Practice Paper 2 (Edexcel IAL Maths)

A snack shop owner has noticed that the sale of energy drinks seems to increase later in the school term. He decides to collect data over the final ten days of a school term to see if the sale of the energy drinks per day, $h$, increases as the number of days until the school holidays, $d$, decreases.

(i) What type of correlation is the snack shop owner testing for?

(ii) State which of the two variables is the explanatory variable.

Over the ten days the snack shop owner collects the following summary statistics:

$\sum d=73$ $\sum d^{2}=805$ $\sum dh=351$

Find the value of the product moment correlation coefficient between $d$ and $h$ correct to 4 decimal places.

The snack shop owner uses this data to calculate the regression line of $d$ on $h$ and uses it to predict the number of energy drinks he will sell on the first day of the new term, when there are still 90 days until the holidays. State two reasons why this is unlikely to give a reliable prediction.

The number of goals scored by the 24 teams that played in the first 44 games of the UEFA Euro cup 2020 can be summarised in the table below.

Estimate the mean number of goals scored by each team.

Find the standard deviation of the number of goals scored by each team.

Hugo, a newly appointed HR administrator for a company, has been asked to investigate the number of absences within the IT department. The department contains 23 employees, and the box plot below summarises the data for the number of days that individual employees were absent during the previous quarter.

An outlier is an observation that falls either more than 1.5 $\times$ (interquartile range) above the upper quartile or less than 1.5 $\times$ (interquartile range) below the lower quartile.

Show that these data have an outlier, and state its value.

For the 23 employees within the department, Hugo has the summary statistics:

$\sum x=286$ and $\sum x^{2}=4238$

Hugo investigates the employee corresponding to the outlier value found in part (a) and discovers that this employee had a long-term illness. Hugo decides not to include that value in the data for the department.

Assuming that there are no other outliers, calculate the mean and standard deviation of the number of days absent for the remaining employees.

A game is played using a fair spinner with four sectors numbered 1 to 4, as well as a fair eight-sided dice with its sides numbered 1 to 8. The spinner is spun and the dice is rolled, and the score in the game is determined as follows:

• if the number on the spinner is higher than the number on the dice, then the score is the sum of the two numbers;
• if the number on the spinner is lower than the number on the dice, then the score is the (positive) difference of the two numbers;
• if the numbers on the spinner and the dice are equal, then the score is the product of the two numbers.

Find the probability of the score in the game being

(i) exactly 7

(ii) 10 or more

(iii) a triangular number (1, 3, 6, 10, 15, 21, …).

Shalit has seven identical red socks and $x$ identical blue socks in his drawer. He takes two socks out of his drawer at random and puts them on.

Find expressions for $a,b,c$ and $d$ on the tree diagram above.

The probability that Shalit took two red socks, given that he took two socks of the same colour is $\frac{7}{8}$. Find the value of $x$.

All the odd numbers from 1 to 99 inclusive are written on individual pieces of paper and placed inside a bag. A piece of paper is picked at random and the odd number $Y$ is recorded.

Find the expectation and standard deviation of $Y$.

The masses of turkeys can be modelled using a normal distribution with a mean of 4.7 kg and a variance of 1.9 kg². Nicholas, a farmer, classes a turkey as ‘holiday ready’ if it weighs more than 5.5 kg.

A turkey is selected at random. Given that it is ‘holiday ready’, find the probability that it weighs less than 6 kg.

Nicholas takes a large sample of ‘holiday ready’ turkeys, estimate the median mass of the turkeys in his sample.

The manager of a local supermarket collected data on the distance a person’s house was from the supermarket, $d$ miles, and the average total cost of the person’s shopping, $c$ dollars. The information is given in the table below.

The manager codes the data such that $x=10d$ and $y=c-30.$

Complete the table below for the values of $x$ and $y$.

Find the mean of $x,\overline{x}$, and show that the mean of $y$ is $\overline{y}=5$

Show that $S_{xx}=24$ and find the value of $S_{xy}.$

The equation of the least squares regression line of $y$ on $x$ is written in the form $y=a+bx,$ where $b=\frac{S_{xy}}{S_{xx}}$ and $a=\overline{y}-b\overline{x}$. Show that $b=\frac{5}{3}$ and find the value of $a$.

By substituting $x=10d$ and $y=c-30$ into your answer for part (d), show that the least squares regression line of $c$ on $d$ is $c=25+\frac{50}{3}d$.

Lucy is working with some grouped, continuous data. For a set of 100 items of data, she has calculated that$\sum xf=357$ and $\sum(x-\overline{x})^{2}f=42$, where $f$ is the frequency for each group.

(i) Give a reason why Lucy has decided to group her data.

(ii) Briefly explain what is meant by $\sum xf$ and $\sum(x-\overline{x})^{2}f$.

To calculate the standard deviation, Lucy could choose to use either of the following two formulae.

standard deviation = $\sqrt{ \frac{sum ( x - \bar{x} )^{2} f}{sum f} }$ or $\sqrt{\frac{\sum x^{2}f}{\sum f}}-\overline{x}^{2}$ where $\overline{x}=\frac{\sum xf}{\sum f}$

(i) Calculate the mean of Lucy’s data.

(ii) Using the appropriate formula from above, calculate the standard deviation of Lucy’s data.

A hotel manager recorded the number of towels that went missing at the end of each day for 12 days. The results are below.

2 4 1 0 3 4

3 9 3 2 4 5

Find the mean and the standard deviation for the number of towels missing at the end of each day. You may use the summary statistics $n=11$, $\sum x=40$, $\sum x^{2}=190$ with the formulae $\overline{x}=\frac{\sum x}{n}$ and $\\sigma = \sqrt{ \frac{sum x^{2}}{n} - ( \bar{x} )^{2}}$.

An outlier is defined as any data value lying outside of 2 standard deviations of the mean. Find any outliers in the data.

The histogram below shows the distribution of masses, in grams, of 80 newly hatched ducklings:

Find the probability that a duckling chosen at random has a mass less than 54 g.

Estimate the probability that a duckling chosen at random has a mass greater than 53 g.

A lifestyle magazine for ancient historians recently conducted a poll of its readership. The following Venn diagram shows some of the results for the numbers of the historians polled who said that they were fans of any, all or none of the following three ancient Near Eastern rulers – the Hittite king Suppiluliuma I ($S$), the Assyrian king Ashurbanipal ($A$), and the Akkadian king Manishtushu ($M$).

180 historians were polled in total. For each historian who was only a fan of Manishtushu, there were 7 who were only fans of Suppiluliuma I. For each two historians who were fans of both Ashurbanipal and Manishtushu but not of Suppiluliuma I, there were nine historians who were fans of all three rulers. Most of the historians were, as expected, fans of at least one of the three rulers, but it was also discovered that being a fan of Ashurbanipal and being a fan of Manishtushu were independent events.

One of the historians who was polled is selected at random. Given that the historian chosen is a fan of at least one of the three rulers, find the exact probability that the historian is a fan of:

(i) Suppiluliuma I

(ii) at least two the three rulers

(iii) exactly two of the three rulers.