Use linear interpolation for median or quartiles

1 The heights \(x \mathrm {~cm}\) of 100 boys in Year 7 at a school are summarised in the table below.

1. Estimate the number of boys who have heights of at least 155 cm .
2. Calculate an estimate of the median height of the 100 boys.
3. Draw a histogram to illustrate the data. The histogram below shows the heights of 100 girls in Year 7 at the same school. \includegraphics[max width=\textwidth, alt={}, center]{ab4d5ab1-e3b7-495f-9142-d37df7e712de-1_868_1361_1015_381}
4. How many more girls than boys had heights exceeding 160 cm ?
5. Calculate an estimate of the mean height of the 100 girls.

6 The heights \(x \mathrm {~cm}\) of 100 boys in Year 7 at a school are summarised in the table below.

1. Estimate the number of boys who have heights of at least 155 cm .
2. Calculate an estimate of the median height of the 100 boys.
3. Draw a histogram to illustrate the data. The histogram below shows the heights of 100 girls in Year 7 at the same school. \includegraphics[max width=\textwidth, alt={}, center]{76283206-687f-45d6-9204-952d60843cf1-3_865_1349_1297_349}
4. How many more girls than boys had heights exceeding 160 cm ?
5. Calculate an estimate of the mean height of the 100 girls.

4. A researcher measured the foot lengths of a random sample of 120 ten-year-old children. The lengths are summarised in the table below.

1. Use interpolation to estimate the median of this distribution.
2. Calculate estimates for the mean and the standard deviation of these data. One measure of skewness is given by $$\text { Coefficient of skewness } = \frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }$$
3. Evaluate this coefficient and comment on the skewness of these data. Greg suggests that a normal distribution is a suitable model for the foot lengths of ten-year-old children.
4. Using the value found in part (c), comment on Greg's suggestion, giving a reason for your answer.

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.

3. The frequency distribution for the lengths of 108 fish in an aquarium is given by the following table. The lengths of the fish ranged from 5 cm to 90 cm .

1. Calculate estimates of the three quartiles of the distribution.
2. On graph paper, draw a box and whisker plot of the data.
3. Hence describe the skewness of the distribution.
4. If the data were represented by a histogram, what would be the ratio of the heights of the shortest and highest bars?

4. The ages of 300 houses in a village are recorded giving the following table of results. Use linear interpolation to estimate for these data

1. the median,
2. the limits between which the middle \(80 \%\) of the ages lie. An estimate of the mean of these data is calculated to be 86.6 years.
3. Explain why the mean and median are so different and hence say which you consider best represents the data.

7. A cyber-cafe recorded how long each user stayed during one day giving the following results.

1. Use linear interpolation to estimate the median and quartiles of these data. The results of a previous study had led to the suggestion that the length of time each user stays can be modelled by a normal distribution with a mean of 72 minutes and a standard deviation of 48 minutes.
2. Find the median and quartiles that this model would predict.
3. Comment on the suitability of the suggested model in the light of the new results.

1000 houses were sold in a small town in a one-year period. The selling prices were as given in the following table:

Selling Price	Number of Houses	Selling Price	Number of Houses
Up to £50 000	60	Up to £150 000	642
Up to £75 000	227	Up to £200 000	805
Up to £100 000	305	Up to £500 000	849
Up to £125 000	414	Up to £750 000	1000

1. Name (do not draw) a suitable type of graph for illustrating this data. [1 mark]
2. Use interpolation to find estimates of the median and the quartiles. [6 marks]
3. Estimate the 37th percentile. [2 marks]

Given further that the lowest price was £42 000 and the range of the prices was £690 000,

1. draw a box plot to represent the data. Show your scale clearly. [4 marks]

In another town the median price was £149 000, and the interquartile range was £90 000.

1. Briefly compare the prices in the two towns using this information. [2 marks]

The following table gives the weights, in grams, of 60 items delivered to a company in a day.

Weight (g)	0 - 10	10 - 20	20 - 30	30 - 40	40 - 50	50 - 60	60 - 80
No. of items	2	11	18	12	9	6	2

1. Use interpolation to calculate estimated values of
   1. the median weight,
   2. the interquartile range,
   3. the thirty-third percentile.
   [7 marks]

Outliers are defined to be outside the range from \(2.5Q_1 - 1.5Q_2\) to \(2.5Q_2 - 1.5Q_1\).

1. Given that the lightest item weighed 3 g and the two heaviest weighed 65 g and 79 g, draw on graph paper an accurate box-and-whisker plot of the data. Indicate any outliers clearly. [5 marks]
2. Describe the skewness of the distribution. [1 mark]

The mean weight was 32.0 g and the standard deviation of the weights was 14.9 g.

1. State, with a reason, whether you would choose to summarise the data by using the mean and standard deviation or the median and interquartile range. [2 marks]

On another day, items were delivered whose weights ranged from 14 g to 58 g; the median was 32 g, the lower quartile was 24 g and the interquartile range was 26 g.

1. Draw a further box plot for these data on the same diagram. Briefly compare the two sets of data using your plots. [6 marks]