A dice manufacturer claims that for a novelty die he produces the probability of scoring thenumbers $1$ to $5$ are all equal, and the probability of a $6$ is two times the probability ofscoring any of the other numbers.

To test the manufacture’s claim one of the novelty dice is rolled $350$ times and the numbersscored on the die are shown in the table below.

A ${\chi }^2$ goodness of fit test is to be used with a $5\%$ significance level.

In a reforested area of pine trees, heights of trees planted in a specific year seem to follow a normal distribution. A sample of 100 such trees is selected to test the validity of this hypothesis. The results of measuring tree heights, to the nearest centimetre, are recorded in the first two columns of the table below.

Describe what is meant by

A survey was conducted on a group of people. The first question asked how many pets they each own. The results are summarized in the following table.

The second question asked each member of the group to state their age and preferred pet. The data obtained is organized in the following table.

A ${\mathrm{\chi <!--\; \chi \; -->}}^2$ test is carried out at the 10 % significance level.

Long term experience shows that if it is sunny on a particular day in Vokram, then the probabilitythat it will be sunny the following day is $0.8$. If it is not sunny, then the probability that it will besunny the following day is $0.3$.

The transition matrix $\mathit{T}$ is used to model this information, where$\mathit{T}=\left(\begin{array}{cc}0.8& 0.3\\ 0.2& 0.7\end{array}\right)$.

The matrix $\mathit{T}$ can be written as a product of three matrices, $\mathit{P}\mathit{D}\mathbf{ }{\mathit{P}}^{-1}$ , where $\mathit{D}$ is adiagonal matrix.

A firm wishes to review its recruitment processes. This question considers the validityand reliability of the methods used.

Every year an accountancy firm recruits new employees for a trial period of one year from alarge group of applicants.

At the start, all applicants are interviewed and given a rating. Those with a rating of eitherExcellent, 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 newemployees will stay with the firm.

Data is collected and recorded in a contingency table.

The next year’s group of applicants are asked to complete a written assessment whichis then analysed. From those recruited as new employees, a random sample of size $18$is selected.

The sample is stratified by department. Of the $91$ new employees recruited that year,$55$ were placed in the national department and $36$ in the international department.

At the end of their first year, the level of performance of each of the $18$ employees in thesample is assessed by their department manager. They are awarded a score between$1$ (low performance) and $10$ (high performance).

The marks in the written assessment and the scores given by the managers are shown inboth the table and the scatter diagram.

The firm decides to find a Spearman’s rank correlation coefficient, $r_s$, for this data.

The same seven employees are given the written assessment a second time, at the end ofthe first year, to measure its reliability. Their marks are shown in the table below.

The written assessment is in five sections, numbered $1$ to $5$. At the end of the year,the employees are also given a score for each of five professional attributes:$\text{V},\text{W},\text{X},\text{Y}$and $\text{Z}$.

The firm decides to test the hypothesis that there is a correlation between the markin a section and the score for an attribute.

They compare marks in each of the sections with scores for each of the attributes.

Steffi the stray cat often visits Will’s house in search of food. Let $X$ be the discrete randomvariable “the number of times per day that Steffi visits Will’s house”.

The random variable $X$ can be modelled by a Poisson distribution with mean 2.1.

Let Y be the discrete random variable “the number of times per day that Steffi is fed at Will’shouse”. Steffi is only fed on the first four occasions that she visits each day.

A factory, producing plastic gifts for a fast food restaurant’s Jolly meals, claims that just $1\%$ ofthe toys produced are faulty.

A restaurant manager wants to test this claim. A box of $200$ toys is delivered to the restaurant.The manager checks all the toys in this box and four toys are found to be faulty.

The restaurant manager performs a one-tailed hypothesis test, at the $10\%$ significance level,to determine whether the factory’s claim is reasonable. It is known that faults in the toysoccur independently.

A Principal would like to compare the students in his school with a national standard.He decides to give a test to eight students made up of four boys and four girls. One ofthe teachers offers to find the volunteers from his class.

The marks out of $40$, for the students who took the test, are:

$25,29,38,37,12,18,27,31.$

For the eight students find

The national standard mark is $25.2$ out of $40$.

Two additional students take the test at a later date and the mean mark for all ten studentsis $28.1$ and the standard deviation is $8.4$.

For further analysis, a standardized score out of $100$ for the ten students is obtained bymultiplying the scores by $2$ and adding $20$.

For the ten students, find

The number of marathons that Audrey runs in any given year can be modelled by a Poisson distribution with mean 1.3 .

Jim writes a computer program to generate 500 values of a variable Z. He obtains the following table from his results.

In this situation, state briefly what is meant by