Box plots & stem & leaf diagrams

Each of 60 students was asked to draw a 20° angle without using a protractor. The size of each angle drawn was measured. The results are summarised in the box plot below.

Find the range for these data.

Find the interquartile range for these data.

The students were then asked to draw a 70° angle. The results are summarised in the table below.

Angle, a, (degrees)	Number of students
55 a 60	6
60 a 65	15
65 a 70	13
70 a 75	11
75 a 80	8
80 a 85	7

Use linear interpolation to estimate the size of the median angle drawn. Give your answer to 1 decimal place.

Show that the lower quartile is 63°

For these data, the upper quartile is 75°, the minimum is 55° and the maximum is 84° An outlier is an observation that falls either more than 1.5 × (interquartile range) above the upper quartile or more than 1.5 × (interquartile range) below the lower quartile.

Show that there are no outliers for these data.

Draw a box plot for these data on the grid on page 3.

State which angle the students were more accurate at drawing. Give reasons for your answer.

The marks of a group of female students in a statistics test are summarised in Figure 1

Write down the mark which is exceeded by 75% of the female students.

The marks of a group of male students in the same statistics test are summarised by the stem and leaf diagram below.

Find the median and interquartile range of the marks of the male students.

An outlier is a mark that is either more than 1.5 × interquartile range above the upper quartile or more than 1.5 × interquartile range below the lower quartile.

In the space provided on Figure 1 draw a box plot to represent the marks of the male students, indicating clearly any outliers.

Compare and contrast the marks of the male and the female students.

Over a long period of time a small company recorded the amount it received in sales per month. The results are summarised below.

An outlier is an observation that falls either 1.5 × interquartile range above the upper quartile or 1.5 × interquartile range below the lower quartile.

On the graph paper below, draw a box plot to represent these data, indicating clearly any outliers.

State the skewness of the distribution of the amount of sales received. Justify your answer.

The company claims that for 75 % of the months, the amount received per month is greater than £10 000. Comment on this claim, giving a reason for your answer.

In a study of how students use their mobile telephones, the phone usage of a random sample of 11 students was examined for a particular week. The total length of calls, y minutes, for the 11 students were 17, 23, 35, 36, 51, 53, 54, 55, 60, 77, 110

Find the median and quartiles for these data.

A value that is greater than Q₃ + 1.5 × (Q₃ – Q₁) or smaller than Q₁ – 1.5 × (Q₃ – Q₁) is defined as an outlier.

Show that 110 is the only outlier.

Using the graph paper on page 15 draw a box plot for these data indicating clearly the position of the outlier.

The value of 110 is omitted.

Show that S_yy for the remaining 10 students is 2966.9

These 10 students were each asked how many text messages, x, they sent in the same week. The values of S_xx and S_xy for these 10 students are S_xx = 3463.6 and S_xy = –18.3.

Calculate the product moment correlation coefficient between the number of text messages sent and the total length of calls for these 10 students.

A parent believes that a student who sends a large number of text messages will spend fewer minutes on calls.

Comment on this belief in the light of your calculation in part (e).

The age in years of the residents of two hotels are shown in the back to back stem and leaf diagram below. Abbey Hotel 8 5 0 means 58 years in Abbey hotel and 50 years in Balmoral hotel Balmoral Hotel

For the Balmoral Hotel,

write down the mode of the age of the residents,

find the values of the lower quartile, the median and the upper quartile.

Find the mean, , of the age of the residents.

One measure of skewness is found using

Evaluate this measure for the Balmoral Hotel.

For the Abbey Hotel, the mode is 39, the mean is 33.2, the standard deviation is 12.7 and the measure of skewness is – 0.454

Compare the two age distributions of the residents of each hotel.