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MATHEMATICS 9709

Cambridge International AS and A Level

Name of student Date
Adm. number Year/grade Stream
Subject Probability & Statistics 1 (S1) Variant(s) P61, P62, P63
Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 7 8 9 10 11 12 Total
Marks 5 4 5 7 6 7 6 9 10 8 8 11 86
Score

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Question 1 Code: 9709/62/M/J/10/1, Topic: Representation of data

The times in minutes for seven students to become proficient at a new computer game were measured. The results are shown below.

$\begin{array}{lllllll}15 & 10 & 48 & 10 & 19 & 14 & 16\end{array}$

$\text{(i)}$ Find the mean and standard deviation of these times. $[2]$

$\text{(ii)}$ State which of the mean, median or mode you consider would be most appropriate to use as a measure of central tendency to represent the data in this case. $[1]$

$\text{(iii)}$ For each of the two measures of average you did not choose in part $\text{(ii)}$, give a reason why you consider it inappropriate. $[2]$

Question 2 Code: 9709/61/M/J/14/1, Topic: The normal distribution

The petrol consumption of a certain type of car has a normal distribution with mean 24 kilometres per litre and standard deviation $4.7$ kilometres per litre. Find the probability that the petrol consumption of a randomly chosen car of this type is between $21.6$ kilometres per litre and $28.7$ kilometres per litre. $[4]$

Question 3 Code: 9709/63/M/J/16/1, Topic: Probability

In a group of 30 adults, 25 are right-handed and 8 wear spectacles. The number who are right-handed and do not wear spectacles is 19.

$\text{(i)}$ Copy and complete the following table to show the number of adults in each category. $[2]$

$$ \begin{array}{|l|c|c|c|} \hline & \text{Wears spectacles} & \text{Does not wear spectacles} & \text{Total} \\ \hline \text{Right-handed} & & & \\ \hline \text{Not right-handed} & & & \\ \hline \text{Total} & & & 30 \\ \hline \end{array} $$

An adult is chosen at random from the group. Event $X$ is 'the adult chosen is right-handed'; event $Y$ is 'the adult chosen wears spectacles'.

$\text{(ii)}$ Determine whether $X$ and $Y$ are independent events, justifying your answer. $[3]$

Question 4 Code: 9709/62/M/J/11/3, Topic: Representation of data

A sample of 36 data values, $x$, gave $\Sigma(x-45)=-148$ and $\Sigma(x-45)^{2}=3089$.

$\text{(i)}$ Find the mean and standard deviation of the 36 values. $[3]$

$\text{(ii)}$ One extra data value of 29 was added to the sample. Find the standard deviation of all 37 values. $[4]$

Question 5 Code: 9709/63/M/J/18/3, Topic: Probability

The members of a swimming club are classified either as 'Advanced swimmers' or 'Beginners'. The proportion of members who are male is $x$, and the proportion of males who are Beginners is $0.7$. The proportion of females who are Advanced swimmers is $0.55$. This information is shown in the tree diagram.

For a randomly chosen member, the probability of being an Advanced swimmer is the same as the probability of being a Beginner.

$\text{(i)}$ Find $x$. $[3]$

$\text{(ii)}$ Given that a randomly chosen member is an Advanced swimmer, find the probability that the member is male. $[3]$

Question 6 Code: 9709/62/M/J/13/4, Topic: Discrete random variables

Robert uses his calculator to generate 5 random integers between 1 and 9 inclusive.

$\text{(i)}$ Find the probability that at least 2 of the 5 integers are less than or equal to $4.$ $[3]$

Robert now generates $n$ random integers between 1 and 9 inclusive. The random variable $X$ is the number of these $n$ integers which are less than or equal to a certain integer $k$ between 1 and 9 inclusive. It is given that the mean of $X$ is 96 and the variance of $X$ is 32.

$\text{(ii)}$ Find the values of $n$ and $k$. $[4]$

Question 7 Code: 9709/62/M/J/17/4, Topic: Probability

Two identical biased triangular spinners with sides marked 1, 2 and 3 are spun. For each spinner, the probabilities of landing on the sides marked 1,2 and 3 are $p, q$ and $r$ respectively. The score is the sum of the numbers on the sides on which the spinners land. You are given that $\mathrm{P}($ score is 6$)=\frac{1}{36}$ and $\mathrm{P}($ score is 5$)=\frac{1}{9}.$ Find the values of $p, q$ and $r$. $[5]$

Question 8 Code: 9709/61/M/J/12/5, Topic: Representation of data

The lengths of the diagonals in metres of the 9 most popular flat screen TVs and the 9 most popular conventional TVs are shown below.

$$ \begin{array}{rlllllllll} \text {Flat screen :} & 0.85 & 0.94 & 0.91 & 0.96 & 1.04 & 0.89 & 1.07 & 0.92 & 0.76 \\ \text {Conventional :} & 0.69 & 0.65 & 0.85 & 0.77 & 0.74 & 0.67 & 0.71 & 0.86 & 0.75 \end{array} $$

$\text{(i)}$ Represent this information on a back-to-back stem-and-leaf diagram. $[4]$

$\text{(ii)}$ Find the median and the interquartile range of the lengths of the diagonals of the 9 conventional TVs. $[3]$

$\text{(iii)}$ Find the mean and standard deviation of the lengths of the diagonals of the 9 flat screen TVs. $[2]$

Question 9 Code: 9709/63/M/J/12/5, Topic: Probability

Suzanne has 20 pairs of shoes, some of which have designer labels. She has 6 pairs of high-heeled shoes, of which 2 pairs have designer labels. She has 4 pairs of low-heeled shoes, of which 1 pair has designer labels. The rest of her shoes are pairs of sports shoes. Suzanne has 8 pairs of shoes with designer labels in total.

$\text{(i)}$ Copy and complete the table below to show the number of pairs in each category. $[2]$

$$ \begin{array}{|l|c|c|c|} \hline & \text{Designer labels} & \text{No designer labels} & \text{Total} \\ \hline \text{High-heeled shoes} & & & \\ \hline \text{Low-heeled shoes} & & & \\ \hline \text{Sports shoes} & & & \\ \hline \text{Total} & & & 20 \\ \hline \end{array} $$ Suzanne chooses 1 pair of shoes at random to wear.

$\text{(ii)}$ Find the probability that she wears the pair of low-heeled shoes with designer labels. $[1]$

$\text{(iii)}$ Find the probability that she wears a pair of sports shoes. $[1]$

$\text{(iv)}$ Find the probability that she wears a pair of high-heeled shoes, given that she wears a pair of shoes with designer labels. $[1]$

$\text{(v)}$ State with a reason whether the events 'Suzanne wears a pair of shoes with designer labels' and 'Suzanne wears a pair of sports shoes' are independent. $[2]$

Suzanne chooses 1 pair of shoes at random each day.

$\text{(vi)}$ Find the probability that Suzanne wears a pair of shoes with designer labels on at most 4 days out of the next 7 days. $[3]$

Question 10 Code: 9709/63/M/J/14/5, Topic: The normal distribution

When Moses makes a phone call, the amount of time that the call takes has a normal distribution with mean $6.5$ minutes and standard deviation $1.76$ minutes.

$\text{(i)}$ $90 \%$ of Moses's phone calls take longer than $t$ minutes. Find the value of $t$. $[3]$

$\text{(ii)}$ Find the probability that, in a random sample of 9 phone calls made by Moses, more than 7 take a time which is within 1 standard deviation of the mean. $[5]$

Question 11 Code: 9709/63/M/J/19/5, Topic: Discrete random variables

On average, $34 \%$ of the people who go to a particular theatre are men.

$\text{(i)}$ A random sample of 14 people who go to the theatre is chosen. Find the probability that at most 2 people are men. $[3]$

$\text{(ii)}$ Use an approximation to find the probability that, in a random sample of 600 people who go to the theatre, fewer than 190 are men. $[5]$

Question 12 Code: 9709/61/M/J/20/7, Topic: Representation of data

The numbers of chocolate bars sold per day in a cinema over a period of 100 days are summarised in the following table.

$$ \begin{array}{|l|c|c|c|c|c|} \hline \text{Number of chocolate bars sold} & 1-10 & 11-15 & 16-30 & 31-50 & 51-60 \\ \hline \text{Number of days} & 18 & 24 & 30 & 20 & 8 \\ \hline \end{array} $$

$\text{(a)}$ Draw a histogram to represent this information. $[5]$

$\text{(b)}$ What is the greatest possible value of the interquartile range for the data? $[2]$

$\text{(c)}$ Calculate estimates of the mean and standard deviation of the number of chocolate bars sold. $[4]$

Worked solutions: P1, P3 & P6 (S1)

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