Draw cumulative frequency graph from frequency table

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.
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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.

2 The manager of a company noted the times spent in 80 meetings. The results were as follows. Draw a cumulative frequency graph and use this to estimate the median time and the interquartile range.

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.

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

1. Draw a cumulative frequency graph on graph paper to illustrate this information.
2. Estimate the number of days when over 30 rooms were occupied.
3. On \(75 \%\) of the days at most \(n\) rooms were occupied. Estimate the value of \(n\).

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.

2 The time taken by a car to accelerate from 0 to 30 metres per second was measured correct to the nearest second. The results from 48 cars are summarised in the following table.

1. On the grid, draw a cumulative frequency graph to represent this information.
   \includegraphics[max width=\textwidth, alt={}, center]{ee1e5987-315b-48eb-8dba-b9d4d34c87c9-03_1207_1406_897_411}
2. 35 of these cars accelerated from 0 to 30 metres per second in a time more than \(t\) seconds. Estimate the value of \(t\).

7 In a UK government survey in 2000, smokers were asked to estimate the time between their waking and their having the first cigarette of the day. For heavy smokers, the results were as follows. Times are given correct to the nearest minute.

1. Assuming that 'At least 60 minutes' means 'At least 60 minutes but less than 240 minutes', calculate estimates for the mean and standard deviation of the time between waking and first cigarette for these smokers.
2. Find an estimate for the interquartile range of the time between waking and first cigarette for these smokers. Give your answer correct to the nearest minute.
3. The meaning of 'At least 60 minutes' is now changed to 'At least 60 minutes but less than 480 minutes'. Without further calculation, state whether this would cause an increase, a decrease or no change in the estimated value of
   (a) the mean,
   (b) the standard deviation,
   (c) the interquartile range.

3 The ages, \(x\) years, of the senior members of a running club are summarised in the table below.

1. Draw a cumulative frequency diagram to illustrate the data.
2. Use your diagram to estimate the median and interquartile range of the data.

3 The heating quality of the coal in a sample of 50 sacks is measured in suitable units. The data are summarised below.

1. Draw a cumulative frequency diagram to illustrate these data.
2. Use the diagram to estimate the median and interquartile range of the data.
3. Show that there are no outliers in the sample.
4. Three of these 50 sacks are selected at random. Find the probability that
   (A) in all three, the heating quality \(x\) is more than 9.5 ,
   \(( B )\) in at least two, the heating quality \(x\) is more than 9.5.

2 Every day, George attempts the quiz in a national newspaper. The quiz always consists of 7 questions. In the first 25 days of January, the numbers of questions George answers correctly each day are summarised in the table below.

1. On the insert, draw a cumulative frequency diagram to illustrate the data.
2. Use your graph to estimate the median length of journey and the quartiles. Hence find the interquartile range.
3. State the type of skewness of the distribution of the data.

8 The heating quality of the coal in a sample of 50 sacks is measured in suitable units. The data are summarised below.

1. Draw a cumulative frequency diagram to illustrate these data.
2. Use the diagram to estimate the median and interquartile range of the data.
3. Show that there are no outliers in the sample.
4. Three of these 50 sacks are selected at random. Find the probability that
   (A) in all three, the heating quality \(x\) is more than 9.5,
   \(( B )\) in at least two, the heating quality \(x\) is more than 9.5. OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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1 The ages, \(x\) years, of the senior members of a running club are summarised in the table below.

1. Draw a cumulative frequency diagram to illustrate the data.
2. Use your diagram to estimate the median and interquartile range of the data.

5. The following grouped frequency distribution summarises the number of minutes, to the nearest minute, that a random sample of 200 motorists were delayed by roadworks on a stretch of motorway.

1. Using graph paper represent these data by a histogram.
2. Give a reason to justify the use of a histogram to represent these data.
3. Use interpolation to estimate the median of this distribution.
4. Calculate an estimate of the mean and an estimate of the standard deviation of these data. One coefficient of skewness is given by $$\frac { 3 ( \text { mean - median } ) } { \text { standard deviation } } .$$
5. Evaluate this coefficient for the above data.
6. Explain why the normal distribution may not be suitable to model the number of minutes that motorists are delayed by these roadworks.

1. The table shows the time, to the nearest minute, spent waiting for a taxi by each of 80 people one Sunday afternoon.

1. Write down the upper class boundary for the \(2 - 4\) minute interval. A histogram is drawn to represent these data. The height of the tallest bar is 6 cm .
2. Calculate the height of the second tallest bar.
3. Estimate the number of people with a waiting time between 3.5 minutes and 7 minutes.
4. Use linear interpolation to estimate the median, the lower quartile and the upper quartile of the waiting times.
5. Describe the skewness of these data, giving a reason for your answer.

7. A college organised a 'fun run'. The times, to the nearest minute, of a random sample of 100 students who took part are summarised in the table below.

1. Give a reason to support the use of a histogram to represent these data.
2. Write down the upper class boundary and the lower class boundary of the class 40-44.
3. On graph paper, draw a histogram to represent these data. END

5. The following grouped frequency distribution summarises the number of minutes, to the nearest minute, that a random sample of 200 motorists were delayed by roadworks on a stretch of motorway.

1. Using graph paper represent these data by a histogram.
2. Give a reason to justify the use of a histogram to represent these data.
3. Use interpolation to estimate the median of this distribution.
4. Calculate an estimate of the mean and an estimate of the standard deviation of these data. One coefficient of skewness is given by $$\frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } } .$$
5. Evaluate this coefficient for the above data.
6. Explain why the normal distribution may not be suitable to model the number of minutes that motorists are delayed by these roadworks.

1. In a particular week, a dentist treats 100 patients. The length of time, to the nearest minute, for each patient's treatment is summarised in the table below.

Draw a histogram to illustrate these data.