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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 | 7 | 6 | 6 | 8 | 9 | 9 | 9 | 10 | 10 | 10 | 9 | 98 |

Score |

Question 1 Code: 9709/62/M/J/15/2, Topic: Representation of data

120 people were asked to read an article in a newspaper. The times taken, to the nearest second, by the people to read the article are summarised in the following table. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text{Time (seconds)} & 1-25 & 26-35 & 36-45 & 46-55 & 56-90 \\ \hline \text{Number of people} & 4 & 24 & 38 & 34 & 20 \\ \hline \end{array} $$Calculate estimates of the mean and standard deviation of the reading times. $[5]$

Question 2 Code: 9709/63/M/J/11/3, Topic: Representation of data

The following cumulative frequency table shows the examination marks for 300 candidates in country $A$ and 300 candidates in country $B$.

$$ \begin{array}{|l|c|c|c|c|c|c|} \hline \text{Mark} & <10 & <20 & <35 & <50 & <70 & <100 \\ \hline \text{Cumulative frequency, A} & 25 & 68 & 159 & 234 & 260 & 300 \\ \hline \text{Cumulative frequency, B} & 10 & 46 & 72 & 144 & 198 & 300 \\ \hline \end{array} $$$\text{(i)}$ Without drawing a graph, show that the median for country $B$ is higher than the median for country $A$.

$\text{(ii)}$ Find the number of candidates in country $A$ who scored between 20 and 34 marks inclusive.

$\text{(iii)}$ Calculate an estimate of the mean mark for candidates in country $A$. $[4]$

Question 3 Code: 9709/61/M/J/17/4, Topic: Representation of data

The times taken, $t$ seconds, by 1140 people to solve a puzzle are summarised in the table.

$$ \begin{array}{|l|c|c|c|c|c|} \hline \text{Time (t seconds)} & 0 \leqslant t < 20 & 20 \leqslant t < 40 & 40 \leqslant t < 60 & 60 \leqslant t < 100 & 100 \leqslant t < 140 \\ \hline \text{Number of people} & 320 & 280 & 220 & 220 & 100 \\ \hline \end{array} $$$\text{(i)}$ On the grid, draw a histogram to illustrate this information. $[4]$

$\text{(ii)}$ Calculate an estimate of the mean of $t$. $[2]$

Question 4 Code: 9709/63/M/J/19/4, Topic: Permutations and combinations

$\text{(i)}$ Find the number of ways a committee of 6 people can be chosen from 8 men and 4 women if there must be at least twice as many men as there are women on the committee. $[3]$

$\text{(ii)}$ Find the number of ways a committee of 6 people can be chosen from 8 men and 4 women if 2 particular men refuse to be on the committee together. $[3]$

Question 5 Code: 9709/62/M/J/11/5, Topic: Representation of data

A hotel has 90 rooms. The table summarises information about the number of rooms occupied each day for a period of 200 days.

$$ \begin{array}{|l|c|c|c|c|c|c|} \hline \text{Number of rooms occupied} & 1-20 & 21-40 & 41-50 & 51-60 & 61-70 & 71-90 \\ \hline \text{Frequency} & 10 & 32 & 62 & 50 & 28 & 18 \\ \hline \end{array} $$$\text{(i)}$ Draw a cumulative frequency graph on graph paper to illustrate this information. $[4]$

$\text{(ii)}$ Estimate the number of days when over 30 rooms were occupied. $[2]$

$\text{(iii)}$ On $75 \%$ of the days at most $n$ rooms were occupied. Estimate the value of $n$. $[2]$

Question 6 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 7 Code: 9709/62/M/J/16/5, Topic: Representation of data

The following are the maximum daily wind speeds in kilometres per hour for the first two weeks in April for two towns, Bronlea and Rogate.

$$ \begin{array}{|l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline \text{Bronlea} & 21 & 45 & 6 & 33 & 27 & 3 & 32 & 14 & 28 & 24 & 13 & 17 & 25 & 22 \\ \hline \text{Rogate} & 7 & 5 & 4 & 15 & 23 & 7 & 11 & 13 & 26 & 18 & 23 & 16 & 10 & 34 \\ \hline \end{array} $$$\text{(i)}$ Draw a back-to-back stem-and-leaf diagram to represent this information. $[5]$

$\text{(ii)}$ Write down the median of the maximum wind speeds for Bronlea and find the interquartile range for Rogate. $[3]$

$\text{(iii)}$ Use your diagram to make one comparison between the maximum wind speeds in the two towns. $[1]$

Question 8 Code: 9709/61/M/J/10/6, Topic: Permutations and combinations

$\text{(i)}$ Find the number of different ways that a set of 10 different mugs can be shared between Lucy and Monica if each receives an odd number of mugs. $[3]$

$\text{(ii)}$ Another set consists of 6 plastic mugs each of a different design and 3 china mugs each of a different design. Find in how many ways these 9 mugs can be arranged in a row if the china mugs are all separated from each other. $[3]$

$\text{(iii)}$ Another set consists of 3 identical red mugs, 4 identical blue mugs and 7 identical yellow mugs. These 14 mugs are placed in a row. Find how many different arrangements of the colours are possible if the red mugs are kept together. $[3]$

Question 9 Code: 9709/62/M/J/13/6, Topic: Permutations and combinations

A town council plans to plant 12 trees along the centre of a main road. The council buys the trees from a garden centre which has 4 different hibiscus trees, 9 different jacaranda trees and 2 different oleander trees for sale.

$\text{(i)}$ How many different selections of 12 trees can be made if there must be at least 2 of each type of tree? $[4]$

The council buys 4 hibiscus trees, 6 jacaranda trees and 2 oleander trees.

$\text{(ii)}$ How many different arrangements of these 12 trees can be made if the hibiscus trees have to be next to each other, the jacaranda trees have to be next to each other and the oleander trees have to be next to each other? $[3]$

$\text{(iii)}$ How many different arrangements of these 12 trees can be made if no hibiscus tree is next to another hibiscus tree? $[3]$

Question 10 Code: 9709/61/M/J/14/6, Topic: Permutations and combinations

Find the number of different ways in which all 8 letters of the word TANZANIA can be arranged so that

$\text{(i)}$ all the letters $\mathrm{A}$ are together, $[2]$

$\text{(ii)}$ the first letter is a consonant ( $\mathrm{T}, \mathrm{N}, \mathrm{Z}$ ), the second letter is a vowel (A, I), the third letter is a consonant, the fourth letter is a vowel, and so on alternately.

4 of the 8 letters of the word TANZANIA are selected. How many possible selections contain$\text{(iii)}$ exactly $1 \mathrm{~N}$ and $1 \mathrm{~A}$, $[2]$

$\text{(iv)}$ exactly $1 \mathrm{~N}$? $[3]$

Question 11 Code: 9709/62/M/J/10/7, Topic: Permutations and combinations

Nine cards, each of a different colour, are to be arranged in a line.

$\text{(i)}$ How many different arrangements of the 9 cards are possible? $[1]$

The 9 cards include a pink card and a green card.

$\text{(ii)}$ How many different arrangements do not have the pink card next to the green card? $[3]$

Consider all possible choices of 3 cards from the 9 cards with the 3 cards being arranged in a line.

$\text{(iii)}$ How many different arrangements in total of 3 cards are possible? $[2]$

$\text{(iv)}$ How many of the arrangements of 3 cards in part $\text{(iii)}$ contain the pink card? $[2]$

$\text{(v)}$ How many of the arrangements of 3 cards in part $\text{(iii)}$ do not have the pink card next to the green card? $[2]$

Question 12 Code: 9709/62/M/J/16/7, Topic: Permutations and combinations

$\text{(a)}$ Find the number of different arrangements which can be made of all 10 letters of the word WALLFLOWER if

$\text{(i)}$ there are no restrictions, $[1]$

$\text{(ii)}$ there are exactly six letters between the two Ws. $[4]$

$\text{(b)}$ A team of 6 people is to be chosen from 5 swimmers, 7 athletes and 4 cyclists. There must be at least 1 from each activity and there must be more athletes than cyclists. Find the number of different ways in which the team can be chosen. $[4]$