Chapter 2 - Bivariate Booklet PDF

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UNIT 3 – FURTHER MATHEMATICS

CHAPTER 2 CLASS WORK Core: BIVARIATE DATA TERM 1, 2021

2A: Response and explanatory variables • response and explanatory variables and their role in investigating associations between variables

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[EXAM Q] Response and Explanatory Variables 2016 Sample Exam 2 Question 3 / 2014 Exam 2 Question 2 The scatterplot below shows the population and area (in square kilometres) of a sample of inner suburbs of a large city. The equation of the least squares regression line for the data in the scatterplot is ����� �� � =�� 5330 + 2680 × ���� a. Write down the response variable. 1 mark

2017 NHT Exam 2 Question 2 Two of the weather indicators collected at the weather station are temperature and relative humidity. The scatterplot below shows relative humidity (%) plotted against temperature (°C) for the 29 days of February in a particular leap year. The measurements were taken at 3 pm each day. A least squares line has been fitted to the scatterplot.

b. The equation of the least squares line is relative ℎ�� ����� = 136 − 4.38 × ����������� i. Write down the response variable. 1 mark

2018 Exam 2 Question 2 The congestion level in a city can also be recorded as the percentage increase in travel time due to traffic congestion in peak periods (compared to non-peak periods). This is called the percentage congestion level. The percentage congestion levels for the morning and evening peak periods for 19 large cities are plotted on the scatterplot below. A least squares line is to be fitted to the data with the aim of predicting evening congestion level from morning congestion level. The equation of this line is ������� ���������� ����� = 8.48 + 0.922 × � ������ ��� ������� ����� b. Name the response variable in this equation. 1 mark

Life span (years) 3.20 4.70 7.60 9.00 9.80 13.7 14.0 16.2 17.0 18.0 20.0 22.4 27.0 28.0 30.0 39.3 40.0 41.0 46.0

Gestation period (days) 19 21 68 28 52 63 60 63 150 31 151 100 180 63 281 Page 2 of 27 252 365 310 336

2019 NHT Exam 2 Question 3 a. A least squares line that enables life span to be predicted from gestation period is fitted to this data. Name the explanatory variable in the equation of this least squares line. 1 mark

2019 Exam 2 Question 4 The relative humidity (%) at 9 am and 3 pm on 14 days in November 2017 is shown in Table 3 below. A least squares line is to be fitted to the data with the aim of predicting the relative humidity at 3 pm (ℎ�� �� ��� 3 ��) from the relative humidit y at 9 am (ℎ� ������ 9 ��). a. Name the explanatory variable. 1 mark

Relative humidity (%) 9 am

3 pm

100

87

99

75

95

67

63

57

81

57

94

74

96

71

81

62

73

53

53

54

57

36

77

39

51

30

41

32

Table 3

Data: Australian Government, Bureau of Meteoro logy , < www. bo m .go v. au />

2B: Associations between two categorical variables Page 3 of 27

Key knowledge • contingency (two-way) frequency tables, two-way frequency tables and their associated bar charts (including percentaged segmented bar charts) and their use in identifying and describing associations between two categorical variables

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[EXAM Q]

Association between Categorical Variables 2016 Sample Exam 2 Question 1 / 2014 Exam 2 Question 1 The segmented bar chart below shows the age distribution of people in three countries, Australia, India and Japan, for the year 2010. a.Write down the percentage of people in Australia who were aged 0–14 years in 2010. 1 mark

b. In 2010, the population of Japan was 128 000 000. How many people in Japan were aged 65 years and over in 2010? 1 mark

Source: Australian Bureau of Statistics, 3201.0 - Population by Age and Sex, Australian States and Territories, June 2010

c. From the graph on page 2, it appears that there is no association between the percentage of people in the 15–64 age group and the country in which they live. Explain why, quoting appropriate percentages to support your explanation. 1 mark

A 2016 Exam 1 Question 1 The blood pressure (low, normal, high) and the age (under 50 years, 50 years or over) of 110 adults were recorded. The results are displayed in the two-way frequency table below. The percentage of adults under 50 years of age who have high blood pressure is closest to A. 11% B. 19% C. 26% D. 44% E. 58%

50 years Under Blood pressure 50 years or over low

15

5

normal

32

24

high

11

23

Total 58

52

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Use the following information to answer 2017 Exam 1 Questions 5–7. A study was conducted to investigate the association between the number of moths caught in a moth trap (less than 250, 250–500, more than 500) and the trap type (sugar, scent, light). The results are summarised in the percentaged segmented bar chart below.

2017 Exam 1 Question 5 There were 300 sugar traps. The number of sugar traps that caught less than 250 moths is closest to A. 30 B. 90 C. 250 D. 300 E. 500 2017 Exam 1 Question 6 The data displayed in the percentaged segmented bar chart supports the contention that there is an association between the number of moths caught in a moth trap and the trap type because A.most of the light traps contained less than 250 moths. B.15% of the scent traps contained 500 or more moths. C.the percentage of sugar traps containing more than 500 moths is greater than the percentage of scent traps containing less than 500 moths. D. 20% of sugar traps contained more than 500 moths while 50% of light traps contained less than 250 moths. E.20% of sugar traps contained more than 500 moths while 10% of light traps contained more than 500 moths. 2018 NHT Exam 2 Question 4 Table 2 A sample of 96 birds are grouped according to their beak size (small, medium, large). c. In order to investigate a possible association between beak size and sex, the same birds are grouped by both their beak size (small, medium, large) and their sex (male, female). The results of this grouping are shown in Table 2. Does the information provided above support the contention that beak size is associated with sex? Justify your answer by quoting appropriate percentages. It is sufficient to consider one beak size only when justifying your answer. 2 marks

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Sex Beak size Male

Female

small

1

23

medium

26

16

large

27

3

Total 54

42

2019 NHT Exam 2 Question 5 A random sample of 12 mammals drawn from a population of 62 types of mammals was categorised according to two variables. likelihood of attack (1 = low, 2 = medium, 3 = high) exposure to attack during sleep (1 = low, 2 = medium, 3 = high) The data is shown in the following table. Likelihood of attack 2

2

1

3

2

3

Exposure to attack

3

1

1

1

3

3

Likelihood of attack 1

3

1

1

3

3

Exposure to attack

1 3 1 1 3 3 a. Use this data to complete the two-way frequency table below. 1 mark Exposure to attack during sleep medium (= 2)

high (= 3)

low (= 1)

0

0

medium (= 2)

0

high (= 3)

0

Likelihood of attack

low (= 1)

The following two-way frequency table was formed from the data generated when the entire population of 62 types of mammals was similarly categorised. Exposure to attack during sleep low

medium

high

31

8

2

medium

2

0

2

high

1

1

15

Likelihood of attack low

b. i. How many of these 62 mammals had both a high likelihood of attack and a high exposure to attack during sleep? 1 mark

ii. Of those mammals that had a medium likelihood of attack, what percentage also had a low exposure to attack during sleep? 1 mark

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iii. Does the information in the table above support the contention that likelihood of attack is associated with exposure to attack during sleep? Justify your answer by quoting appropriate percentages. It is sufficient to consider only one category of likelihood of attack when justifying your answer. 2 marks

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2C – Associations between a numerical variable and a categorical variable

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Association between Numerical and Categorical Variables 2016 Sample Exam 1 Question 9 / 2014 Exam 1 Question 7 The parallel boxplots below summarise the distribution of population density, in people per square kilometre, for the inner suburbs and the outer suburbs of a large city. Which one of the following statements is not true? A.More than 50% of the outer suburbs have population densities below 2000 people per square kilometre. B. More than 75% of the inner suburbs have population densities below 6000 people per square kilometre. Page 11 of 27

C.Population densities are more variable in the outer suburbs than in the inner suburbs. D. The median population density of the inner suburbs is approximately 4400 people per square kilometre. E.Population densities are, on average, higher in the inner suburbs than in the outer suburbs. 2016 Exam 2 Question 2 The weather station also records daily maximum temperatures. b. The boxplots below display the distribution of maximum daily temperature for the months of May and July. i. Describe the shapes of the distributions of daily temperature (including outliers) for July and for May. 1 mark July May ii. Determine the value of the upper fence for the July boxplot. 1 mark

iii. Using the information from the boxplots, explain why the maximum daily temperature is associated with the month of the year. Quote the values of appropriate statistics in your response. 1 mark

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2017 NHT Exam 2 Question 1 d. The distribution of the amount of energy generated by the solar array for the months of April, May and June for the last 22 years is displayed in the parallel boxplots below. The parallel boxplots suggest that the amount of energy generated is associated with the month of the year. Explain why, quoting the values of an appropriate statistic. 2 marks

2017 Exam 2 Question 2 The back-to-back stem plot below displays the wingspan, in millimetres, of 32 moths and their place of capture (forest or grassland). b. Write down the modal wingspan, in millimetres, of the moths captured in the forest. 1 mark

c. Use the information in the back-to-back stem plot to complete the table below. 2 marks Wingspan (mm) Place of capture minimum Q1

median (M) Q3

maximum

forest

20

21

52

24

30

grassland

18

32

45

d. Show that the moth captured in the forest that had a wingspan of 52 mm is an outlier. 2 marks

e. The back-to-back stem plot suggests that wingspan is associated with place of capture. Explain why, quoting the values of an appropriate statistic. 2 marks

2018 NHT Exam 1 Question 6 The parallel boxplots below display the distribution of height for three groups of athletes: rowers, netballers and basketballers. Page 13 of 27

Which one of the following statements is not true? A.The shortest athlete is a netballer. B.The rowers have the least variable height. C.More than 25% of the netballers are shorter than all rowers. D. The basketballers are the tallest athletes in terms of median height. E.More than 50% of the basketballers are taller than any of the rowers or netballers. 2019 Exam 2 Question 3 The five-number summary for the distribution of minimum daily temperature for the months of February, May and July in 2017 is shown in Table 2. The associated boxplots are shown below the table.

Table 2. Five-number summary for minimum daily temperature

Explain why the information given above supports the contention that � ������ ����� ��� �������� is associa ted wit h the ����ℎ. Refer to the values of an appropriate statistic in your response. 2 marks

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Minimum �� February 5.9 9.5

Median �� 10.9 13.9

Maximum

May

3.3

6.0

7.5

9.8

12.7

July

1.6

3.7

5.0

5.9

7.7

�����

22.2

2D Analysing scatterplots and the association between two numerical variables The value of Pearson’s product-moment correlation coefficient is used to determine the strength of linear relationships.

Relationships

0.75 ≤ r < 1 0.5 ≤ r < 0.75 0.25 ≤ r < 0.5 -0.25 ≤ r < 0.25 -0.5 ≤ r < -0.25 -0.75 ≤ r < -0.5 -1 ≤ r ≤ -0.75

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2D EXAM QS: Scatterplots: Association between Numerical Variables 2016 Exam 2 Question 3 The data in the table below shows a sample of actual temperatures and apparent temperatures recorded at the weather station. A scatterplot of the data is also shown. The data will be used to investigate the association between the variables apparent temperature and actual temperature.

Apparent temperature (°C)

Actual temperature (°C)

24.7

28.5

24.3

27.6

24.9

27.7

23.2

26.9

24.2

26.6

22.6

25.5

21.5

24.4

20.6

23.8

19.4

22.3

18.4

22.1

a.

Use the scatterplot to describe the association between apparent temperature and actual temperature in terms of strength, direction and form. (1 mark)

17.6 2019 NHT Exam 2 Question 4 20.9 The scatterplot below plots the variable life span, in years, against the variable sleep time, in hours, for a sample of 19 types of mammals. 18.7

21.2

On the assumption that the association between sleep time and life span is linear, a least squares line is fitted to this data with sleep time as the

18.2

20.5

explanatory variable. The equation of this least squares line is = 42.1 − 1.90 × The coefficient of determination is 0.416 b. Describe the linear association between life span and sleep time in terms of strength and direction. 2 marks

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2E: Correlation Coefficient 2018 NHT Exam 1 Question 9 The table below shows the lean body mass (), percentage body fat () and body mass index () of a sample of 12 professional athletes. The Pearson correlation coefficient, , between lean body mass (LBM) and percentage body fat (PBF) is closest to A. −0.235

B. −0.124

C. 0.124

D. 0.235

E. 0.352

LBM (kg) PBF (%) BMI (kg/��) 63.3

19.8

20.6

58.6

21.3

20.7

55.4

19.9

21.9

57.2

23.7

21.9

53.2

17.6

19.0

53.8

15.6

21.0

60.2

20.0

21.7

48.3

22.4

20.6

54.6

18.0

22.6

53.4

15.1

19.4

61.9

18.1

21.2

48.3

23.3

22.0

Life span (years) Gestation period (days) 2019 NHT Exam 2 Question 3 The life span, in years, and gestation period, in days, for 3.20 19 19 types of mammals are displayed in the table below. 4.70 21 c. Write the value of the correlation coefficient rounded to three decimal places. 1 mark

7.60

68

9.00

28

�= 2019 Exam 2 Question 4

9.80

52

13.7

63

14.0

60

16.2

63

17.0

150

18.0

31

20.0

151

22.4

100

27.0

180

28.0

63

30.0

281

39.3

252

40.0

365

41.0

310

46.0

336

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Table 3 Relative humidity (%) 9 am

3 pm

100

87

Data: Australian Government, Bureau of Meteorology, <

The relative humidity (%) at 9 am and 3 pm on 14 days in November 2017 is shown in Table 3 below.

www.bom.gov.au/>

A least squares line is to be fitted to the data with the aim of predicting ℎ the relative humidity at 3 pm (ℎ3 ) from the relative humidity at 9 am ℎ (ℎ9 ). c. Determine the value of the correlation coefficient for this data set. Round your answer to three decimal places. 1 mark

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2F: Correlation, causality, and the coefficient of

determination

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2F exams Q: Correlation vs. Causation 2016 Sample Exam 1 Question 12 A large study of secondary-school male students shows that there is a negative association between the time spent playing sport each week and the time spent playing computer games. From this information, it can be concluded that

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A. male students who spend a lot of time playing computer games do not play sport. B. encouraging male students to spend less time playing sport will increase the time they spend playing computer games. C. encouraging male students to spend more time playing sport will reduce the time they spend playing computer games. D. male students who tend to spend more time playing sport tend to spend less time playing computer games. E. male students who tend to spend more time playing sport tend to spend more time playing computer games. 2016 Exam 1 Question 12 There is a strong positive association between a country’s Human Development Index and its carbon dioxide emissions. From this information, it can be concluded that A. increasing a country’s carbon dioxide emissions will increase the Human Development Index of the country. B. decreasing a country’s carbon dioxide emissions will increase the Human Development Index of the country. C. this association must be a chance occurrence and can be safely ignored. D. countries that have higher human development indices tend to have higher levels of carbon dioxide emissions. E. countries that have higher human development indices tend to have lower levels of carbon dioxide emissions. 2017 Exam 1 Question 12 Data collected over a period of 10 years indicated a strong, positive association between the number of stray cats and the number of stray dogs reported each year ((= 0.87) in a large, regional city. A positive association was...