2.02g Calculate mean and standard deviation

1 For \(n\) values of the variable \(x\), it is given that $$\Sigma ( x - 50 ) = 144 \quad \text { and } \quad \Sigma x = 944 .$$ Find the value of \(n\).

5 A fair three-sided spinner has sides numbered 1, 2, 3. A fair five-sided spinner has sides numbered \(1,1,2,2,3\). Both spinners are spun once. For each spinner, the number on the side on which it lands is noted. The random variable \(X\) is the larger of the two numbers if they are different, and their common value if they are the same.

1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 7 } { 15 }\). \includegraphics[max width=\textwidth, alt={}, center]{a3b3ebd1-db9e-4552-9abe-bfdeba786d02-08_69_1569_541_328}
2. Draw up the probability distribution table for \(X\).
3. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

6 The annual salaries, in thousands of dollars, for 11 employees at each of two companies \(A\) and \(B\) are shown below.

1. Represent the data by drawing a back-to-back stem-and-leaf diagram with company \(A\) on the left-hand side of the diagram.
2. Find the median and the interquartile range of the salaries of the employees in company \(A\). [3]
   A new employee joins company \(B\). The mean salary of the 12 employees is now \(\\) 38500$.
3. Find the salary of the new employee.

5 The times taken by 200 players to solve a computer puzzle are summarised in the following table.

1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{1a27e2ca-9be5-48a0-a1aa-01844573f4d4-08_1397_1198_808_516}
2. Calculate an estimate of the mean time taken by these 200 players.
3. Find the greatest possible value of the interquartile range of these times.

7 The heights, in cm, of the 11 basketball players in each of two clubs, the Amazons and the Giants, are shown below.

1. State an advantage of using a stem-and-leaf diagram compared to a box-and-whisker plot to illustrate this information.
2. Represent the data by drawing a back-to-back stem-and-leaf diagram with Amazons on the left-hand side of the diagram.
3. Find the interquartile range of the heights of the players in the Amazons.
   Four new players join the Amazons. The mean height of the 15 players in the Amazons in now 191.2 cm . The heights of three of the new players are \(180 \mathrm {~cm} , 185 \mathrm {~cm}\) and 190 cm .
4. Find the height of the fourth new player.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 A sports club has a volleyball team and a hockey team. The heights of the 6 members of the volleyball team are summarised by \(\Sigma x = 1050\) and \(\Sigma x ^ { 2 } = 193700\), where \(x\) is the height of a member in cm . The heights of the 11 members of the hockey team are summarised by \(\Sigma y = 1991\) and \(\Sigma y ^ { 2 } = 366400\), where \(y\) is the height of a member in cm .

1. Find the mean height of all 17 members of the club.
2. Find the standard deviation of the heights of all 17 members of the club.

3 The times taken to travel to college by 2500 students are summarised in the table.

1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{d69f6a47-7c88-46b3-9e8f-07727106e987-04_1201_1198_1050_516} From the data, the estimate of the mean value of \(t\) is 31.44 .
2. Calculate an estimate of the standard deviation of the times taken to travel to college.
3. In which class interval does the upper quartile lie?
   It was later discovered that the times taken to travel to college by two students were incorrectly recorded. One student's time was recorded as 15 instead of 5 and the other's time was recorded as 65 instead of 75 .
4. Without doing any further calculations, state with a reason whether the estimate of the standard deviation in part (b) would be increased, decreased or stay the same.

1 A summary of 50 values of \(x\) gives $$\Sigma ( x - q ) = 700 , \quad \Sigma ( x - q ) ^ { 2 } = 14235$$ where \(q\) is a constant.

1. Find the standard deviation of these values of \(x\).
2. Given that \(\Sigma x = 2865\), find the value of \(q\).

4 The times taken, in minutes, to complete a cycle race by 19 cyclists from each of two clubs, the Cheetahs and the Panthers, are represented in the following back-to-back stem-and-leaf diagram. Key: 7 |9| 1 means 97 minutes for Cheetahs and 91 minutes for Panthers

1. Find the median and the interquartile range of the times of the Cheetahs.
   The median and interquartile range for the Panthers are 103 minutes and 14 minutes respectively.
2. Make two comparisons between the times taken by the Cheetahs and the times taken by the Panthers.
   Another cyclist, Kenny, from the Cheetahs also took part in the race. The mean time taken by the 20 cyclists from the Cheetahs was 99 minutes.
3. Find the time taken by Kenny to complete the race.

1 A summary of 20 values of \(x\) gives $$\Sigma ( x - 30 ) = 439 , \quad \Sigma ( x - 30 ) ^ { 2 } = 12405 .$$ A summary of another 25 values of \(x\) gives $$\sum ( x - 30 ) = 470 , \quad \sum ( x - 30 ) ^ { 2 } = 11346 .$$

1. Find the mean of all 45 values of \(x\).
2. Find the standard deviation of all 45 values of \(x\).

4 The back-to-back stem-and-leaf diagram shows the annual salaries of 19 employees at each of two companies, Petral and Ravon. Key: 2 | 31 | 5 means \\(31 200 for a Petral employee and \\)31500 for a Ravon employee.

1. Find the median and the interquartile range of the salaries of the Petral employees.
   The median salary of the Ravon employees is \(\\) 33800\(, the lower quartile is \)\\( 32000\) and the upper quartile is \(\\) 34400$.
2. Represent the data shown in the back-to-back stem-and-leaf diagram by a pair of box-and-whisker plots in a single diagram. \includegraphics[max width=\textwidth, alt={}, center]{f979a442-da05-410b-84dc-3da3286514a0-07_707_1395_477_335}
3. Comment on whether the mean or the median would be a better representation of the data for the employees at Petral.

7 Helen measures the lengths of 150 fish of a certain species in a large pond. These lengths, correct to the nearest centimetre, are summarised in the following table.

1. Draw a cumulative frequency graph to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{f7c0e35d-1889-4e5b-b094-f467052a66cf-10_1593_1296_790_466}
2. 40\% of these fish have a length of \(d \mathrm {~cm}\) or more. Use your graph to estimate the value of \(d\).
   The mean length of these 150 fish is 15.295 cm .
3. Calculate an estimate for the variance of the lengths of the fish.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 A driver records the distance travelled in each of 150 journeys. These distances, correct to the nearest km , are summarised in the following table.

1. Draw a cumulative frequency graph to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{3f05dc2a-b466-40bc-9f5f-0fd2bff120c8-06_1593_1397_852_415}
2. For 30\% of these journeys the distance travelled is \(d \mathrm {~km}\) or more. Use your graph to estimate the value of \(d\).
3. Calculate an estimate of the mean distance travelled for the 150 journeys.

3 The times taken, in minutes, by 150 students to complete a puzzle are summarised in the table.

1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{d1a3524c-a3b5-45fe-86a7-5cbda087efcd-06_1193_1489_886_328}
2. Calculate an estimate for the mean time for these students to complete the puzzle.
3. In which class interval does the lower quartile of the times lie?

5 Anil is taking part in a tournament. In each game in this tournament, players are awarded 2 points for a win, 1 point for a draw and 0 points for a loss. For each of Anil's games, the probabilities that he will win, draw or lose are \(0.5,0.3\) and 0.2 respectively. The results of the games are all independent of each other. The random variable \(X\) is the total number of points that Anil scores in his first 3 games in the tournament.

1. Show that \(\mathrm { P } ( X = 2 ) = 0.114\).
2. Complete the probability distribution table for \(X\).

   \(x\)	0	1	2	3	4	5	6
   \(\mathrm { P } ( \mathrm { X } = \mathrm { x } )\)			0.114	0.207	0.285		0.125

3. Find the value of \(\operatorname { Var } ( X )\).

6 The times, \(t\) minutes, taken by 150 students to complete a particular challenge are summarised in the following cumulative frequency table.

1. Draw a cumulative frequency graph to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{033ceb76-8fd4-4a89-ab05-5e20039d1c8d-08_1689_1195_744_516}
2. \(24 \%\) of the students take \(k\) minutes or longer to complete the challenge. Use your graph to estimate the value of \(k\).
3. Calculate estimates of the mean and the standard deviation of the time taken to complete the challenge.

2 A bag contains 5 red balls and 3 blue balls. Sadie takes 3 balls at random from the bag, without replacement. The random variable \(X\) represents the number of red balls that she takes.

1. Show that the probability that Sadie takes exactly 1 red ball is \(\frac { 15 } { 56 }\).
2. Draw up the probability distribution table for \(X\).
3. Given that \(\mathrm { E } ( X ) = \frac { 15 } { 8 }\), find \(\operatorname { Var } ( X )\).

7 A particular piece of music was played by 91 pianists and for each pianist, the number of incorrect notes was recorded. The results are summarised in the table.

1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{9f0f0e3c-7baf-42eb-a4fb-9ce61922c3cd-10_1488_1493_785_365}
2. State which class interval contains the lower quartile and which class interval contains the upper quartile. Hence find the greatest possible value of the interquartile range.
3. Calculate an estimate for the mean number of incorrect notes.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 A summary of 40 values of \(x\) gives the following information: $$\Sigma ( x - k ) = 520 , \quad \Sigma ( x - k ) ^ { 2 } = 9640$$ where \(k\) is a constant.

1. Given that the mean of these 40 values of \(x\) is 34 , find the value of \(k\).
2. Find the variance of these 40 values of \(x\).

3 The times taken, in minutes, by 360 employees at a large company to travel from home to work are summarised in the following table.

1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{217c5a58-2966-4b86-b3b6-9d1676d2979c-04_1198_1200_836_516}
2. Calculate an estimate of the mean time taken by an employee to travel to work.

4 The times, to the nearest minute, of 150 athletes taking part in a charity run are recorded. The results are summarised in the table.

1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{e8c2b51e-d788-4917-829e-1b056a24f520-08_1493_1397_936_415}
2. Calculate estimates for the mean and standard deviation of the times taken by the athletes.

4 The weights, \(x \mathrm {~kg}\), of 120 students in a sports college are recorded. The results are summarised in the following table.

1. Draw a cumulative frequency graph to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{82c36c11-878c-47d1-a07f-fbf8b2a22d97-06_1390_1389_660_418}
2. It is found that \(35 \%\) of the students weigh more than \(W \mathrm {~kg}\). Use your graph to estimate the value of \(W\).
3. Calculate estimates for the mean and standard deviation of the weights of the 120 students. [6]

3 The time taken, in minutes, to walk to school was recorded for 200 pupils at a certain school. These times are summarised in the following table.

1. Draw a cumulative frequency graph to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-04_1217_1509_705_278}
2. Use your graph to estimate the median and the interquartile range of the data. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-05_2723_35_101_20}
3. Calculate an estimate for the mean value of the times taken by the 200 pupils to walk to school.

6 Teams of 15 runners took part in a charity run last Saturday. The times taken, in minutes, to complete the course by the runners from the Falcons and the runners from the Kites are shown in the table.

1. Draw a back-to-back stem-and-leaf diagram to represent this information, with the Falcons on the left-hand side.
2. Find the median and the interquartile range of the times for the Falcons.
   Let \(x\) and \(y\) denote the times, in minutes, of a runner from the Falcons and a runner from the Kites respectively. It is given that $$\sum x = 792 , \quad \sum x ^ { 2 } = 43504 , \quad \sum y = 783 , \quad \sum y ^ { 2 } = 42223 .$$
3. Find the mean and the standard deviation of the times taken by all 30 runners from the two teams.

4 On a certain day, the heights of 150 sunflower plants grown by children at a local school are measured, correct to the nearest cm . These heights are summarised in the following table.

1. Draw a cumulative frequency graph to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-06_1600_1301_760_383}
2. Use your graph to estimate the 30th percentile of the heights of the sunflower plants. \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-07_2723_35_101_20}
3. Calculate estimates for the mean and the standard deviation of the heights of the 150 sunflower plants.