Calculate range and interquartile range

1 The times taken, in minutes, by 80 people to complete a crossword puzzle are summarised by the box and whisker plot below.
\includegraphics[max width=\textwidth, alt={}, center]{acb05873-e441-4b95-9732-6ebd5ae79fa6-2_147_848_507_612}

1. Write down the range and the interquartile range of the times.
2. Determine whether any of the times can be regarded as outliers.
3. Describe the shape of the distribution of the times.

1 The stem-and-leaf diagram shows the heights, in metres to the nearest 0.1 m , of a random sample of trees of species \(A\). means 6.4 m

1. Find the median and interquartile range of the heights.
2. The heights, in metres to the nearest 0.1 m , of a random sample of trees of species \(B\) are given below.
   \(\begin{array} { l l l } 7.6 & 5.2 & 8.5 \end{array}\)
   5.2
   6.3
   6.3
   6.8
   7.2
   6.7
   7.3
   5.4
   7.5
   7.4
   6.0
   6.7 In the answer book, complete the back-to-back stem-and-leaf diagram.
3. Make two comparisons between the heights of the two species of tree.

1. The company Seafield requires contractors to record the number of hours they work each week. A random sample of 38 weeks is taken and the number of hours worked per week by contractor Kiana is summarised in the stem and leaf diagram below.

Key : 3|2 means 32 The quartiles for this distribution are summarised in the table below.

1. Find the values of \(w , x\) and \(y\) Kiana is looking for outliers in the data. She decides to classify as outliers any observations greater than $$Q _ { 3 } + 1.0 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$
2. Showing your working clearly, identify any outliers that Kiana finds.
3. Draw a box plot for these data in the space provided on the grid opposite.
4. Use the formula $$\text { skewness } = \frac { \left( Q _ { 3 } - Q _ { 2 } \right) - \left( Q _ { 2 } - Q _ { 1 } \right) } { \left( Q _ { 3 } - Q _ { 1 } \right) }$$ to find the skewness of these data. Give your answer to 2 significant figures. Kiana's new employer, Landacre, wishes to know the average number of hours per week she worked during her employment at Seafield to help calculate the cost of employing her.
5. Explain why Landacre might prefer to know Kiana's mean, rather than median, number of hours worked per week. Turn over for a spare grid if you need to redraw your box plot.

6. Hospital records show the number of babies born in a year. The number of babies delivered by 15 male doctors is summarised by the stem and leaf diagram below.

1. Find the median and inter-quartile range of these data.
2. Given that there are no outliers, draw a box plot on graph paper to represent these data. Start your scale at the origin.
3. Calculate the mean and standard deviation of these data. The records also contain the number of babies delivered by 10 female doctors.

   34	30	20	15	6
   32	26	19	11	4

   The quartiles are 11, 19.5 and 30 .
4. Using the same scale as in part (b) and on the same graph paper draw a box plot for the data for the 10 female doctors.
5. Compare and contrast the box plots for the data for male and female doctors.

2. The number of caravans on Seaview caravan site on each night in August last year is summarised in the following stem and leaf diagram.

1. Find the three quartiles of these data. During the same month, the least number of caravans on Northcliffe caravan site was 31. The maximum number of caravans on this site on any night that month was 72 . The three quartiles for this site were 38,45 and 52 respectively.
2. On graph paper and using the same scale, draw box plots to represent the data for both caravan sites. You may assume that there are no outliers.
3. Compare and contrast these two box plots.
4. Give an interpretation to the upper quartiles of these two distributions.

6. Three swimmers Alan, Diane and Gopal record the number of lengths of the swimming pool they swim during each practice session over several weeks. The stem and leaf diagram below shows the results for Alan.

1. Find the three quartiles for Alan's results. The table below summarises the results for Diane and Gopal.

   	Diane	Gopal
   Smallest value	35	25
   Lower quartile	37	34
   Median	42	42
   Upper quartile	53	50
   Largest value	65	57

2. Using the same scale and on the same sheet of graph paper draw box plots to represent the data for Alan, Diane and Gopal.
3. Compare and contrast the three box plots.

2. A botany student counted the number of daisies in each of 42 randomly chosen areas of 1 m by 1 m in a large field. The results are summarised in the following stem and leaf diagram.

1. Write down the modal value of these data.
2. Find the median and the quartiles of these data.
3. On graph paper and showing your scale clearly, draw a box plot to represent these data.
4. Comment on the skewness of this distribution. The student moved to another field and collected similar data from that field.
5. Comment on how the student might summarise both sets of raw data before drawing box plots.
   (1 mark)

2. The box plot shows the times, \(t\) minutes, it takes a group of office workers to travel to work.
\includegraphics[max width=\textwidth, alt={}, center]{7d45bacd-20ac-49b4-8f3f-613edf3739f9-04_365_1237_351_356}

1. Find the range of the times.
2. Find the interquartile range of the times.
3. Using the quartiles, describe the skewness of these data. Give a reason for your answer. Chetna believes that house prices will be higher if the time to travel to work is shorter. She asks a random sample of these office workers for their house prices \(\pounds x\), where \(x\) is measured in thousands, and obtains the following statistics $$\mathrm { S } _ { x x } = 5514 \quad \mathrm {~S} _ { x t } = 10 \quad \mathrm {~S} _ { t t } = 1145.6$$
4. Calculate the product moment correlation coefficient between \(x\) and \(t\).
5. State, giving a reason, whether or not your correlation coefficient supports Chetna's belief. Adam and Betty are part of the group of office workers and they have both moved house. Adam's time to travel to work changes from 32 minutes to 36 minutes. Betty's time to travel to work changes from 38 minutes to 58 minutes. Outliers are defined as values that are more than 1.5 times the interquartile range above the upper quartile.
6. Showing all necessary calculations, determine how the box plot of times to travel to work will change and draw a new box plot on the grid on page 5. \includegraphics[max width=\textwidth, alt={}, center]{7d45bacd-20ac-49b4-8f3f-613edf3739f9-05_499_1413_2122_180}