Practice Paper 1 (Cambridge International A-Level Mathematics)

The probability of a student in a primary school library returning his or her books on time had been found to be 0.35. Joanna, the school librarian, has started a new incentive scheme and believes that more students are now returning their books on time because of it. She conducts a hypothesis test using the null hypothesis $H_{0}:p=0.35$ to test her belief

(i) State a suitable alternative hypothesis to test Joanna’s belief that more students are now returning their books on time.

(ii) Write down the conditions under which Joanna could use a binomial probability distribution to model this problem.

Joanna takes a random sample of 30 students who have checked out books and finds that under the new incentive scheme 15 of them return their books on time. She calculates the following probabilities for the random variable $X \sim B ( 30 , 0.35 )$ :

$P(X=14)=0.06112$

$P(X=15)=0.03511$

$P(X\geq16)=0.03008$

Write down the values of $P(X\geq15)$ and $P(X\geq14).$

Write a conclusion for the hypothesis test, in context, if Joanna had chosen a significance level of:

(i) 5%

(ii) 10%

There is a large cohort of students studying the statistics course at the University of Bernoulli. The students’ marks on an exam can be modelled by a normal distribution with mean $μ$ and standard deviation 12. The lecturer, Jacob, is trying to demonstrate the power of hypothesis testing. He knows the true value of $μ$ to be 59 but he keeps this a secret from the students.

Jacob asks the students to use the mean of a random sample of 25 exam results to test the null hypothesis $H_{0} : mu = 55$ against the alternate hypothesis $H_{1} : mu > 55$. Jacob calculates the probability of the test producing a Type II error to be 0.325.

(i) The students calculate the probability that their test will result in a Type I error, find this probability.

(ii) Write down the actual probability of a Type I error on this test.

A hypothesis test uses a sample of data in an experiment to test a statement made about the value of a population parameter ($p$).

Explain, in the context of hypothesis testing, what is meant by:

(i) ‘sample of data’,

(ii) ‘population parameter’

(iii) ‘null hypothesis’,

(iv) ‘alternative hypothesis’,

(v) ‘a Type I error’,

(vi) ‘a Type II error’.

Evaluate the following definite integral, giving your answer in terms of $k$.

$\int_{3}^{9} k ( x^{2} + 3 ) d x$

Hence find the value of $k$ given that $f ( x )$ is a probability density function where

$f ( x ) = \{ \begin{gathered} k ( x^{2} + 3 ) 3 \text{less} \text{or} \text{equal} \text{than} x \text{less} \text{or} \text{equal} \text{than} 9 \\ 0 otherwise \end{gathered} )$

The random variable $X$ has a mean of 40 and a variance of 36.

Find the mean and variance of:

(i) $2X+5$

(ii) $3X-10$

(iii) $100-2X$

It is known that $X$ follows a Normal distribution.

Find $P(2X+1<75)$.

In a computer game, enemies appear at random and at a constant average rate of 2.5 per minute. You may assume that enemies appear independently of one another.

Find the probability that exactly 10 enemies appear during a randomly chosen 5-minute period.

Find the probability that at least 3 enemies appear during a randomly chosen 90-second period.

The probability that at least one enemy appears during a $k$-minute period is 0.999.

Find the value of $k$.

The random variable $X \sim P o \left(\lambda\right)$ can be approximated by $Y \text{tilde} N ( mu , \text{sigma} ^{2} )$ when certain conditions are fulfilled.

State the condition for $λ$ which is required to use this approximation.

For each of the following random variables, $X$:

• State, with reasons, whether $X$ can be approximated by a normal distribution,
• If appropriate, write down the normal approximation to $X$ in the form $Y \text{tilde} N ( mu , \text{sigma} ^{2} )$, giving the values of $μ\mathrm{and}σ$.

(i) $X \text{tilde} P o ( 100 )$

(ii) $X \text{tilde} P o ( 4 )$

(iii) $X \text{tilde} P o ( 50 )$

The wingspans of a sample of 25 flamingos in a nature reserve were measured to the nearest centimetre.

(i) State one condition needed for the sample to be a random sample.

(ii) A researcher suggests that collecting the data on a weekday morning might cause the sample to be biased. State, with a reason, whether the researcher's suggestion is valid.

The standard normal distribution is given by $Z \sim N ( 0 , 1^{2} )$.

Given that $P(-a<Z<a)=0.95$,

(i) explain why $P(Z<a)=0.975$,

(ii) hence find the value of $a$.

A random sample of 25 observations is taken from $N(μ,6²)$ and the sample mean, ̅x is 15. The formula for the end-points of the 95% confidence interval for the population mean, $μ$ is given by:

̅x\pm z\times\frac{σ}{\sqrt{n}}whereP(-z<Z<z)=0.95

Using your answer to part (a) and the formula above

(i) Calculate the end-points of the 95% confidence interval for $μ$,

(ii) Hence write down the 95% confidence interval for $μ$.

Write down the probability that the confidence interval found in part (b) contains the true population mean, $μ$.

The probability of a wild Asian elephant living past 40 years old is 45%.

Rosie, a zoologist, obtains data on 20 elephants in captivity and records their ages at death. She suspects that the proportion of Asian elephants that live past 40 years old is smaller for those in captivity than those in the wild.

Using a 1% level of significance, find the critical value of a one-tailed hypothesis test to enable Rosie to test her suspicion. Clearly state your hypotheses.

Rosie finds that only one elephant, out of the 20 in captivity, lived past 40 years old.

(i) Write down the conclusion of the hypothesis test based on this information.

(ii) Calculate the probability that this observed result has led to a Type I error.