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WORKSHEET General 2 MathematicsTopic Areas:FOCUS STUDIES
Mathematics and Health Correlation / Body Measurements Life Expectancy Medication
Teacher: PETER HARGRAVESSource: HSC exam questionsExam Equivalent Time: 55.5 minutesWorked Solutions: IncludedNote: Each question has designated marks. Use this information as both a guide to the question's difficultyand as a timing indicator, whereby each mark should equate to 1.5 minutes of working (examination) time.
Questions
1. FS Health, 2011 HSC 8 MC
In which graph would the data have a correlation coefficient closest to – 0.9?
2. FS Health, 2012 HSC 11 MC
Which of the following relationships would most likely show a negative correlation?
(A) The population of a town and the number of hospitals in that town.
(B) The hours spent training for a race and the time taken to complete the race.
(C) The price per litre of petrol and the number of people riding bicycles to work.
(D) The number of pets per household and the number of computers per household.
3. FS Health, 2013 HSC 2 MC
Which graph best shows data with a correlation closest to
0.3?
4. FS Health, 2UG 2008 HSC 12 MC
A scatterplot is shown.
Which of the following best describes the correlation between and ?
(A) Positive
(B) Negative
(C) Positively skewed
(D) Negatively skewed
5. FS Health, 2UG 2014 HSC 4 MC
Young’s formula below is used to calculate the required dosages of medicine for children aged years.
How much of the medicine should be given to an monthold child in a hour period ifeach adult dosage is mL? The medicine is to be taken every hours by both adults andchildren.
(A)
(B)
(C)
(D)
R T
1– 12
Dosage =age of child (in years) × adult dosage
age of child (in years) + 12
18 2445 6
5 mL
20 mL
27 mL
30 mL
6. FS Health, 2UG 2014 HSC 17 MC
A child who weighs needs to be given of paracetamol for every ofbody weight. Every of a particular medicine contains of paracetamol.
What is the correct dosage of this medicine for the child?
(A)
(B)
(C)
(D)
7. FS Health, 2UG Med S3
Bronwyn needs to have litres of intravenous liquid given to her over a period of hours.
What is the required flow rate in mL per minute? (2 marks)
14 kg 15 mg 2 kg10 mL 120 mg
5.6 mL
8.75 mL
11.43 mL
17.5 mL
3.0 4
Tony sees an eruption that lasts minutes. Based on the data in the graph, what isthe minimum time that he can expect to wait for the next eruption? (1 mark)Julia saw two consecutive eruptions, one hour apart. Based on the data in thegraph, what was the longest possible duration of the first eruption that she saw? (1mark)What does the graph suggest about the relationship between the duration of aneruption and the time to the next eruption? (1 mark)
8. Data, 2UG 2012 HSC 29a
Tourists visit a park where steam erupts from a particular geyser.
The brochure for the park has a graph of the data collected for this geyser over a period oftime.
The graph shows the duration of an eruption and the time until the next eruption, timed fromthe end of one eruption to the beginning of the next.
(i)
(ii)
(iii)
(NB. Changes in the Syllabus now have correlation in the Focus Study:Health category and hence this question is classified twice, under"Data" and "FS Health")
9. FS Health, 2UG Med S2
A medication is available in both tablet and liquid form. A tablet contains of the activeingredient while the liquid form contains per . Michael likes taking tablets andGeorgia prefers liquid medicines. If they each need of the active ingredient, whatdosages do they take? (3 marks)
4
50 mg60 mg 10 mL
0.2 g
Determine the gradient of the line of best fit shown on the graph. (1 mark)
Explain the meaning of the gradient in the context of the data. (1 mark)
Determine the equation of the line of best fit shown on the graph. (2 marks)
Use the line of best fit to predict the height of a typical yearold male. (1 mark)
Why would this model not be useful for predicting the height of a typical yearoldmale? (1 mark)
10. FS Health, HSC Supplement 1
A patient is to receive of pain killer medication by intravenous drip that will take to administer.
Given , calculate the amount of drops per minute the machine must be seton. (2 marks)
11. FS Health, 2UG 2013 HSC 28b
Ahmed collected data on the age ( ) and height ( ) of males aged to years.
He created a scatterplot of the data and constructed a line of best fit to model the relationshipbetween the age and height of males.
(i)
(ii)
(iii)
(iv)
(v)
1.8 L1.5 hours
1 mL = 4 drops
a h 11 16
17
45
A line of best fit has been drawn on the graph.
Describe the correlation between the height and mass of this child, as shown in thegraph. (1 mark)
Find the equation of this line. (2 marks)
12. FS Health, 2UG 2009 HSC 28b
The height and mass of a child are measured and recorded over its first two years.
This information is displayed in a scatter graph.
(i)
(ii)
13. FS Health, 2UG 2014 HSC 30b
The scatterplot shows the relationship between expenditure per primary school student, as a
, , , , , , , , , , , , , ,
The expenditures per primary school student for the countries in the scatterplotare:
For the given data, the correlation coefficient, , is . What does this indicateabout the relationship between expenditure per primary school student and lifeexpectancy for the countries? (1 mark)
For the data representing expenditure per primary school student, is and is .What is the interquartile range? (1 mark)
Another country has an expenditure per primary school student of of itsGDP. Would this country be an outlier for this set of data? Justify your answer withcalculations. (2 marks)
Complete the table below by calculating the mean, , and the standard deviation,
, of these data. Calculate both values to two decimal places.
The table also shows the mean, , and the standard deviation, , oflife expectancy for the same countries. (2 marks)
percentage of a country’s Gross Domestic Product (GDP), and the life expectancy in years for countries.
(i)
(ii)
(iii)
(iv)
15
r 0.83
15
QL 8.4QU 22.5
47.6%
15
5.9 7 7.6 8.4 11.2 11.2 13.7 17.1 18.7 21.1 22 22.5 23.224.9 27.6
xσx
y σy
15
. (2 marks)
Using the values from the table in part (iv), show that the equation of the leastsquares line of best fit is
On the scatterplot, draw the leastsquares line of best fit,
. (2 marks)
Using this line, or otherwise, estimate the life expectancy in a country which has anexpenditure per primary school student of of its GDP. (1 mark)
Why is this line NOT useful for predicting life expectancy in a country which hasexpenditure per primary school student of of its GDP? (1 mark)
(v)
(vi)
(vii)
(viii)
Copyright © 200914 The State of New South Wales (Board of Studies, Teaching and Educational Standards NSW)
y = 1.29x + 49.9
y = 1.29x + 49.9
18%
60%
♦ Mean mark 43%
♦ Mean mark 42%
Worked Solutions
1. FS Health, 2011 HSC 8 MC
2. FS Health, 2012 HSC 11 MC
3. FS Health, 2013 HSC 2 MC
4. FS Health, 2UG 2008 HSC 12 MC
5. FS Health, 2UG 2014 HSC 4 MC
Data needs to show a strong negative correlation
(i.e. top left to lower right.)
⇒ D
Increased hours training should reduce the
time to complete a race.
⇒ B
A is correct since the data slopes bottom
left to top right.
D also slopes correctly but exhibits a higher
correlation co-efficient.
⇒ A
Correlation is positive.
NB. The skew does not refer to
correlation.
⇒ A
Dosage =1.5 × 451.5 + 12
= 5 mLSince 1 dosage every 6 hrs
In 24 hours,
Medicine given = 4 × 5 = 20 mL
⇒ B
♦ Mean mark 47%
6. FS Health, 2UG 2014 HSC 17 MC
7. FS Health, 2UG Med S3
8. Data, 2UG 2012 HSC 29a
(i)
(ii)
(iii)
Paracetamol needed
= × 15 mg142
= 105 mg
Since 120 mg in 10 mL
⇒ 105 mg is contained in
× 10 mL = 8.75 mL105120
⇒ B
Total liquid = 3.0 × 1000
= 3000 mL
Total minutes = 4 × 60
= 240
∴ Flow Rate =3000240
= 12.5 mL/minute
70 minutes (refer to graph)
3 minutes
Locate 60 minutes on y-axis and look for longest
duration (i.e. the largest x-axis value) for that
given y-value.
It suggests that the longer an eruption lasts for,
the longer you will wait until the next one, or
there is a positive correlation between the length
of an eruption and the time until the next one.
9. FS Health, 2UG Med S2
10. FS Health, HSC Supplement 1
Michael - tablets
0.2 g = 200 mg
∴ # Tablets =20050
= 4
Georgia - liquid
60 mg in 10 mL
⇒ 1 mg = = 0.166... mL1060
⇒ 200 mg = 200 × 0.166...
= 33.33... mL
= 33.3 mL (to 1 d.p.)
∴ Michael needs to take 4 tablets
and Georgia needs to take 33.3 mL.
Total drops required = 1800 × 4
= 7200 drops
Time (in minutes) = 1.5 × 60
= 90 minutes
⇒ Drops per minute =720090
= 80∴ The machine must be set to 80 drops per minute.
♦♦ Mean marks of 38%, 26%and 25% respectively for parts(i)(iii).MARKER'SCOMMENT: Interpretinggradients has been consistentlyexamined in recent history andalmost always poorlyanswered. Work hard tounderstand this area.
11. FS Health, 2UG 2013 HSC 28b
(i)
(ii)
(iii)
(iv)
(v)
Gradient =riserun
=176 − 146
16 − 11
=305
= 6
Males should grow 6cm per year
between the ages 11-16.
Gradient = 6
Passes through (11, 146)
Using y −y1 = m (x − )x1
h −146 = 6 (a −11)
= 6a −66
∴ h = 6a + 80
If a = 17
h = (6 × 17) + 80
= 182
∴ Typical 17 year old is expected to be 182cm
People slow and eventually stop growing
after they become adults.
♦ Mean mark 48%.
♦♦♦ Mean mark 18%. MARKER'S COMMENT: Manystudents had difficulty due to thefact the horizontal axis startedat and not theorigin.
♦ Mean mark 35%
12. FS Health, 2UG 2009 HSC 28b
(i)
(ii)
13. FS Health, 2UG 2014 HSC 30b
(i)
(ii)
(iii)
The correlation between height and
mass is positive and strong.
Using (40, 1.2) and (80, 10.4)P1 P2
H = 40cm
Gradient = −y2 y1
−x2 x1
=10.4 − 1.2
80 − 40
=9.240
= 0.23Line passes through (40, 1.2)P1
Using y − y1 = m (x − )x1
y −1.2 = 0.23 (x −40)
= 0.23x −9.2
y = 0.23x −8
∴ Equation of the line is M = 0.23H − 8
It indicates there is a strong positive
correlation between the two variables
IQR = −QU QL
= 22.5 − 8.4
= 14.1
An outlier on the upper side must be more than
+ 1.5 × IQRQu
= 22.4 + (1.5 × 14.1)
= 43.55%
∴ A country with an expenditure of 47.6% is an outlier.
♦♦ Mean mark 30%Note that the Formulae andData Sheet gives you thegradient and interceptformulae you need for thisquestion.
♦♦ Mean mark 32%Only a minority of studentscould draw the line of best fit onthe graph. This can simply bedone using 2 points the intercept (49.9) and another onechosen.
(iv)
(v)
(vi)
Need to show y = 1.29x + 49.9
y
Gradient = r ×σy
σx
= 0.83 ×10.947.03
= 1.2916...
= 1.29 (2 d.p.)
y intercept = − (gradient × )y x
= 70.73 − (1.29 × 16.14)
= 49.909...
= 49.9 (1 d.p.)∴ y = 1.29x + 49.9 ... as required.
y
♦♦ Mean mark 39%
♦♦♦ Mean mark 0%. Thetoughest question on the 2014paper.COMMENT: Examiners oftenlike to highlight linearrelationships that only work in agiven range and have studentscomment on outliers where therelationship breaks down.
(vii)
(viii)
Copyright © 2015 M2 Mathematics Pty Ltd (SmarterMaths.com.au)
Life expectancy ≈ 73.1 years (see dotted line)
Alternative Solution
When x = 18
y = 1.29(18) + 49.9 = 73.12 years
At 60% GDP, the line predicts a life
expectancy of 127.3. This line of best
fit is only predictive in a lower range
of GDP expenditure.
