2.02g Calculate mean and standard deviation

7 A random sample of 97 people who own mobile phones was used to collect data on the amount of time they spent per day on their phones. The results are displayed in the table below.

1. Calculate estimates of the mean and standard deviation of the time spent per day on these mobile phones.
2. On graph paper, draw a fully labelled histogram to represent the data.

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

1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{3ada18de-c4f7-4049-9032-46b796be83c3-12_1203_1399_833_415}
2. What is the greatest possible value of the interquartile range for the data?
3. Calculate estimates of the mean and standard deviation of the number of chocolate bars sold.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1 Two cricket teams kept records of the number of runs scored by their teams in 8 matches. The scores are shown in the following table.

1. Find the mean and standard deviation of the scores for team \(A\). The mean and standard deviation for team \(B\) are 130.75 and 29.63 respectively.
2. State with a reason which team has the more consistent scores.

2 The following table shows the results of a survey to find the average daily time, in minutes, that a group of schoolchildren spent in internet chat rooms. The mean time was calculated to be 27.5 minutes.

1. Form an equation involving \(f\) and hence show that the total number of children in the survey was 26 .
2. Find the standard deviation of these times.

1 The length of time, \(t\) minutes, taken to do the crossword in a certain newspaper was observed on 12 occasions. The results are summarised below. $$\Sigma ( t - 35 ) = - 15 \quad \Sigma ( t - 35 ) ^ { 2 } = 82.23$$ Calculate the mean and standard deviation of these times taken to do the crossword.

4 The lengths of time in minutes to swim a certain distance by the members of a class of twelve 9 -year-olds and by the members of a class of eight 16 -year-olds are shown below.

1. Draw a back-to-back stem-and-leaf diagram to represent the information above.
2. A new pupil joined the 16 -year-old class and swam the distance. The mean time for the class of nine pupils was now 13.6 minutes. Find the new pupil's time to swim the distance.

5 As part of a data collection exercise, members of a certain school year group were asked how long they spent on their Mathematics homework during one particular week. The times are given to the nearest 0.1 hour. The results are displayed in the following table.

1. Draw, on graph paper, a histogram to illustrate this information.
2. Calculate an estimate of the mean time spent on their Mathematics homework by members of this year group.

6 During January the numbers of people entering a store during the first hour after opening were as follows.

1. Find the values of \(a\) and \(b\).
2. Draw a cumulative frequency graph to represent this information. Take a scale of 2 cm for 10 minutes on the horizontal axis and 2 cm for 50 people on the vertical axis.
3. Use your graph to estimate the median time after opening that people entered the store.
4. Calculate estimates of the mean, \(m\) minutes, and standard deviation, \(s\) minutes, of the time after opening that people entered the store.
5. Use your graph to estimate the number of people entering the store between ( \(m - \frac { 1 } { 2 } s\) ) and \(\left( m + \frac { 1 } { 2 } s \right)\) minutes after opening.

4 The numbers of rides taken by two students, Fei and Graeme, at a fairground are shown in the following table.

1. The mean cost of Fei's rides is \(\\) 2.50\( and the standard deviation of the costs of Fei's rides is \)\\( 0\). Explain how you can tell that the roller coaster and the water slide each cost \(\\) 2.50\( per ride. [2]
2. The mean cost of Graeme's rides is \)\\( 3.76\). Find the standard deviation of the costs of Graeme's rides.

2 The heights, \(x \mathrm {~cm}\), of a group of 82 children are summarised as follows. $$\Sigma ( x - 130 ) = - 287 , \quad \text { standard deviation of } x = 6.9 .$$

1. Find the mean height.
2. Find \(\Sigma ( x - 130 ) ^ { 2 }\).

6 There are 5000 schools in a certain country. The cumulative frequency table shows the number of pupils in a school and the corresponding number of schools.

1. Draw a cumulative frequency graph with a scale of 2 cm to 100 pupils on the horizontal axis and a scale of 2 cm to 1000 schools on the vertical axis. Use your graph to estimate the median number of pupils in a school.
2. \(80 \%\) of the schools have more than \(n\) pupils. Estimate the value of \(n\) correct to the nearest ten.
3. Find how many schools have between 201 and 250 (inclusive) pupils.
4. Calculate an estimate of the mean number of pupils per school.

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

1. Find the mean and standard deviation of the 36 values.
2. One extra data value of 29 was added to the sample. Find the standard deviation of all 37 values.

1 Red Street Garage has 9 used cars for sale. Fairwheel Garage has 15 used cars for sale. The mean age of the cars in Red Street Garage is 3.6 years and the standard deviation is 1.925 years. In Fairwheel Garage, \(\Sigma x = 64\) and \(\Sigma x ^ { 2 } = 352\), where \(x\) is the age of a car in years.

1. Find the mean age of all 24 cars.
2. Find the standard deviation of the ages of all 24 cars.

5 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.

1. Represent this information on a back-to-back stem-and-leaf diagram.
2. Find the median and the interquartile range of the lengths of the diagonals of the 9 conventional TVs.
3. Find the mean and standard deviation of the lengths of the diagonals of the 9 flat screen TVs.

1 The ages, \(x\) years, of 150 cars are summarised by \(\Sigma x = 645\) and \(\Sigma x ^ { 2 } = 8287.5\). Find \(\Sigma ( x - \bar { x } ) ^ { 2 }\), where \(\bar { x }\) denotes the mean of \(x\).

2 The heights, \(x \mathrm {~cm}\), of a group of young children are summarised by $$\Sigma ( x - 100 ) = 72 , \quad \Sigma ( x - 100 ) ^ { 2 } = 499.2 .$$ The mean height is 104.8 cm .

1. Find the number of children in the group.
2. Find \(\Sigma ( x - 104.8 ) ^ { 2 }\).

1 A summary of 30 values of \(x\) gave the following information: $$\Sigma ( x - c ) = 234 , \quad \Sigma ( x - c ) ^ { 2 } = 1957.5 ,$$ where \(c\) is a constant.

1. Find the standard deviation of these values of \(x\).
2. Given that the mean of these values is 86 , find the value of \(c\).

2 A summary of the speeds, \(x\) kilometres per hour, of 22 cars passing a certain point gave the following information: $$\Sigma ( x - 50 ) = 81.4 \quad \text { and } \quad \Sigma ( x - 50 ) ^ { 2 } = 671.0 .$$ Find the variance of the speeds and hence find the value of \(\Sigma x ^ { 2 }\).

6 The weights, \(x\) kilograms, of 144 people were recorded. The results are summarised in the cumulative frequency table below.

1. On graph paper, draw a cumulative frequency graph to represent these results.
2. 64 people weigh more than \(c \mathrm {~kg}\). Use your graph to find the value of \(c\).
3. Calculate estimates of the mean and standard deviation of the weights.

6 The times taken by 57 athletes to run 100 metres are summarised in the following cumulative frequency table.

1. State how many athletes ran 100 metres in a time between 10.5 and 11.0 seconds.
2. Draw a histogram on graph paper to represent the times taken by these athletes to run 100 metres.
3. Calculate estimates of the mean and variance of the times taken by these athletes.

4 The heights, \(x \mathrm {~cm}\), of a group of 28 people were measured. The mean height was found to be 172.6 cm and the standard deviation was found to be 4.58 cm . A person whose height was 161.8 cm left the group.

1. Find the mean height of the remaining group of 27 people.
2. Find \(\Sigma x ^ { 2 }\) for the original group of 28 people. Hence find the standard deviation of the heights of the remaining group of 27 people.

5 The table shows the mean and standard deviation of the weights of some turkeys and geese.

1. Find the mean weight of the 27 birds.
2. The weights of individual turkeys are denoted by \(x _ { t } \mathrm {~kg}\) and the weights of individual geese by \(x _ { g } \mathrm {~kg}\). By first finding \(\Sigma x _ { t } ^ { 2 }\) and \(\Sigma x _ { g } ^ { 2 }\), find the standard deviation of the weights of all 27 birds.

7 The amounts spent by 160 shoppers at a supermarket are summarised in the following table.

1. Draw a cumulative frequency graph of this distribution.
2. Estimate the median and the interquartile range of the amount spent.
3. Estimate the number of shoppers who spent more than \(\\) 115$.
4. Calculate an estimate of the mean amount spent.

4 The monthly rental prices, \(\\) x$, for 9 apartments in a certain city are listed and are summarised as follows. $$\Sigma ( x - c ) = 1845 \quad \Sigma ( x - c ) ^ { 2 } = 477450$$ The mean monthly rental price is \(\\) 2205$.

1. Find the value of the constant \(c\).
2. Find the variance of these values of \(x\).
3. Another apartment is added to the list. The mean monthly rental price is now \(\\) 2120.50$. Find the rental price of this additional apartment.

1 Kadijat noted the weights, \(x\) grams, of 30 chocolate buns. Her results are summarised by $$\Sigma ( x - k ) = 315 , \quad \Sigma ( x - k ) ^ { 2 } = 4022$$ where \(k\) is a constant. The mean weight of the buns is 50.5 grams.

1. Find the value of \(k\).
2. Find the standard deviation of \(x\).