4.1 Statistics Toolkit

A supermarket wants to gather data from its shoppers on how far they have travelled to shop there. One lunchtime an employee is stationed at the door of the shop for half an hour and instructed to ask every customer how far they have travelled.

(i)      State the sampling method the employee is using.

(ii)     Give one advantage and one disadvantage of using this method.

State and briefly describe an alternative method of non-random sampling that the employee could use to obtain the required data for a sample of 30 customers.

A pharmacy sells face masks in a variety of sizes.  Their sales over a week are recorded in the table below:

	Kids		Adults
Size	Small	Large	S	M	L	XL
Frequency	29	4	8	24	15	4

Write down the mode for this data.

Explain why, in this case, the mode from part (i) would not be particularly helpful to the shop owner when reordering masks.

The lengths ( cm) of a sample of nine otters, measured to the nearest centimetre by a wildlife research team, are:

76        77        91        65        63        83        92        61        88

Calculate the mean and standard deviation of the nine recorded lengths.

Jeanette works for a conservation charity who rescue orphaned otters.  Over many years she records the weight () of each otter when it first arrives.  The data is illustrated in the following box and whisker diagram:

Using the box plot above:

Write down the median weight of the otters.

Write down the lower quartile.

Otters are then weighed weekly to track their growth.  Summary data on the weights () of otters after one month is shown in the table below:

	Weight
Smallest weight	125
Range	48
Median	152
Upper Quartile	164
Interquartile Range	33

On the grid, draw a box plot for the information given above.

The heights, in metres, of a flock of 20 flamingos are recorded and shown below:

0.4       0.9       1.0       1.0       1.2       1.2       1.2       1.2       1.2       1.2

1.3       1.3       1.3       1.4       1.4       1.4       1.4       1.5       1.5       1.6

An outlier is an observation that falls either more than 1.5  (interquartile range) above the upper quartile or less than 1.5  (interquartile range) below the lower quartile. 

(i)        Find the values of ,  and 

(ii)       Find the interquartile range.

(iii)      Identify any outliers.

Using your answers to part (a), draw a box plot for the data.

120 competitors enter an elimination race for charity.  Runners set off from the same start running as many laps of the course as possible.  Their total distance is tracked and the competitor who runs the furthest over a 6-hour period is the winner.  The distances runners achieved are recorded in the table below:

Distance  (miles)	Frequency
25 ≤ d < 30	8
30 ≤ d < 35	10
35 ≤ d < 40	32
40 ≤ d < 45	54
45 ≤ d < 50	10
50 ≤ d < 55	6

On the grid below, draw a cumulative frequency graph for the information in the table.

Use your graph to find an estimate for the median and interquartile range.

Police check the speed of vehicles travelling along a stretch of highway.  The cumulative frequency curve below summarises the data for the speeds, in kmph, of 80 vehicles:

Use the graph to find an estimate for the median speed.

The speed limit for this section of road is 80 kmph.

Vehicles travelling above the speed limit are issued with a speeding ticket. Those travelling more than 10% over the speed limit are pulled over.  Use the graph to estimate the percentage of vehicles that the police pull over.

The following cumulative frequency curve shows the number of hours, , students took to complete their online driving course.  The data is taken from 80 students, randomly selected from a large sample over a 12 month period.

Find the median number of hours spent completing the online driving course.

Find the number of students whose online course time was within 1 hour of the median.

Calculate the interquartile range.

The same information is represented by the following table.

Find the value of  and the value of .

It is known that 10% of students take longer than  hours to complete the online driving course.

Find the value of .

For her IA, Mia decides to investigate the popularity of the social media platform SMEsocial amongst 16 to 18 year olds. She thinks of two possible methods for collecting data to analyse.

Method 1: Use visitor statistics from the SMEsocial website.

Method 2: Survey pupils from her school on their usage of SMEsocial.

State one advantage and one disadvantage of method 2 outlined above.

Mia decides to go ahead with the survey of pupils in her school and designs a questionnaire.  She randomly asks 15 students to complete it.

State the type of sampling method used.

One of the questions that Mia asks on the survey is the following:

“How long do you spend on SMEsocial each week?”

State one criticism of the question used by Mia.

Describe one way in which Mia could increase the reliability of her investigation.

Describe one way that Mia could improve the validity of her data.

Nils believes that people with longer legs can run quicker.  He decides to test his hypothesis by measuring the outer leg length of 10 boys aged between 10 and 12, and then using a data logger to record the time taken for each participant to run 100 metres.  The results are shown on the scatter diagram below.

Nils notices that the scatter graph appears to support his hypothesis. Recalling that the world record for running 100 metres is 9.58 seconds, however, he believes that there may be a problem with the data that he has collected.

Based on the information given in the question and the data, explain what the data suggests about the students.

Explain what could be wrong with the data collected and how it may have occurred.

State what Nils should do to confirm the reliability of the data.

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.