Practice Paper 1 (Edexcel IAL Maths)

A class is asked to collect a sample of bivariate data. They collect data on the shoe size, S, and the arm span, A cm, of 20 randomly selected boys from the class.

Explain what is meant by the term ‘bivariate data’.

The class plot the data in a scatter diagram and find the equation of the regression line of* A* on S to be A=4.5 S + 133. These are both plotted in the diagram below.

(i) Interpret the value 4.5 in the context of the question.

(ii) Interpret the value 133 in the context of the question.

(iii) Explain how the sign of the coefficient of S in the equation is related to the correlation shown in the scatter diagram.

Some entomologists were studying the amount of time two different species of butterflies spent cocooned. The table shows the means and standard deviations of the time spent cocooned, measured in days, by 15 Monarch butterflies and 25 Common Blue butterflies.

Given that the overall mean time for all 40 butterflies was 11.93 days, calculate the mean number of days the Monarch butterflies spent cocooned and complete the table.

Calculate the overall standard deviation of the time spent cocooned by all 40 butterflies.

There are 180 dogs in a rescue shelter. The histogram below shows the highest sound level reached by each individual dog’s bark, measured in decibels (dB).

Write down the underlying feature associated with each of the bars in a histogram.

Estimate how many dogs had a bark which ranged between 99 dB and 107 dB.

A bag contains 10 black tokens and 6 white tokens. A token is drawn from the bag and its colour recorded, and then a fair coin is flipped. If the coin lands on heads then a second token is drawn from the bag without replacing the first token. If the coin lands on tails then the first token is replaced in the bag before a second token is drawn.

Draw a tree diagram to represent this experiment.

Find the probability that the second token drawn is white.

$A,B$and $C$ are three events such that $P(C)=0.37,P(A^{'}\cap B^{'}\cap C)=0.18,P(A\cap B\cap C)=0.09,$ and $P((A\cup B\cup C)^{'})=0.21$. In addition the following two relations hold:

$P ( A \cap B \cap C^{'} ) ∶ P ( A \cap B^{'} \cap C ) ∶ P ( A^{'} \cap B \cap C ) = 3 : 4 : 1 P ( A ) ∶ P ( B ) = 9 : 10$

Draw a Venn diagram to illustrate the probabilities.

Find:

(i) $P(A\cup B^{'}\cup C)$

(ii) $P ( A \cup ( B \cap C^{'} ) \vert A \cup C )$

(iii) $P ( B \cup C \vert ( A \cup B \cup C ) \cup ( A \cup B \cup C )^{'} )$

The random variable $X$ has the probability function

$\mathrm{P} ( X = x ) = \{ \begin{gathered} \frac{1}{k} & x = 1 , 2 , 3 , 4 , 5 , 6 \\ 0 otherwise. \end{gathered} )$

(i) Write down the value of $k$.

(ii) Write down the name of this probability distribution.

(iii) Find the values of $E(X)$ and $\mathrm{Var}(X)$.

For the random variable,$X \sim N ( 32 , 9 )$,

(i) Write down the mean and standard deviation.

(ii) Draw a sketch of the graph and clearly label the mean.

(i) With the help of your diagram, explain how you should know, without carrying out any calculations, that P$(X\leq34)>0.5.$

(ii) With the help of your diagram, explain how you should know, without carrying out any calculations, that P$(X\geq38)<0.5.$

Use the formula $Z=\frac{X-μ}{σ}$ to find appropriate $z$-values and then use the table of $z$-values to calculate, to three decimal places,

(i) $P(X\leq34)$

(ii) $P(X\geq38)$

(iii) $P(34\leq X\leq38).$

Explain what is measured by the Pearson product moment correlation coefficient.

The product moment correlation coefficient between two variables is denoted $r$. Five different values of $r$, rounded to four decimal places, are given below:

$r_{1}=0.0000\,r_{2}=0.9812\,r_{3}=-1.0000\,r_{4}=0.7652\,r_{5}=-0.7098$

Match each of the following four scatter graphs, showing observations from different bivariate data sets, to one of the values of $r$ given above. You should use each given value of $r$ no more than once.

Sketch a scatter graph for the remaining value of $r$ from the list above.

Zisien measures the speeds, $x$ miles per hour, of a number of cars passing her house one day. She knows that the speed limit is 30 miles per hour so she decides to use the coding $y$ = $x$ – 30 when she records the data.

She finds that $Σy$=13.4 and $Σy^{2}$= 1470.

Is it more likely that more of the cars in Zisien’s sample were going over or under the speed limit that day? Give a reason for your answer.

Given that the mean speed of the cars in Zisien’s sample, $\overline{x}$, is 30.67 miles per hour, calculate the standard deviation of $x$.

Zisien’s sister, Ying, used the code $z$ = $x$ – 20 to record the data for the same cars.

(i) Calculate the value of $Σz$.

(ii) Ying decides to work out the median of her coded data and discovers that it is 9.4. Use this information to comment on your answer to part (a).