Practise interpreting scatter plots and correlation, using y-on-x regression for prediction, and judging residuals, extrapolation and causation limits.
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.
For the remaining ten responses in the table, Juliet calculates the mean happiness score to be 52.5.
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…)
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…)
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...)
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.