PART 1:
1. What sort of data are each of these? Nominal, discrete or continuous?
2.
(a)
The suburb a person lives in
_________________
(b)
The speed of a car
_________________
(c)
The make and model of a car
_________________
(d)
The weights of pet animals
_________________
(e)
The number of goals scored
_________________
The table shows the weekly rents in a survey of 50 one bedroom units around Thornbury.
Rent ($)
180–
Frequency
3
200–
6
220–
12
240–
10
260–
3
280–
5
300–
3
320-
6
340-
2
Relative frequency
% relative frequency
1.00
100%
50
(a)
Complete the relative frequency and % relative frequency columns
(b)
What percentage of flats have
(i)
rents less than $300 a week?
(ii)
rents of at least $240 a week?
(iii)
rents between $220 and $300 a week?
(c)
If another 30 flats were surveyed, based on this survey, how many would you expect
to have rents less than $300 a week?
3. What type of shape does each frequency histogram have? Briefly explain what the shape
suggests about the data.
(a)
Hours
worked
(b)
(c )
Weight loss
for dieters
4.
The stem and leaf plot shows the grams of sugar in a serve of various breakfast cereals,
rounded to the nearest gram.
Kids cereals
9987
87764442
998877665542
6522
60
Key 15 = 15
0
1
2
3
4
5
Adults cereals
889
0568
00288
566889
00224446688
2
(a)
How many cereals were surveyed? ____________
(b)
Complete the table.
Sugar
Content
Kids
cereals
Adults
cereals
Lowest
Highest
Mode
(c)
(i)
Change the stem and leaf plots into frequency tables.
KIDS
ADULTS
0-9
0-9
10 - 19
10 - 19
20 - 29
20 - 29
30 - 39
30 - 39
40 - 49
40 - 49
50 - 59
50 - 59
Will the class interval of grams of sugar be open or closed? Give reasons.
(ii)
Show the sugar content as line graphs, both drawn on the same graph.
PART 2
1. Find the sum of x, (that is x), the total frequency (n), the mean ( x ) and the median for the
daily minimum temperatures in Canada, in °C.
−2, 5, 10.5, −5, 6, 5, 5.5, −13, 14.5, −3, 0, −6.5, 8, 2
2. Find the mean ( x ) and the total frequency (n) for the following grouped data.
(a)
Pets in a
family
1
2
3
5
8
Number of
families
12
15
10
2
1
(b)
Hours
of TV
1
2
3
4
5
Number of
teenagers
4
5
18
15
8
3.
We use the mid-point when the data is in intervals. Fill in the mid-points in these tables
and then find the mean and the total frequency, n.
Temp.
Mid-point
1 – 10
11 – 20
21 – 30
31 – 40
Weight (kg)
Mid-point
Number of
days
3
14
15
8
Number of
pets.
0–
5–
10–
15–
4.
5.
4
6
12
3
Over 14 games the number of goals we scored was
4, 9, 13, 11, 12, 3, 5, 10, 12, 8, 7, 12, 14, 15
(a)
Write the goal scores in order from lowest to highest
(b)
What was our median goal score?
(c)
What was our mean goal score?
(d)
Here the median was a better measure of the average score than the mode or mean.
Why?
The table below shows the weights of 20 pets.
Weight (kg)
(x)
5
7
8
9
11
14
Total
(a)
Number of pets
(f)
3
2
5
7
2
1
20
x f
Weight up to
Cumulative
frequency
Up to 5 kg
Up to 7 kg
Up to 8 kg
Up to 9 kg
Up to 11 kg
Up to 14 kg
Complete the cumulative frequency column. Use it to find the median weight.
(b)
Work out the mean weight.
(c)
What is the mode?
(d)
Out of the median, mean and mode which do you think is the best measure of
average weight? Why?
6.
The table shows the result of a survey of 90 people who were asked
"How much did you spend the last time you went out?"
Amount spent
How many
Spent less than
Cumulative
% cumulative
(frequency)
frequency
frequency
Less
than
$10
$0 − $19
12
Less than $20
$20 − $39
21
Less than $30
$40 − $59
16
Less than $40
15
$60 − $79
Less than $50
17
$80 − $99
Less than $60
9
$100 − $119
Total
90
(a) Complete the cumulative frequency and % cumulative frequency columns.
(b)
Plot the cumulative frequency graph and read off the median.
(c)
Complete the table below, and use it to work out the mean.
Amount spent
$0 − $19
$20 − $39
$40 − $59
$60 − $79
$80 − $99
$1000 − $119
Total
Mid-point
(x)
Frequency
(f)
12
21
16
15
17
9
90
Total spent for group
x f
(d)
7.
For this data, what is your preferred measure of average? Why?
The histogram below shows the numbers of tickets bought by 53 people in a lottery.
(a)
Based on the shape of the
histogram is the mean or the
median the better measure of the
average number of tickets bought
by each person? Why?
12
Frequency
10
8
6
Mean = 4.2 tickets
Median = 3 tickets
4
2
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Tickets bought
(b)
8.
Use your preferred measure of average to answer this. If another 15 people buy
tickets, how many tickets in total would we expect them to buy?
Show approximately where you would expect the mode, median and mean to be located
in the following histograms.
(b)
frequency
frequency
(a)
goal scores
waiting time
9.
Activity
(a)
What happens to the shape of the frequency distribution pattern when you use larger
number of measurements?
(b)
What would happen to the pattern if you didn’t shake the coins well each time?
(c)
If you use large numbers of measurements will you often see big differences between
the mean and median, or unusual results? Explain.
PART 3
1.
For each of the following data sets.
(a)
Find the median, QL , QU and IQR.
(b)
Identify any outliers by showing the 1.5 x IQR calculations.
(c)
Draw a box plot. Be sure to include a scaled line with your boxplot.
Data set 1
Temperatures over 14 days at a ski resort
3, −2, 5, 1, –2, 0, 10, 2, 3, 2, 0, −1, 3, 0
Data set 2
Weights of 27 students.
Key 7|3 = 73 kg
Stem
5
6
7
8
Leaf
269
1235566777789
11233678
027
Data set 3
Number of phone lines operated by 50 companies.
Number of phone
lines
1
2
4
5
8
10
12
15
Number of companies
(frequency)
1
2
3
3
8
20
8
5
50
Cumulative
frequency
Fill in the cumulative
frequency column
before finding the
median and the
quartiles.
2.
There are four box plots shown below. Two of them relate to the histograms shown.
Match the histograms with their correct box plots. Give reasons.
Histogram 1
Box plot 1
Box plot 2
Goal score
Histogram 2
Box plot 3
Box plot 4
Light bulb
hours of life
3.
(a)
Draw the shapes of histograms that match these box plots.
(b)
Goal score
0
1
2
3
4
5
6
7
8
9
10
11
12
Sally’s team
Brett’s team
(a)
Sally’s team
Fill in this table
Brett’s team
Range
QL
QU
IQR
Median
(b)
When drawing the box plot we forgot to check for outliers. Find any outliers.
(c)
Which team scored better? Why?
(d)
(i)
In how many games did Sally’s team score more than 6 goals?
(ii)
In how many games did Brett’s team score more than 5 goals?
PART 4
1.
Fill in the following tables, then work out the mean, variance and standard deviation of
each sample. Show your workings.
Sample A:
Weights of 12 pets
Weight (kg), x
x2
1.4
7
9
11.
2
9.3
20.
5
7
10
12.
6
5
4.1
6
Sample B:
Number of cars seen each hour over a 24 hour period
Cars seen (x)
0
Total
Number of hours (f)
2
3
4
7
4
13
6
25
5
30
3
xf
x2 f
Sample C:
The daily temperatures over our 60 day camping holiday
Set up your own table for this one and show how you work out the mean and standard deviation.
Temperature,
°C
0−
10 −
20 −
30 −
40 −
Total
x
Number of days
4
13
21
19
3
60
xf
x2 f
2.
A survey of 100 teams found the mean score to be 14 goals per game with a standard
deviation of 4.3 goals.
(a)
(b)
The histogram of the survey is almost symmetrical. We can therefore conclude that
(i)
about 68% of teams score between __________ and ____________ goals.
(ii)
about 95% of teams score between _________ and ___________ goals.
(iii)
about 99.7% of teams score between ___________ and ___________
goals.
(i)
Is a score of 3 goals an unusual score? Why?
(ii)
Is a score of 18 goals an unusual score? Why?
3.
Number
of
seedlings
The histogram shows the number of seedlings which grow using a growth solution.
Show on the histogram
(i)
the 68% confidence interval
(ii)
the 95% confidence interval
(iii)
the 99.7% confidence interval
10
8
Mean dose of growth solution
= 8.9 units
6
4
Std dev. = 3.7 units
2
0
0
2
4
6
8
10
12
14
Units of
growth
16
solution
PART 5
1. For each of the following sequences
(a) Find a formula for the nth term, tn
(b) Evaluate t5 and t11
(i)
17, 13, 9, …
(ii)
–5, 2, 9, 16, …
2.
In an arithmetic sequence the third term is 1 and the eighth term is 9
(a) Find the term, a, and the common difference d
(b)
3.
Find which term is 25 (Find n where tn = 25)
Use your graphics calculator or use the formulae to find:
(a) the first 10 terms of the sequence with the rule tn = 5n + 1
(b)
the 35th term of the same sequence.
(c)
the sum of the first 12 terms of the sequence.
4.
An employee receives $12000 in her 1st year and is to be given a yearly increase of
$1050.
(a) How much will she receive in her 8th year?
(b)
5.
How much will she have earned altogether
A builder wishes to tile a section of a roof in the shape of an isosceles triangle (like
the ladder rungs in Lesson 3). In order to fix the tiles, it is necessary to lay timber
battens across the roof parallel to one another. If the shortest batten is 1 metre long,
the longest batten is 5 metres long and 16 battens are required, find:
1m
(a)
the amount by which adjacent battens differ in length.
5m
(b)
the total length of the batten required.
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