Practice Paper 2 (Edexcel IAL Maths)

It is known that there is a 0.35% chance of picking up a virus when downloading a file from a particular website. After a security update, a security analyst believes that the chances of picking up a virus have been reduced. In a sample of 1542 files downloaded from the website, 1 virus is detected. The analyst wishes to use a hypothesis test with its significance level appropriately chosen to allow her to report with as much certainty as possible that her belief is correct.

Consider tests at the 1%, 5% and 10% significance levels, and explain which of those significance levels the analyst should use for the test in her report.

A spinner with seven equal sections labelled 0, 1, 1, 3, 3, 3 and 3 is being used in a game show where contestants spin the spinner twice and the two individual values it lands on, $S_{1}$ and $S_{2}$, are recorded.

List all the possible samples that could result when a contestant spins the spinner twice.

Assuming that the spinner is fair, find the sampling distribution for the sum of $S_{1}\mathrm{and}S_{2}$ $( \text{sum} S )$.

If the sum of the two spins is 0, the contestant wins a grand prize. After a day of playing the game, the grand prize has been won six times and the game show host thinks that someone may have tampered with the spinner so as to make a grand prize win more likely. A hypothesis test is carried out at the 5% level of significance to test if the spinner is fair.

Write suitable null and alternative hypotheses for the test.

Given that the game was played 100 times on the day in question,

(i) suggest a suitable approximating distribution that could be used to carry out the hypothesis test

(ii) describe the circumstances under which the null hypothesis would be rejected.

The random variable $Y \sim B ( 5 , p )$.

(i) Find the value of $p$ when $E(Y)=3$.

(ii) Find the value of $p$ when $P(Y=5)=0.32768.$

(iii) Find the possible values of $p$ when $\mathrm{Var}(Y)=1.05$.

The continuous random variable $X$ has probability density function

$\mathrm{f} ( x ) = \{ \begin{gathered} \frac{3}{16} ( x^{2} - 1 ) & 2 \text{less} \text{or} \text{equal} \text{than} x \text{less} \text{or} \text{equal} \text{than} 3 \\ 0 otherwise. \end{gathered} )$

Find $\int \frac{3}{16} x ( x^{2} - 1 ) d x.$

Hence find the value of $E(X)$ using the formula

$\mathrm{E} ( X ) = \int x \mathrm{f} ( x ) \mathrm{d} x.$

Find $\int \frac{3}{16} x^{2} ( x^{2} - 1 ) \mathrm{d} x.$

Hence find the value of $E(X²)$ using the formula

$\mathrm{E} ( X ² ) = \int x^{2} \mathrm{f} ( x ) \mathrm{d} x.$

Hence find the value of $\mathrm{Var}(X)$using the formula $\mathrm{Var}(X)=E(X^{2})-(E(X))².$

The continuous random variable $T$ represents the times, in hours, that children at a boarding school take to tidy their rooms. The cumulative distribution function of $T$ is

F ( t ) = { | 0 t less than 0 | | k ( 9 t ^2 - t ^ 4 ) | 0 <= t <= 2 | | 1 t <= 2 | | close

Show that $k=\frac{1}{20}$ .

Find the probability that a randomly selected child takes longer than 90 minutes to tidy their room.

Find the probability density function $f(t)$ of $T$.

Find the mean time taken for a child to tidy their room.

Amira has a bad Internet connection at her house. Her internet disconnects on average 5 times each day.

Define a suitable distribution to model the number of times the internet at Amira’s house disconnects during a day. State any assumptions you make.

Find the probability that during a random day the internet at Amira’s house disconnects:

(i) exactly four times

(ii) at most three times

(iii) no fewer than two times.

As part of a marketing promotion, 47% of packets of a particular brand of crisps contain a zombie toy as a prize. A random sample of 100 packets is taken.

Find the probability that exactly 49 of the packets contain a prize. Give your answer to 6 decimal places

Write down the normal distribution that could be used to approximate the distribution for the number of the 100 packets that contain a prize.

(i) Use the normal approximation to approximate the probability that exactly 49 of the packets contain a prize.

(ii) Find the percentage error when using a normal approximation to calculate the probability that exactly 49 of the packets contain a prize. Give your answer correct to two significant figures.

Roger’s bunnies are well trained and almost always leave their droppings in a designated area, however, Roger occasionally finds that they have left their droppings in his garden too.

Given that over a six-hour period the probability there are no droppings found in the garden is $4.53999\times10^{-5}$ correct to 6 significant figures, find the mean number of droppings left per hour in Roger’s garden.

Roger’s bunnies’ new treat craze is popcorn and Roger believes that the bunnies leave less droppings in the garden after he has given them this treat. Roger decides to test his theory by giving his bunnies the popcorn and then seeing how many droppings are left in the garden over a three-hour period.

Given that the probability that Roger incorrectly rejects the null hypothesis is 0.0404277 correct to 6 significant figures, find the critical region for Roger’s hypothesis test.

Roger finds that two droppings were left in his garden during the three-hour period. Stating your hypotheses clearly, write down the conclusion of Roger’s hypothesis test.

Explain what you understand by the significance level of a hypothesis test.

(i) Give an advantage of using a lower significance level for a hypothesis test.

(ii) Give a disadvantage of using a lower significance level for a hypothesis test.

For each of the following scenarios, explain whether a 1%, 5% or 10% level of significance would be most appropriate.

(i) A shopkeeper takes a sample of 10 cartons of milk to test whether the amount of milk in a carton has decreased.

(ii) A doctor takes a sample of 100 patients to test whether there is an improvement to the recovery rate of an illness when a new drug is used, compared with the current best treatment regime.

(iii) A manager takes a sample of 100 employees to test whether their level of job satisfaction has changed after new working hours have been introduced.

The random variable $X \sim B ( 50 , 0.75 )$. Find:

(i) $P(X\leq52\mathrm{and}X>31)$

(ii) $P(X<29\mathrm{or}X>38)$

(iii) $P(40>X\geq30)$.

The continuous random variable $X$ has probability density function

$\mathrm{f} ( x ) = \{ \begin{gathered} \frac{k}{x^{3}} \\ 0 \end{gathered} ) \begin{gathered} 2 \text{less} \text{or} \text{equal} \text{than} x \text{less} \text{or} \text{equal} \text{than} a \\ otherwise. \end{gathered}$

Show that $k=\frac{8a^{2}}{a^{2}-4}.$

Given that $E(X)=\frac{20}{7},$ show that $a=5$ and hence find the value of $k$.

Find the exact value of $\mathrm{Var}(X)$ .