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IB Maths AI HL 4.2 Statistics and Probability Ahl Content Questions

Practise IB Mathematics AI HL 4.2 by analysing advanced probability models, distributions, inference, regression and technology output.

Syllabus
First assessment 2021
Course
Mathematics: applications and interpretation HL
Level
HL

Exam points

  • Design valid data-collection instruments, categorise variables and evaluate reliability, validity and test assumptions.
  • Fit and compare non-linear regression models using residual sums and R², interpreting parameters and limitations.
  • Transform expectations and variances and use linear combinations, estimators, sample means and the central limit theorem.
  • Construct and interpret normal/t confidence intervals for population means, choosing the correct known/unknown-σ method.
  • Select Poisson/binomial/normal/t models and conduct advanced tests for means, proportions or correlation, including error risks.

Covered syllabus points

AHL 4.12 (HL)—Data collection and validityAHL 4.13 (HL)—Non-linear regressionAHL 4.14 (HL)—Random variable transformationsAHL 4.15 (HL)—Sampling distributions and CLTAHL 4.16 (HL)—Confidence intervalsAHL 4.17 (HL)—Poisson distributionAHL 4.18 (HL)—Advanced hypothesis testingAHL 4.19 (HL)—Transition matrices and Markov chains

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Study 4.2 Statistics and probability - AHL content conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

Question 1

[Maximum number: 4]

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.

Question (a)

(a)

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

Juliet's results are summarized in the following table.

Table for Question (a) — IB Maths AI HL
[ 1 ]

Question (b)

(b)

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
[ 3 ]

Question (i)

(i)

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

[ 2 ]

Question (ii)

(ii)

Hence compare the two models.

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

[ 1 ]

Question 2

[Maximum number: 13]

Taylor is playing a computer game in which they shoot at spaceships and battleships. The number of spaceships they hit per minute can be modelled by a Poisson distribution with mean 4.2. The number of battleships they hit per minute can be modelled by a Poisson distribution with a mean of 2.3. Any single hit occurs independently of all others.

Question (a)

(a)

Find the probability Taylor hits

[ 5 ]

Question (i)

(i)

at most 10 spaceships in 2 minutes.

[ 2 ]

Question (ii)

(ii)

a total of more than 10 spaceships and battleships in one minute.

[ 3 ]

Question (b)

(b)

Every spaceship that is hit earns Taylor 3 points and every battleship 5 points. Let T be the total points earned in one minute.

Find

[ 3 ]

Question (i)

(i)

E(T).

[ 1 ]

Question (ii)

(ii)

Var⁡(T)\quad \operatorname{Var}(T).

[ 2 ]

Question (c)

(c)

State one reason why the distribution of T cannot be Poisson.

[ 1 ]

Question (d)

(d)

Taylor intends to play the game for one hour.

Use the central limit theorem to find the probability that Taylor's mean score per minute is greater than 25 .

[ 4 ]
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