Describe one way in which Juliet could improve the reliability of her investigation.

Describe one criticism that can be made about the validity of Juliet’s investigation.

Juliet classifies response $\text{K}$ as an outlier and removes it from the data. Suggest one possible justification for her decision to remove it.

Calculate the mean annual income for these remaining responses.

Determine the value of $r$, Pearson’s product-moment correlation coefficient, for these remaining responses.

State why the hypothesis test should be one-tailed.

State the null and alternative hypotheses for this test.

The critical value for this test, at the $5\%$ significance level, is $0.549$. Juliet assumes that the population is bivariate normal.

Determine whether there is significant evidence of a positive correlation between annual income and happiness. Justify your answer.

Use Juliet’s data to find the value of $a$ and of $b$.

Interpret, referring to income and happiness, what the value of $a$ represents.

Find the value of $c$, of $d$ and of $e$.

Find the coefficient of determination for each of the two models she considers.

Hence compare the two models.

Juliet decides to use the coefficient of determination to choose between these two models.

Comment on the validity of her decision.

State the name of the test which Juliet should use.

State the null and alternative hypotheses for this test.

Perform the test, using a $5\%$ significance level, and state your conclusion in context.

Any one from:                R1

increase sample size / increase response rate / repeat process
check whether sample is representative
test-retest participants or do a parallel test
use a stratified sample
use a random sample

Note: Do not condone:
Ask different types of doctor
Ask for proof of income
Ask for proof of being a doctor
Remove anonymity
Remove response $\mathrm{K}$.

[1 mark]

Any one from:                R1

non-random sampling means a subset of population might be responding
self-reported happiness is not the same as happiness
happiness is not a constant / cannot be quantified / is difficult to measure
income might include external sources
Juliet is only sampling doctors in her city
correlation does not imply causation
sample might be biased

Note: Do not condone the following common but vague responses unless they make a clear link to validity:
Sample size is too small
Result is not generalizable
There may be other variables Juliet is ignoring
Sample might not be representative

[1 mark]

because the income is very different / implausible / clearly contrived              R1

Note: Answers must explicitly reference "income" to get credit.

[1 mark]

$(\$) 90 200$           (M1)A1

[2 marks]

$r=0.558 \left(0.557723\dots \right)$                  A2

[2 marks]

EITHER
only looking for change in one direction                R1

OR
only looking for greater happiness with greater income                R1

OR
only looking for evidence of positive correlation                R1

[1 mark]

${\text{H}}_0:\rho =0; {\text{H}}_1:\rho >0$               A1A1

Note: Award A1 for $\rho$ seen (do not accept $r$), A1 for both correct hypotheses, using their $\rho$ or $r$. Accept an equivalent statement in words, however reference to “correlation for the population” or “association for the population” must be explicit for the first A1 to be awarded.

Watch out for a null hypothesis in words similar to “Annual income is not associated with greater happiness”. This is effectively saying $\rho \le 0$ and should not be condoned.

[2 marks]

METHOD 1 – using critical value of $\mathit{r}$

$0.558>0.549 \left(0.557723\dots >0.549\right)$      R1

(therefore significant evidence of) a positive correlation          A1

Note: Do not award R0A1.

METHOD 2 – using $\mathit{p}$-value

$0.0469<0.05 \left(0.0469463\dots <0.05\right)$         A1

Note: Follow through from their $r$-value from part (c)(ii).

(therefore significant evidence of) a positive correlation          A1

Note: Do not award A0A1.

[2 marks]

$a=0.000126 \left(0.000125842\dots \right), b=41.1 \left(41.1490\dots \right)$         A1

[1 mark]

EITHER
the amount the happiness score increases for every $\$1$ increase in (annual) income       A1

OR
rate of change of happiness with respect to (annual) income       A1

Note: Accept equivalent responses e.g. an increase of $1.26$ in happiness for every $\$10000$ increase in salary.

[1 mark]

$c=-2.06\times {10}^{-9} \left(-2.06191\dots \times {10}^{-9}\right)$,

$d=7.05\times {10}^{-4} \left(7.05272\dots \times {10}^{-4}\right)$,

$e=12.6 \left(12.5878\dots \right)$       A1

[1 mark]

for quadratic model: $R^2=0.659 \left(0.659145\dots \right)$       A1

for linear model: $R^2=0.311 \left(0.311056\dots \right)$       A1

Note: Follow through from their $r$ value from part (c)(ii).

[2 marks]

EITHER
quadratic model is a better fit to the data / more accurate      A1

OR
quadratic model explains a higher proportion of the variance      A1

[1 mark]

EITHER
not valid, $R^2$ not a useful measure to compare models with different numbers of parameters     A1

OR
not valid, quadratic model will always have a better fit than a linear model     A1

Note: Accept any other sensible critique of the validity of the method. Do not accept any answers which focus on the conclusion rather than the method of model selection.

[1 mark]

(single sample) $t$-test    A1

[1 mark]

EITHER

${\text{H}}_0:\mu =80 000; {\text{H}}_1:\mu \ne 80 000$             A1

OR

${\text{H}}_0:$ (sample is drawn from a population where) the population mean is $\$80 000$
${\text{H}}_1:$ the population mean is not $\$80 000$             A1


Note: Do not allow FT from an incorrect test in part (f)(i) other than a $z$-test.

[1 mark]

$p=0.610 \left(0.610322\dots \right)$             A1

Note: For a $z$-test follow through from part (f)(i), either $0.578$ (from biased estimate of variance) or $0.598$ (from unbiased estimate of variance).

$0.610>0.05$            R1

EITHER

no (significant) evidence that mean differs from $\$80 000$            A1

OR

the sample could plausibly have been drawn from the quoted population         A1


Note: Allow R1FTA1FT from an incorrect $p$-value, but the final A1 must still be in the context of the original research question.

[3 marks]