IB Maths AI HL 4 Statistics & Probability

[Maximum number: 24]

Juliet is a sociologist who wants to investigate if income affects happiness amongst doctors. This question asks you to review Juliet's methods and conclusions.
Juliet obtained a list of email addresses of doctors who work in her city. She contacted them and asked them to fill in an anonymous questionnaire. Participants were asked to state their annual income and to respond to a set of questions. The responses were used to determine a happiness score out of 100 . Of the 415 doctors on the list, 11 replied.

(a)

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

[ 1 ]

(b)

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

Juliet's results are summarized in the following table.

[ 1 ]

(c)

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

[ 1 ]

(d)

For the remaining ten responses in the table, Juliet calculates the mean happiness score to be 52.5.

[ 4 ]

(i)

Calculate the mean annual income for these remaining responses.

[ 2 ]

(ii)

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

Juliet decides to carry out a hypothesis test on the correlation coefficient to investigate whether increased annual income is associated with greater happiness.

[ 2 ]

$\quad r=0.558(0.557723 \ldots)$

(e)

State why the hypothesis test should be one-tailed.

[ 1 ]

(f)

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.

[ 2 ]

$\quad \mathrm{H}_{0}: \rho=0 ; \mathrm{H}_{1}: \rho>0$
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 $\rho \leq 0$ and should not be condoned.

(g)

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

[ 2 ]

METHOD 1 - using critical value of r
$0.558>0.549(0.557723 \ldots>0.549)$
R1
(therefore significant evidence of) a positive correlation
A1
Note: Do not award R0A1.

METHOD 2 - using p-value
$0.0469<0.05(0.0469463 \ldots<0.05)$

Note: Follow through from their r-value from part (c)(ii).
(therefore significant evidence of) a positive correlation
A1
Note: Do not award A0A1.

(h)

Juliet wants to create a model to predict how changing annual income might affect happiness scores. To do this, she assumes that annual income in dollars, X, is the independent variable and the happiness score, Y, is the dependent variable.

She first considers a linear model of the form

Y=a X+b

[ 7 ]

(i)

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

[ 1 ]

$a=0.000126(0.000125842 \ldots), b=41.1(41.1490 \ldots)$
A1

(ii)

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

Juliet then considers a quadratic model of the form

Y=c X^{2}+d X+e

[ 1 ]

(iii)

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

[ 1 ]

$\quad c=-2.06 \times 10^{-9}\left(-2.06191 \ldots \times 10^{-9}\right)$ ,
$d=7.05 \times 10^{-4}\left(7.05272 \ldots \times 10^{-4}\right)$ ,
e=12.6 (12.5878...)
A1

(iv)

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

[ 2 ]

for quadratic model: $R^{2}=0.659(0.659145 \ldots)$
A1
for linear model: $R^{2}=0.311(0.311056 \ldots)$

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

(v)

Hence compare the two models.

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

[ 1 ]

(vi)

Comment on the validity of her decision.

After presenting the results of her investigation, a colleague questions whether Juliet's sample is representative of all doctors in the city.

A report states that the mean annual income of doctors in the city is $\$ 80000$ . Juliet decides to carry out a test to determine whether her sample could realistically be taken from a population with a mean of $\$ 80000$ .

[ 1 ]

EITHER
not valid, $R^{2}$ not a useful measure to compare models with different numbers of parameters
OR
not valid, quadratic model will always have a better fit than a linear model
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.

(i)

State the name of the test which Juliet should use.

[ 1 ]

(j)

State the null and alternative hypotheses for this test.

[ 1 ]

EITHER
$\mathrm{H}_{0}: \mu=80000 ; \mathrm{H}_{1}: \mu \neq 80000$
A1

OR
$\mathrm{H}_{0}$ : (sample is drawn from a population where) the population mean is $\$ 80000$ $\mathrm{H}_{1}$ : the population mean is not $\$ 80000$
A1
Note: Do not allow FT from an incorrect test in part (f)(i) other than a z-test.

(k)

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

[ 3 ]

$\quad p=0.610(0.610322 \ldots)$
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 $80000
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.

IB Maths AI HL 4 Statistics and Probability

Covered syllabus points

Covered syllabus points

4 Statistics and probability question 1

Question (a)

Question (b)

Question (c)

Question (d)

Question (i)

Question (ii)

Question (e)

Question (f)

Question (g)

Question (h)

Question (i)

Question (ii)

Question (iii)

Question (iv)

Question (v)

Question (vi)

Question (i)

Question (j)

Question (k)