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.

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:

The lengths (l 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 (g) of each otter when it first arrives. The data is illustrated in the following box and whisker diagram:

Using the box plot above:

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

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.

Find the values of , and .

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:

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, h, 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 p and the value of q.

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

Find the value of .

It is known that over a 12 month period, 4000 students in total sat the online driving course.

Estimate the number of students over a 12 month period who took less than 3 hours to complete the course.

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

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

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

Find the value of

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

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 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

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.