IB Maths AI HL 4.2 Statistics and Probability Ahl Content Questions

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.

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.

A random sample of eight packets of Apollo coffee granules are selected from a supermarket shelf.
The weights of the coffee granules present in each packet are as follows:
$222 \mathrm{~g} \quad 226 \mathrm{~g} \quad 221 \mathrm{~g} \quad 228 \mathrm{~g} \quad 227 \mathrm{~g} \quad 225 \mathrm{~g} \quad 222 \mathrm{~g} \quad 223 \mathrm{~g}$

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

Juliet's results are summarized in the following table.

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.

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 .

A firm wishes to review its recruitment processes. This question considers the validity and reliability of the methods used.
Every year an accountancy firm recruits new employees for a trial period of one year from a large group of applicants.
At the start, all applicants are interviewed and given a rating. Those with a rating of either Excellent, Very good or Good are recruited for the trial period. At the end of this period, some of the new employees will stay with the firm.
It is decided to test how valid the interview rating is as a way of predicting which of the new employees will stay with the firm.
Data is collected and recorded in a contingency table.

For the data in this table, test the null hypothesis, $\mathrm{H}_{0}: \rho=0$, against the alternative hypothesis, $\mathrm{H}_{1}: \rho>0$, at the 5 % significance level. You may assume that all the requirements for carrying out the test have been met.

A geneticist uses a Markov chain model to investigate changes in a specific gene in a cell as it divides. Every time the cell divides, the gene may mutate between its normal state and other states.
The model is of the form

where $X_n$ is the probability of the gene being in its normal state after dividing for the nth time, and $Z_n$ is the probability of it being in another state after dividing for the nth time, where $n\in\mathbb{N}$.
Matrix $\boldsymbol{M}$ is found to be $\left(\begin{smallmatrix}0.94&b\\0.06&0.98\end{smallmatrix}\right)$.