Practice Paper 2 (Cambridge International A-Level Mathematics)

A meteorologist records the highest wind speed each day, in knots, at a weather station. The box-and-whisker plot shows the results for 15 days in June.

(i) Give one reason why a box-and-whisker plot is a suitable way to represent these data.

(ii) The highest wind speed recorded in June, 68 knots, is an extreme value. Give a reason why the median is likely to be more suitable than the mean as a measure of the average wind speed in June.

The box-and-whisker plot for 15 days in July has been added to the same diagram.

Make two comparisons between the highest wind speeds in June and the highest wind speeds in July.

A golfer records her scores for her last 8 rounds of golf, which are given below.

73, 75, 81, 72, 76, 99, 80, 78

(i) Find the median, the lower quartile and the upper quartile of these scores.

(ii) Find the interquartile range.

Find the mean of these scores, and give a reason why the median is likely to be more suitable than the mean as a measure of the golfer's average score.

A café owner records the number of customers served between 12 pm and 2 pm on each weekday for three weeks. The results are as follows.

(i) State the mode and find the median of the number of customers served.

(ii) For these data, $\sum x = 851$ and $\sum x^{2} = 48965$. Calculate the mean and the standard deviation of the number of customers served.

State what feature of the data accounts for the mean being greater than the median.

Isabel has two boxes of chocolates. Box $A$ contains 4 caramels and 6 toffees, and box $B$ contains 5 caramels and $x$ toffees. Isabel chooses a chocolate at random from box $A$ and places it in box $B$. Her friend then chooses a chocolate at random from box $B$.

Show that the probability that Isabel's friend chooses a caramel is $\frac{27}{5(x+6)}$.

Given that the probability that Isabel's friend chooses a toffee is $\frac{11}{20}$, find the value of $x$.

In an online chess tournament, each match begins with a standard game, which is either won outright, lost outright or drawn. If the standard game is drawn, a single sudden-death game is played to decide the winner of the match. A sudden-death game cannot be drawn. A player who wins a match goes on to play another match.

In his first match, the probability that Jamie wins outright is 0.7 and the probability that he loses outright is 0.1.

If Jamie wins his first match outright, the probabilities that he wins or loses his second match outright are again 0.7 and 0.1.

If Jamie wins his first match after a sudden-death game, the probability that he wins his second match outright is 0.55 and the probability that he loses it outright is 0.3.

The probability that Jamie wins any sudden-death game is $p$.

Given that the probability that Jamie wins both of his first two matches is 0.6508, find the value of $p$.

Find the probability that Jamie won both of his first two matches outright, given that he won both of these matches.

Find the number of different arrangements of the 10 letters in the word POSITIVITY.

Find the number of different arrangements of the 10 letters in the word POSITIVITY in which

(i) the three Is are all together,

(ii) the two Ts are not next to each other.

An arrangement of the 10 letters in the word POSITIVITY is chosen at random.

Find the probability that the four vowels, O, I, I and I, are not all together.

Farmer Kate has a herd of 50 alpacas. From previous testing of the herd, she knows that each alpaca has a probability of 0.95 of testing negative for tuberculosis (TB). She chooses a random sample of 8 alpacas from the herd and tests them for TB.

$N$ is the number of alpacas in the sample that test negative for TB, and $T$ is the number of alpacas in the sample that test positive for TB.

(i) State the distribution of $N$, giving the values of its parameters.

(ii) State the distribution of $T$, giving the values of its parameters.

Find the probability that the number of alpacas in the sample that test positive for TB is

(i) zero,

(ii) more than 2.

The lengths of the leaves of a certain type of tree are normally distributed with mean $μ$ cm and standard deviation 6 cm. The probability that a randomly chosen leaf has a length less than 51 cm is 0.8665.

Find the value of $μ$.

The masses of peaches grown in an orchard are normally distributed with mean 93 grams and standard deviation $σ$ grams. The probability that a randomly chosen peach has a mass greater than 101 grams is 0.0735.

Find the value of $σ$.

The discrete random variable $X$ has the probability distribution shown in the following table.

Without calculating the value of $p$, explain why $\text{E} ( X ) = 2$.

Find the value of $p$.

Find $\text{Var} ( X )$.

Find \text{P}\left(X > \text{E}(X)\right).

For each of the following, state with a reason whether the random variable in question is a discrete random variable or a continuous random variable.

(i) 100 red squirrels from the wild are sampled. The random variable $A$ is the tail length of a randomly selected red squirrel.

(ii) 100 students sit a test which is marked out of 50. The random variable $B$ is the number of marks achieved by a randomly selected student.

(iii) 100 men are in a shoe shop. The random variable $C$ is the shoe size of a randomly selected man.

The following histogram shows the distribution of results when a large number of measurements of the specified random variable $D$ are made.

State with a reason whether a normal distribution would be appropriate for modelling the random variable.

A taxi company records the times, to the nearest minute, that customers wait for their taxis to arrive. The times for a random sample of 20 customers are as follows.

Find the median and the interquartile range of these waiting times.

On the grid, draw a box-and-whisker plot to represent these waiting times.

The times, in days, spent cocooned by 15 Monarch butterflies and 25 Common Blue butterflies were recorded. The table shows the means and the standard deviations of the times.

Given that the mean time for all 40 butterflies is 11.93 days, find the mean time for the Monarch butterflies and complete the table.

Find the standard deviation of the times spent cocooned by all 40 butterflies.

The times taken, $t$ seconds, by 80 students to run one lap of a sports hall are represented in the cumulative frequency graph.

Complete the frequency table.

Estimate the mean and the interquartile range of the times.

Any times under 32 seconds or over 90 seconds are recorded as unusual. Use the graph to estimate the percentage of the students' times that are recorded as unusual.