Practice Paper 2 (Cambridge International A-Level Mathematics)

On a particular Pokémon game, it is possible to create offspring through breeding. Sometimes when two Pokémon are bred together, their offspring possess a hidden ability.

One Pokémon fan website,$A$, claims that the probability of producing offspring with a hidden ability is 0.6. Another website,$B$, claims the probability is higher.

Maya conducts an experiment and finds that 17 out of 20 of her Pokémon’s offspring possess a hidden ability.

Use Maya’s sample to test the claim of website $A$ against website $B$ at the 5% significance level, stating your null and alternative hypotheses clearly.

Website $B$ claims that the probability is 0.8.

Using a two-tailed hypothesis test, test the claim at the 5% significance level and make a statement about which website is more likely to be correct.

The critical regions of a two-tailed hypothesis test to test $B's$ claim are $X\leq12\mathrm{and}X\geq20$. Given that under the null hypothesis $P(X\geq12)=0.990018$, find the probability of the hypothesis test resulting in a Type I error.

Cyd is a fan of jazz music. The length of a jazz song, $L$, follows a normal distribution with a standard deviation of 0.71 minutes. Cyd reads a headline that states that the mean length of a jazz song is 4 minutes, Cyd claims that the mean length of a jazz song is less than 4 minutes. To test her claim, she takes a random sample of 40 songs and calculates the sample mean.

Stating your hypotheses clearly, find the rejection region for ̅L for Cyd’s test using a 5% level of significance.

Cyd decides to include more songs in her sample, what effect would this have on the rejection region?

Cyd includes songs in her sample and calculates the sample mean as 3.95 minutes.

Given that this sample mean is in the rejection region, find the minimum possible value for the sample size $n$.

Explain whether it was necessary to use the Central Limit theorem for any part of this question.

A memory experiment involves having participants read a list of 20 words for two minutes and then recording how many of the words they can recall. Peter, a psychologist, claims that more than 60% of teenagers can recall all the words. Peter takes a random sample of 40 teenagers and records how many of them recall all the words.

(i) State a suitable null hypothesis to test the psychologist’s claim.

(ii) State a suitable alternative hypothesis to test the psychologist’s claim.

Given that the critical value for the test is $x=19$, state the outcome of the test if

(i) 18 out of the 40 teenagers recall all the words

(ii) 19 out of the 40 teenagers recall all the words

(iii) 20 out of the 40 teenagers recall all the words.

A meeting at a company lasts $T$ hours. The random variable $T$ can be modelled by the probability density function

$f ( t ) = \{ \begin{gathered} \frac{3}{4} t ( t - 2 )^{2} 0 \text{less} \text{or} \text{equal} \text{than} t \text{less} \text{or} \text{equal} \text{than} 2 \\ 0 otherwise \end{gathered} )$

Find the mean time of a meeting and find the standard deviation of times.

Show that the median length of a meeting is 0.771 hours, correct to 3 significant figures.

By using differentiation, find the mode of the times.

If $X$ is normally distributed, then $aX+b$ is normally distributed for any constants $a$ and $b$.

Using the formulae $E(aX+b)=aE(X)+b\mathrm{and}Var(aX+b)=a^{2}Var(X)$, show that, if $X \sim N ( \\mu , \\sigma ^{2} )$ then:

$a X + b \sim N ( a \\mu + b , a^{2} \\sigma ^{2} )$

The random variable $X \sim N ( 50 , 16 )$, write down the distribution for the random variable:

(i) $2X+10$

(ii) $3X-20$

(iii) $50-X$

The random variable $Y \sim N ( 15 , 5^{2} )$

(i) Find $P(Y<18)$

(ii) Write down the distribution of $4Y+40$

(iii) Find $P(4Y+40<112)$

The number of mistakes made by a student, Priya, in a 20-minute revision period is modelled by the distribution $\text{Po} ( 1 . 2 )$. The number of mistakes made by a different student, Qays, in a 30-minute revision period is modelled by the distribution $\text{Po} ( 2 . 2 )$.

Stating a necessary assumption, find the probability that, in a randomly chosen 1-hour revision period, Priya makes exactly 2 mistakes and Qays makes exactly 1 mistake. Give your answer in the form $\frac{a}{b} \text{e}^{c}$, where $a$, $b$ and $c$ are integers.

Given that Priya and Qays make a total of exactly 9 mistakes in a randomly chosen 1-hour revision period, find the probability that Priya makes exactly 5 mistakes.

Given instead that Priya makes exactly 5 mistakes in a randomly chosen 1-hour revision period, find the probability that Priya and Qays make a total of exactly 9 mistakes in that period.

In the week before a test, Priya has ten 30-minute revision periods.

Estimate the number of these revision periods during which Priya makes at least one mistake.

The number of occurrences in a fixed period of time is denoted as $X$. Write down the conditions that are needed so that $X$ can be modelled as a Poisson distribution.

For each of the following scenarios, identify the name of the distribution (if any) which is the most appropriate to model the specified random variables.

(i) A manager receives emails randomly and independently at a constant rate of 15 per hour. The random variable $A$ is the number of emails she receives in a two-hour period.

(ii) A fair die has six sides labelled 1 to 6. The random variable $B$ is the number of times that the die is rolled until it lands on ‘3’.

(iii) It is known that on average 23 in 100 people have blonde hair. A hairdresser has 20 customers per day. The random variable $C$ is the number of customers with blonde hair.

(iv) A machine breaks down at a rate of two times per week. Breakdowns are independent of each other. The random variable $D$ is the number of times that the machine breaks down in a year.

A school has 40 staff and 490 students. Before this year's summer festival, the headteacher wants to survey a sample of the staff and students about their opinions of the festival.

The headteacher first suggests asking the staff and students who are helping to run the festival.

Give a reason why this method is unsatisfactory.

Instead, the headteacher numbers all 530 staff and students from 1 to 530 and uses a calculator to generate random digits. The first 20 random digits are as follows.

1 2 7 3 9 4 0 6 5 1 2 7 8 8 3 4 8 8 2 0

The numbers of the first two people in the sample are 127 and 394.

Use these digits to find the numbers of the next two people in the sample.

The price, $\pounds P$, for an hour tutoring session in a local town can be modelled by a normal distribution with mean $μ$ and standard deviation $σ$. A random sample of 20 one-hour tutoring sessions was taken, and the 99% confidence interval for was calculated to be £29.42 to £41.68.

Write down the mean price of the 20 tutoring sessions in the sample.

Show that $σ=10.64$, correct to two decimal places.

Find the minimum sample size needed so that the width of the 99% confidence interval for $μ$ is less than 5.

Historical records show that England wins 45% of Ashes test cricket series. Rory believes that in the recent past that proportion has decreased. To test this, he looks at their results in the last 17 series.

Using a 5% level of significance identify the critical region to enable Rory to test his claim.

England have won five series in the last 17. Use your answer in part (a) to state, with a reason, whether this sample supports Rory’s claim.

State an assumption made when using a binomial model.