label=(\alph*)]
\item Use Pearson's product-moment correlation coefficient to test, at the $5 \%$ significance level, whether there is positive association between openness and creativity.
\item State what the value of Pearson's product-moment correlation coefficient shows about a scatter diagram illustrating the data.
\item A student suggests that there is a way to obtain a more accurate measure of the correlation. Before carrying out the test it would be better to standardise the test scores so that they have the same mean and variance.

Explain whether you agree with this suggestion.
\end{enumerate}

\hfill \mbox{\textit{OCR Further Statistics AS 2023 Q5 [9]}}

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