Question

Paper 1

18
Exercise 1B
1 For each dataset find
i
iii
a)
the mode ii the median
the mean iv the range.
Price of the cheapest season ticket
at each Premiership football club in
2013-14.
£1014 £335
£499 £595
£550
£544
£501
£365
£710
£299
£532
£383
£499
£608
£459
£400
£449
£795
£349
£640
b) Estimated annual earnings, in millions
of pounds, of players in England's
national team.
£1.4m £1.0m £1.6m £4.7m
£5.3m £3.1m £3.4m £2.6m
£2.6m £0.8m £3.6m £15.5m
£2.9m £4.2m £5.0m £3.9m
£6.2m £6.0m £5.2m
2 Stacy is writing a report about the rates of
pay of workers at a warehouse.
She finds this table which gives the mean
rate of pay for packers and fork-lift truck
drivers.
Number of Mean wage
workers (£ per hour)
42
£9.50
14
£13.50
Stacy says that the overall mean must be
£11.50 per hour.
Packers
Fork-lift truck drivers
a) Explain why Stacy is wrong.
b) Find the overall mean wage.
Start by finding the total wages for the 56
workers.
3 Jack's mean score from 8 matches on the
cricket team is 45. The coach says he must
increase his mean score to 50 in the next
match to keep his place on the team.
How many runs must Jack score in the
next match?
Analysis of data
4 The bar chart shows the number of students
studying A level sciences at a school.
Students on A level courses
students
Number
15
Male Female
TUTT
r
10
0
Biology Chemistry Physics Psychology Sociology
Subject
Sam says the average class size is 23, Tao
says it is 20 and Lily says it is 19.
a) Explain how they can all be correct.
b) Which average do you think best
represents the data? Give reasons.
5
a) For each dataset find the median and
interquartile range.
i Number of minutes that the
school bus is late each morning
in 4 weeks:
2 0 1 0 1 0 19 3 0 1
002001 4120
ii Number of plants that germinate
from 10 packets of seeds:
42 39 37 40 43 13 45
38 41 44
iii Amount spent on lunch by each
student on a college course:
£2.55 £4.36 £2.97 £9.10
£2.87 £3.45 £1.99 £2.75
£3.52
b) Explain how you can tell that the
median and interquartile range will
give more useful values than the mean
and range for these datasets.
6
Find the mean and standard deviation of
each dataset.
a) Marks in a test out of 50:
46 34 45 49 32 44 47 35
29 38 46 29

Answer 1

1. For the dataset of the price of the cheapest season ticket at each Premiership football club in 2013-14, we are asked to find:

i) the mode (most frequently occurring value),

ii) the median (middle value when the data is arranged in order),

iii) the mean (average),

iv) the range (difference between the maximum and minimum values).

a) Mode: In this dataset, the mode is the value that appears most frequently. We can see that the values £499 and £550 both appear twice, so these are the modes.

b) Median: To find the median, we need to arrange the data in order. After sorting the data, the middle value is £532, so the median is £532.

c) Mean: To find the mean, we sum up all the values and divide by the total number of values. Adding up all the values, we get a sum of £9,171. Dividing this by the number of values (18), we find the mean to be approximately £509.50.

d) Range: The range is the difference between the maximum and minimum values. In this dataset, the cheapest season ticket price ranges from £299 to £1014, resulting in a range of £715.

2. a) Stacy is wrong because she cannot simply calculate the overall mean by taking the mean of the individual means. The overall mean is influenced by the number of workers in each category.

b) To find the overall mean wage, we need to calculate the total wages for packers and fork-lift truck drivers. For packers, the total wages would be 42 workers multiplied by £9.50 per hour, which is £399. For fork-lift truck drivers, the total wages would be 14 workers multiplied by £13.50 per hour, which is £189. To find the overall mean wage, we divide the total wages (£588) by the total number of workers (56), resulting in an overall mean wage of approximately £10.50 per hour.

3)To find out how many runs Jack must score in the next match to increase his mean score from 45 to 50, we can follow these steps:

• Calculate the current total runs:

Current total runs = Mean score x Number of matches

Current total runs = 45 x 8 = 360

• Calculate the target total runs:

Target total runs = Desired mean score x Total number of matches

Target total runs = 50 x 9 = 450

• Find the runs needed in the next match:

Runs needed = Target total runs - Current total runs

Runs needed = 450 - 360 = 90

Therefore, Jack must score 90 runs in the next match to increase his mean score from 45 to 50 and secure his place on the cricket team.

4.a) Sam, Tao, and Lily can all be correct in their statements about the average class size, despite having different values, due to the possibility of different methods of calculating the average. For example, Sam may have calculated the overall average class size, Tao may have calculated the average for each subject and then found the overall average, while Lily may have considered the median class size.

b) Considering the nature of the data and the purpose of finding a representative average, the median class size may be the most suitable measure. This is because the median takes into account the varying class sizes across different subjects and is less influenced by outliers or extreme values. It provides a value that represents the central tendency of the data.

5. a) i) For the dataset representing the number of minutes that the school bus is late each morning in 4 weeks, the median is 1 minute, and the interquartile range is 1 minute.

ii) For the dataset representing the number of plants that germinate from 10 packets of seeds, the median is 40 plants, and the interquartile range is 3 plants.

iii) For the dataset representing the amount spent on lunch by each student on a college course, the median is £2.97, and the interquartile range is £1.08.

b) The median and interquartile range are more useful than the mean and range in these datasets because they are less affected by outliers. Outliers are extreme values that can significantly skew the mean and range. The median and interquartile range provide a better representation of the typical values in the dataset.

6) To find the mean and standard deviation of the given dataset, follow these steps:

1. Calculate the mean:

- Add up all the numbers in the dataset: 46 + 34 + 45 + 49 + 32 + 44 + 47 + 35 + 29 + 38 + 46 + 29 = 484.

- Divide the sum by the total number of values (12 in this case): 484 ÷ 12 = 40.33 (rounded to two decimal places).

- The mean of the dataset is approximately 40.33.

2. Calculate the standard deviation:

- Calculate the deviation from the mean for each value by subtracting the mean from each value:

Deviation = Value - Mean.

- Square each deviation to eliminate negative values: (46 - 40.33)^2, (34 - 40.33)^2, (45 - 40.33)^2, and so on.

- Find the sum of all the squared deviations: Add up all the squared deviations.

- Divide the sum by the total number of values: Divide the sum by 12.

- Take the square root of the result to find the standard deviation.

Using a calculator or spreadsheet, the standard deviation is found to be approximately 6.17 (rounded to two decimal places).