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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

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/62/M/J/10/1, Topic: Representation of data The times in minutes for seven students to become proficient at a new computer game were measured. The results are shown below.$\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]\$

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