Statistics Toolkit (DP IB Maths: AI HL)

The following cumulative frequency curve shows the distance travelled, in kilometres, to work by 160 people in Cape Town, South Africa, during 2021.

Rounding your answer to the nearest half kilometre, use the graph to find the

median distance.

lower quartile.

Draw a box-and-whisker diagram to represent this sample.

Using your answers from part (a), calculate the maximum distance that can be travelled by someone and still not be considered an outlier.

The following cumulative frequency curve shows the amount of water, in litres, that 60 people drink over a month.

State whether the data is discrete or continuous.

Complete the following frequency table.

Litres of water,
Frequency
Cumulative Frequency

The following cumulative frequency curve shows the number of hours spent gaming per week by 120 high school students.

Find the

median

Calculate the percentage of students that spent less than 17 hours gaming per week.

The 120 students were chosen randomly by sampling 60 senior students and 60 junior students. The school has the same number of senior and junior students.

Write down the sampling method used.

The histogram below shows the weights of kiwifruit, each measured to the nearest gram.

Write down the modal weight of the kiwifruits.

Find the median weight of the kiwifruits.

Write down two inequalities that represent the weight, , of a kiwifruit that is considered an outlier.

A group of people who use a gym participated in a research survey and the ages of the participants were recorded in the following table:

It is known that .

Write down

Determine the class in which the lower quartile lies.

Calculate the mean age of participants between the ages of 30 and 80.

The participants in this survey were chosen by selecting every person who entered the gym who was not wearing headphones.

Write down the type of sampling method used.

The table below shows the distribution of deliveries made by a group of food delivery drivers on a working day in Berlin.

Find

Determine if a delivery driver who made 4 deliveries would be considered an outlier.

The delivery drivers were selected for the survey by ordering their names alphabetically, then selecting every 20th number.

Identify the sampling technique used in the sampling method.

The following table shows the number of passes made by 11 players on a rugby team:

Write down

Find the interquartile range.

Determine if any of the players would be considered an outlier, and if so, state which player(s).

The table below shows the number of TVs school students have at their home:

Find

Determine if a student who has seven TVs would be considered an outlier.

The students were selected for the survey by randomly selecting student ID numbers, using a random number generator.

Identify the sampling technique used in the sampling method.

The number of days off taken by employees in a company during a two-year period was recorded. The data was organised into a box and whisker diagram as shown below:

For this data, write down

Steven claims that this box and whisker diagram can be used to infer that the percentage of employees who took between 14 and 30 days off is greater than the percentage of employees who took 25 days or more off.

State whether Steven is correct. Justify your answer.

The table below shows the points scored per game from two basketball players, Karo and Anna, across 9 games:

State a statistical measure that would be helpful for a coach who wants to measure the consistency of players scoring performances.

For both players, find

Determine whether the mean or median is a better representation of Karo’s scoring ability. Justify your answer.

Ashika runs a maths revision course three months before the final exam. In order to determine the current levels of the students Ashika gives them a practice exam the day before the course starts. On the first day of the course Ashika gives the students another test which has similar questions to the first test.

State whether Ashika is testing the reliability or validity of her method for determining the current levels of the students. State the name of the test she uses.

Ashika wants to determine whether the course improves the students’ levels so she asks the following question to each of the students after the revision course:

“Do you think that this course will help you to achieve at least a level 5 in your maths exam?”

Explain whether this question has content validity. Give two reasons for your answer.

Explain whether asking the question anonymously would increase its validity.

After the final exam, Ashika analyses how well the levels from the test at the start of her revision course predict the levels in the final exam.

State the name of the validity test that Ashika is using.

A group of Netflix subscribers participated in a research survey and the ages of participants were recorded in the following table.

Age, in years
Number of participants	11	62	56		12

It is known that .

Write down

the modal class

Determine the class in which the upper quartile lies.

Using the mid-interval values the mean of the data can be estimated to be 39.95.

Find the value of .

The participants in this survey were chosen by randomly selecting people entering a supermarket. However, to be more efficient, the surveyor only selected people who were in groups of at least 3.

Write down the type of sampling method used.

25 people are invited to the premier of the movie “ICE”, and they are asked to give the movie a score out of 1-10. The table below shows the distribution of the scores.

Score	1	2	3	4	5	6	7	8	9	10
Frequency	1	2	1	4	5	5	1		3

It is known that .

Find the value of and  the value of .

Draw a bar chart of the data on the grid below.

A shoe store wants to know which shoe size is the most popular and so they record the shoe sizes of 30 customers.

Score	7	7.5	8	8.5	9	9.5	10	11	12	13
Frequency	5	1	4		3	3	2	1		1

State whether the above data is continuous or discrete.

It is known that .

Find the value of

.

Write down

the mean

the median

State which statistical measure is most useful for the shoe store. Justify your answer.

A data set has a mean of 22 and a standard deviation of 6.

Each element of the data set has 4 subtracted from it.

Find the value of

the new mean

After each element has 4 subtracted from it, each element is divided by   .

Find the value of

the new mean

The following box and whisker diagram shows the number of social media posts made by a group of content creators over a week.

State whether the data is discrete or continuous.

It is given that  and .

Calculate the value of

A content creator made  posts, where . Given that is an outlier find the maximum value of .

The following table displays the percentage change of an investor’s portfolio over 12 months.

Month	Percentage change
1
2
3
4
5
6
7
8
9
10
11
12

Calculate

the mean

State which statistical measure, calculated in part (a), gives an indication of the volatility of the investor’s portfolio.

The investor’s portfolio value at the beginning of the 12 months was $6000.

Calculate the value of the portfolio at the end of the 12 months. Give your answer to 2 decimal places.

At a swimming competition the mean time of the first four swimmers is 28.2 seconds. The time for the fifth and sixth swimmers are then recorded and the mean time of the first six swimmers is 29.8 seconds. The difference between the fifth and sixth swimmer’s time is 0.4 seconds.

Find the time achieved by

the fifth swimmer

The first swimmers time is 25.7 seconds.

Calculate the range in times of the swimmers.

The table below shows the average temperature, , in a city over a normal year

(not a leap year).

Temperature
Frequency	124		109

It is given that .

Calculate the values of

Using your GDC, estimate the value of

the mean

It Is given that the mean temperature of the city over the year is .

Calculate the percentage error between your estimate of the mean temperature, found in part (b) (i), and the actual mean temperature.

Ann is the product manager of a bicycle company and she is testing a new type of tyre. Ann wants to test whether using the new tyres allows cyclists to reach higher speeds. She asks 10 volunteers to cycle 30 miles and to record their times. An extract of the data is shown below:

It is known that the recording devices had two settings for displaying time. Ann checks and finds that volunteer C took 1 hour and 30 minutes whereas volunteer D took 1 hour and 25 minutes.

Explain why Ann will need to verify the time of volunteer A.

Ann runs the experiment again but this time she records the times herself. She asks the volunteers to repeat the process three times.

State three factors which Ann should control to try to ensure that her measurements are reliable.

It is later discovered that the volunteers completed all three journeys during the same day. Explain how this would affect the reliability of the measurements.