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Name of student | Date | ||||

Adm. number | Year/grade | Stream | |||

Subject | Probability & Statistics 1 (S1) | Variant(s) | P62 | ||

Start time | Duration | Stop time |

Qtn No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Total |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Marks | 4 | 5 | 5 | 5 | 6 | 7 | 6 | 9 | 7 | 9 | 10 | 9 | 82 |

Score |

Question 1 Code: 9709/62/M/J/18/1, Topic: Representation of data

Each of a group of 10 boys estimates the length of a piece of string. The estimates, in centimetres, are as follows.

$$ \begin{array}{llllllllll} 37 & 40 & 45 & 38 & 36 & 38 & 42 & 38 & 40 & 39 \end{array} $$$\text{(i)}$ Find the mode. $[1]$

$\text{(ii)}$ Find the median and the interquartile range. $[3]$

Question 2 Code: 9709/62/M/J/10/2, Topic: The normal distribution

The lengths of new pencils are normally distributed with mean $11 \mathrm{~cm}$ and standard deviation $0.095 \mathrm{~cm}$.

$\text{(i)}$ Find the probability that a pencil chosen at random has a length greater than $10.9 \mathrm{~cm}$. $[2]$

$\text{(ii)}$ Find the probability that, in a random sample of 6 pencils, at least two have lengths less than $10.9 \mathrm{~cm}$. $[3]$

Question 3 Code: 9709/62/M/J/11/2, Topic: The normal distribution

In Scotland, in November, on average $80 \%$ of days are cloudy. Assume that the weather on any one day is independent of the weather on other days.

$\text{(i)}$ Use a normal approximation to find the probability of there being fewer than 25 cloudy days in Scotland in November (30 days). $[4]$

$\text{(ii)}$ Give a reason why the use of a normal approximation is justified. $[1]$

Question 4 Code: 9709/62/M/J/12/2, Topic: Discrete random variables

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

$$ \begin{array}{|c|c|c|c|} \hline x & 2 & 4 & 6 \\ \hline \mathrm{P}(X=x) & 0.5 & 0.4 & 0.1 \\ \hline \end{array} $$Two independent values of $X$ are chosen at random. The random variable $Y$ takes the value 0 if the two values of $X$ are the same. Otherwise the value of $Y$ is the larger value of $X$ minus the smaller value of $X$.

$\text{(i)}$ Draw up the probability distribution table for $Y$. $[1]$

$\text{(ii)}$ Find the expected value of $Y$. $[4]$

Question 5 Code: 9709/62/M/J/12/4, Topic: Representation of data

The back-to-back stem-and-leaf diagram shows the values taken by two variables $A$ and $B$.

$\text{(i)}$ Find the median and the interquartile range for variable $A$. $[3]$

$\text{(ii)}$ You are given that, for variable $B$, the median is $0.171$, the upper quartile is $0.179$ and the lower quartile is $0.164$. Draw box-and-whisker plots for $A$ and $B$ in a single diagram on graph paper. $[3]$

Question 6 Code: 9709/62/M/J/15/4, Topic: Probability

Nikita goes shopping to buy a birthday present for her mother. She buys either a scarf, with probability $0.3$, or a handbag. The probability that her mother will like the choice of scarf is $0.72$. The probability that her mother will like the choice of handbag is $x$. This information is shown on the tree diagram. The probability that Nikita's mother likes the present that Nikita buys is $0.783$.

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

$\text{(ii)}$ Given that Nikita's mother does not like her present, find the probability that the present is a scarf. $[4]$

Question 7 Code: 9709/62/M/J/18/4, Topic: Discrete random variables

Mrs Rupal chooses 3 animals at random from 5 dogs and 2 cats. The random variable $X$ is the number of cats chosen.

$\text{(i)}$ Draw up the probability distribution table for $X$. $[4]$

$\text{(ii)}$ You are given that $\mathrm{E}(X)=\frac{6}{7}$. Find the value of $\operatorname{Var}(X)$. $[2]$

Question 8 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 9 Code: 9709/62/M/J/18/5, Topic: Representation of data

The lengths, $t$ minutes, of 242 phone calls made by a family over a period of 1 week are summarised in the frequency table below.

$$ \begin{array}{|l|c|c|c|c|c|} \hline \text{Length of phone call (t minutes)} & 0 < t \leqslant 1 & 1 < t \leqslant 2 & 2 < t \leqslant 5 & 5 < t \leqslant 10 & 10 < t \leqslant 30 \\ \hline \text{Frequency} & 14 & 46 & 102 & a & 40 \\ \hline \end{array} $$$\text{(i)}$ Find the value of $a$. $[1]$

$\text{(ii)}$ Calculate an estimate of the mean length of these phone calls. $[2]$

$\text{(iii)}$ On the grid, draw a histogram to illustrate the data in the table. $[4]$

Question 10 Code: 9709/62/M/J/16/6, Topic: The normal distribution

The time in minutes taken by Peter to walk to the shop and buy a newspaper is normally distributed with mean $9.5$ and standard deviation $1.3$.

$\text{(i)}$ Find the probability that on a randomly chosen day Peter takes longer than $10.2$ minutes. $[3]$

$\text{(ii)}$ On $90 \%$ of days he takes longer than $t$ minutes. Find the value of $t$. $[3]$

$\text{(iii)}$ Calculate an estimate of the number of days in a year ( 365 days) on which Peter takes less than $8.8$ minutes to walk to the shop and buy a newspaper. $[3]$

Question 11 Code: 9709/62/M/J/19/6, Topic: Representation of data

$\text{(i)}$ Give one advantage and one disadvantage of using a box-and-whisker plot to represent a set of data. $[2]$

$\text{(ii)}$ The times in minutes taken to run a marathon were recorded for a group of 13 marathon runners and were found to be as follows.

$$ \begin{array}{lllllllllllll} 180 & 275 & 235 & 242 & 311 & 194 & 246 & 229 & 238 & 768 & 332 & 227 & 228 \end{array} $$State which of the mean, mode or median is most suitable as a measure of central tendency for these times. Explain why the other measures are less suitable. $[3]$

$\text{(iii)}$ Another group of 33 people ran the same marathon and their times in minutes were as follows.

$$ \begin{array}{lllllllllll} 190 & 203 & 215 & 246 & 249 & 253 & 255 & 254 & 258 & 260 & 261 \\ 263 & 267 & 269 & 274 & 276 & 280 & 288 & 283 & 287 & 294 & 300 \\ 307 & 318 & 327 & 331 & 336 & 345 & 351 & 353 & 360 & 368 & 375 \end{array} $$$\text{(a)}$ On the grid below, draw a box-and-whisker plot to illustrate the times for these 33 people. $[4]$

$\text{(b)}$ Find the interquartile range of these times. $[1]$

Question 12 Code: 9709/62/M/J/17/7, Topic: Probability

During the school holidays, each day Khalid either rides on his bicycle with probability $0.6$, or on his skateboard with probability $0.4$. Khalid does not ride on both on the same day. If he rides on his bicycle then the probability that he hurts himself is $0.05$. If he rides on his skateboard the probability that he hurts himself is $0.75$.

$\text{(i)}$ Find the probability that Khalid hurts himself on any particular day. $[2]$

$\text{(ii)}$ Given that Khalid hurts himself on a particular day, find the probability that he is riding on his skateboard. $[2]$

$\text{(iii)}$ There are 45 days of school holidays. Show that the variance of the number of days Khalid rides on his skateboard is the same as the variance of the number of days that Khalid rides on his bicycle. $[2]$

$\text{(iv)}$ Find the probability that Khalid rides on his skateboard on at least 2 of 10 randomly chosen days in the school holidays. $[3]$