Practice Paper 1 (Edexcel IAL Maths)

A mathematics teacher makes silly mistakes at the average rate of 20 per day and she teaches 5 lessons, each of equal length, per day.

(i) State a suitable sampling distribution for the number of silly mistakes the mathematics teacher makes within one lesson.

(ii) Find the probability that the mathematics teacher makes less than 2 silly mistakes during the lesson.

(iii) After a few weeks, the teacher believes the average number of mistakes made in a day has changed. Write down suitable null and alternative hypotheses that could be used to test for a change in the number of silly mistakes she makes per day.

From previous research, Marta has found that in general there is a 15% chance that any given customer ordering food at her restaurant will choose a salad. She wants to test whether people are more inclined to eat salads when it is sunny out.

(i) Clearly defining the value of the population parameter $( p )$, state a suitable null hypothesis that Marta could use for this test.

(ii) State a suitable alternative hypothesis that Marta could use for this test.

(iii) Give an example of a test statistic that Marta could use to carry out this test.

After carrying out the test, Marta had evidence to conclude that people are more likely to eat salads when the sun is out. State whether she accepted or rejected the null hypothesis you have written in part (a)(i).

A biased coin has probability 0.8 of landing on heads. Sunita and Mark model the probabilities of obtaining $X$ heads when the coin is tossed 10 times using the random variable $XB(10,p_{1})$.

(i) Explain why $p_{1}=0.8$ in this case.

(ii) Sunita decides to use her calculator to determine any probabilities. She determines $P(X=4)$ using the calculation

$^{10}C_{0}\times(0.8)^{4}(1-0.8)^{10-4}$

Use Sunita’s calculation to find $P(X=4)$ to four decimal places.

Mark decides to use statistical tables to determine any probabilities.

(i) Explain why Mark will not be able to use the random variable $X \sim B ( 10 , 0.8 )$ with statistical tables.

(ii) Mark says that instead of considering the number of heads obtained he will consider the number of tails obtained, $Y$, instead. He will use the random variable $Y \sim B ( 10 , p_{2} )$. Find the value of $p_{2}$ and explain how you found it.

Sunita and Mark use their methods to calculate the probability that the coin lands on heads at least 9 times.

(i) Mark will use tables to find $P(Y\leq y)$. State the value of $y$ and find $P(Y\leq y)$ from statistical tables, writing down all four decimal places given.

(ii) Sunita will use her calculator to find $P(X\geq9)$. Using either a calculation similar to the one given in part (a)(ii) or the statistical features of your calculator, find $P(X\geq9)$, to four decimal places.

The graph below shows the probability density function of a continuous random variable, $X$.

By finding relevant areas under the graph, find:

(i) $P(0<X<0.1)$

(ii) $P(X>0.15)$

(iii) $P(0.05<X<0.12)$.

The continuous random variable, $X$, has probability density function $\mathrm{f} ( x )$ given by

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

The cumulative distribution function $F(x)$ is given by

F ( x ) = { | 0 x <= 0 | | g ( x ) | 0 less than x less than 1 | | h ( x ) | 1 <= x <= 3 | | 1 x greater than 3 | | close

where $g(x)$ and $h(x)$ are functions.

Explain why

(i) $g(0)=0$

(ii) $h(3)=1$

(iii) $g(1)=h(1)$.

Use integration and part (a) to find $g(x)$ and $h(x)$.

A random variable $X \sim P o ( 6 )$.

Find the exact value, in terms of $e$, of:

(i) $P(X=3)$

(ii) $P(X=4)$

(iii) $P(X=5)$

Use your answers to part (a) to find the exact value of:

(i) $\frac{6}{4}P(X=3)$

(ii) $\frac{6}{5}P(X=4)$

Compare your answers to part (a) and part (b), then use the pattern to write down an expression, in terms of $k$, for

$\frac{6}{k+1}P(X=k)$

where $k$ is an integer and $k\geq0$.

The random variable$X \sim B ( 100 , 0.36 )$is approximated by $Y \sim N ( \\mu , \\sigma ^{2} )$.

Find the value of $μ$ and show that $σ=4.8$.

Explain why a continuity correction must be incorporated when using this approximation.

Use continuity corrections to find the value of $k$ in each of the following approximations:

(i) $P(X\leq30)\approx P(Y<k)$

(ii) $P(X<30)\approx P(Y<k)$

(iii) $P(X\geq30)\approx P(Y>k)$

(iv) $P(X>30)\approx P(Y>k)$

A study of ladybirds in the UK found that 65% of all ladybirds are found to be of the seven-spot species. Alex believes that more than 65% of the ladybirds in his garden are of the seven-spot species. He conducts a hypothesis test at the 5% significance level by collecting a sample of 25 ladybirds from his garden and counting the number of them that are seven-spot ladybirds.

(i) Alex uses the random variable $X \sim B ( n , p )$ to represent the number of seven-spot ladybirds in his sample. Explain what and represent in the context of Alex’ experiment.

(ii) State an assumption Alex has made in order to use the distribution in part (a)(i).

(iii) State suitable null and alternative hypotheses that Alex could use to test his belief that more than 65% of the ladybirds in his garden are seven-spot ladybirds.

Alex finds that 21 out of the 25 ladybirds in his sample are seven-spot ladybirds.

(i) The probability of Alex’s observed value or that of a more extreme value is 0.03205. Write this in the form $P(X\geq a)=b$.

(ii) Alex has not yet calculated the critical value for his hypothesis test. Explain why he does not need to do this to come to a conclusion for his test.

Chase buys a board game which contains a six-sided dice. He rolls the dice 150 times and obtains the number six on 15 occasions. Chase wishes to test his belief that the dice is not fair.

(i) State a suitable null hypothesis to test Chase’s belief.

(ii) State a suitable alternative hypothesis for a two-tailed test.

Given that $P(X\leq15)=0.01452$ when $X \sim B ( 150 , \frac{1}{6} )$, test Chase’s belief that the dice is not fair, using a 2% level of significance.

The random variable $X \sim B ( 40 , 0.15 )$. Find:

(i) $P(X<10)$

(ii) $P(X>7)$

(iii) $P(3\leq X<14)$

(iv) $P(5<X<12)$.