Practise IB Mathematics AI HL 4.1 by analysing probability models, distributions, regression, inference and technology output with explicit assumptions.
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.
Describe one way in which Juliet could improve the reliability of her investigation.
Any one from:
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 K.
Juliet classifies response K as an outlier and removes it from the data. Suggest one possible justification for her decision to remove it.
because the income is very different/implausible/clearly contrived
Note: Answers must explicitly reference "income" to get credit.
For the remaining ten responses in the table, Juliet calculates the mean happiness score to be 52.5.
Calculate the mean annual income for these remaining responses.
($) 90200
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.
r=0.558(0.557723…)
State why the hypothesis test should be one-tailed.
EITHER
only looking for change in one direction
OR
only looking for greater happiness with greater income
OR
only looking for evidence of positive correlation
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.
H0:ρ=0;H1:ρ>0
Note: Award A1 for ρ seen (do not accept r ), A1 for both correct hypotheses, using their ρ 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 ρ≤0 and should not be condoned.
Determine whether there is significant evidence of a positive correlation between annual income and happiness. Justify your answer.
METHOD 1 - using critical value of r
0.558>0.549(0.557723…>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…<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.
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
Use Juliet's data to find the value of a and of b.
a=0.000126(0.000125842…),b=41.1(41.1490…)
A1
Interpret, referring to income and happiness, what the value of a represents.
Juliet then considers a quadratic model of the form
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
Note: Accept equivalent responses e.g. an increase of 1.26 in happiness for every $ 10000 increase in salary.
Find the value of c, of d and of e.
c=−2.06×10−9(−2.06191…×10−9),
d=7.05×10−4(7.05272…×10−4),
e=12.6 (12.5878...)
A1
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.
EITHER
not valid, R2 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.
State the name of the test which Juliet should use.
(single sample) t-test
A1
State the null and alternative hypotheses for this test.
EITHER
H0:μ=80000;H1:μ=80000
A1
OR
H0 : (sample is drawn from a population where) the population mean is $80000H1 : 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.
Perform the test, using a 5 % significance level, and state your conclusion in context.
p=0.610(0.610322…)
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.