further mathematics unit 3 sac 1mcmullenmaths.weebly.com/.../0/8/10083938/car_sac_final.docx · web...

Further Mathematics Unit 3 SAC 1 2012

BUYING A CAR

NAME: ______________________ TEACHER: ___________________________

Duration: 4 x 50mins

Materials allowed: 1 x Bound reference (to be collected at the end of each session); CAS calculator or Graphics calculator; ruler; pens/pencils

Outcomes: Mark allocation noted in brackets (total marks: 40)

Outcome 1 (15 marks)Define and explain key terms and concepts as specified in the content from the areas of study, and use this knowledge to apply related mathematical procedures to solve routine application problems.

Outcome 2 (20marks)Use mathematical concepts and skills developed in the ‘Data analysis’ area of study to analyse a practical and extended situation, and interpret and discuss the outcomes of this analysis in relation to key features of that situation.

Outcome 3 (5 marks)Select and appropriately use technology to develop mathematical ideas, produce results and carry out analysis in situations requiring problem-solving, modelling or investigative techniques or approaches in the area of study ‘Data analysis’ and the selected module from the ‘Applications’ area of study.

1


Students need to demonstrate:

Comprehensive and correct use of mathematical conventions, symbols and terminology in all formulations, presentations, manipulations, computations and descriptions;

Thorough and relevant definition and explanation of key concepts with comprehensive identification of conditions or restrictions that apply in different contexts;

Consistent application of accurate mathematical skills and techniques to obtain correct results;

Comprehensive and detailed identification of important information, variables and constraints with appropriate selection of values for development of the mathematics relevant to the task and context;

Comprehensive and appropriate use of key statistical concepts and approaches to solve problems;

Thorough analysis, interpretation and discussion of results with comprehensive consideration of the validity and limitations of any models;

Critical and appropriate selection of technology for efficient and systematic production of solutions and presentations for given contexts;

Skilled use of technology to enable thorough analysis and interpretation of results in tabular, graphical and numerical forms.

Outline of SAC

Students will be given the entire SAC at the beginning and will have four periods to complete this SAC.

At the end of each period the student must: 1. Submit their SAC, 2. Submit their bound reference, 3. Sign to state that they have submitted both these items, 4. Clear their CAS calculators.

Students will be given their SAC and bound reference back at the start of each lesson.

Completion of SAC

The total marks allocated is 77. Each question relates to a specific outcome and therefore your final marks relate to the three outcomes on the previous page. Marks will be adjusted so that they reflect the marks allocated for each outcome and therefore students’ final mark will be out of 40.

The final mark will be given to students once calculated based on the outcomes and after comparing the results of all three classes.

After SACs have been marked, teachers will decide if a student has demonstrated enough knowledge throughout their SAC to be given a satisfactory. If a student can’t demonstrate through her SAC that she understands the content covered then the student must demonstrate her knowledge by either: 1. Showing her teacher her up-to-date workbook; or, 2. Completing a short test.

At the end of the SAC students must sign an agreement form stating that they have submitted their SAC and that all work is their own.

2


Name: ____________________________________ Teacher: ________________________________________

All answers correct to two decimal places

Buying a CarNew cars are safer to drive, but they are expensive to buy. This SAC explores the prices of cars of various ages and various makes and attempts to find a satisfactory relationship between the age of a car and its price.

QUESTION ONEAndrew is an 18 year old who would like to buy a recent model car because of the additional safety features. He goes onto carsguide.com.au and collates the following information about the prices of 2009 model cars that are bought now. All vehicles are standard automatic petrol sedans and the prices are the averages for a private sale for that model.

2009 Car model Price $(AUS) 2009 Car model Price $(AUS)BMW 3 35i $73400 Kia Cerato $11800Dodge Avenger $16000 Kia Rio $10100Ford Falcon $15500 Mazda 3 $16700Ford Focus $14400 Mercedes Benz

E200K $58400Ford Mondeo $17400 Mercedes Benz

E280 $64300Holden Barina $11600 Mercedes Benz

E500 $97500Holden Berlina $22100 Nissan Tiida $12200Holden Caprice $38100 Subara Liberty $20300Holden Commodore $16830 Toyota Camry

Altise $15400Holden Statesman $37800 Toyota Corolla $14900Honda Accord V6 $23500 Toyota Yaris $11800Honda Civic $17300 Volvo S40 $28000Hyundai Accent $11700 Volvo S60 $28100Hyundai Elantra $15400 Volvo S80 $44400Hyundai Sonata $16600

3


4


(1A) Complete the following frequency table for the 2009 car price data by filling in the missing frequencies. (1 mark)

Price Range Frequency$0<-$10000 0

$10000<-$20000

17$20000<-$30000

5$30000<-$40000$40000<-$50000

1$50000<-$60000

1$60000<-$70000

1$70000<-$80000$80000<-$90000

0$90000<-$100000

1

(1B) Construct a histogram of the 2009 car price data on the grid below.(3 marks)

5


(1C) Describe the 2009 Car price data in terms of shape, centre and spread.

(3 marks)

(1D) There are a number of outliers in the 2009 Car price data. Determine the value of these outliers by calculating the lower and upper boundary. (3 marks)

6


QUESTION TWOAndrew is wondering about whether some vehicles hold their value better than others. He explores the current prices of a number of used-vehicles that were manufactured in 2005 and 2009. The table below collates this information includes the mean and standard deviation of the prices for each year:

2009 Car Price ($)

2005 Car Price ($)Ford Falcon 15,500 7,700

Ford Focus 14,400 7,300Holden Berlina 22,100 9,700Holden Caprice 38,100 15,400Holden Commodore

16,830 7,300Holden Statesman

37,800 12,800Honda Accord V6 23,500 11,600Honda Civic 17,300 9,600Hyundai Elantra 15,400 7,500Hyundai Sonata 16,600 6,600Kia Cerato 11,800 7,700Kia Rio 10,100 6,000Mazda 3 16,700 11,500Subara Liberty 20,300 12,400Toyota Camry Altise

15,400 7,250Toyota Corolla 14,900 9,000Volvo S40 28,000 14,800Volvo S60 28,100 14,800Average 20157.22 9941.67Standard deviation

8099.91 3080.98

(2A) Randomly select one of the cars from this data, stating below the method that you used to select this vehicle.

(1 mark)

7


(2B) Calculate the standardised score for your selected vehicle for the 2009 data correct to 2 decimal places. (2 marks)

(2C) For the standardised score obtained in (2B):(i) Determine the approximate percentage (or range) for the

number of cars below this value(ii) Explain your reasoning for your answer

(2 marks)

(2D) Calculate the standardised score for the same vehicle for the 2005 car data correct to 2 decimal places.

(2 marks)

8


(2E) On the normal graph provided, roughly place your values found in (2B) for 2009 and (2D) for 2005 and comment on what the 2 standard scores imply for your chosen car.

(2 marks)

(2F) Andrew is thinking of buying a car in the top 16% price range. Determine the minimum amount that he would spend on a car in:

(i) 2009(ii) 2005 (2 marks)

9


(2G) Calculate the five number summaries for the prices of cars from each of the years 2009 and 2005. (2 marks)

(2H) Construct parallel boxplots for the price data from 2005 and 2009 on the grid below (include any outliers).

(4 marks)

10


(2I) Compare the boxplots for the years 2005 and 2009 and make a conclusion regarding the price of cars from 2005 and 2009.

(4 marks)

11


QUESTION THREEAndrew is interested in buying a Mitsubishi Magna sedan. The scatter plot shows the relationship between the age of a Magna sedan and its price on carsguide.com.au.

7 9 11 13 15 17 19 21 23 25 271000150020002500300035004000450050005500

Age of Magna (years)

Pric

e ($

)

(3A) Describe the relationship between the age of the Magna and the price in dollars in terms of strength, direction and form.

(3 marks)

(3B) The value of the coefficient of determination for the least squares relationship between the price and the age is 0.9035, correct to 4 decimal places. Interpret the meaning of this value in terms of the given data.

(1 mark)

12


(3C) Determine the value of Pearson’s correlation coefficient for this data correct to 2 decimal places. (1 mark)

(3D) Interpret the meaning of the value calculated in part (3C). (1 mark)

(3E) Given the following information calculate the equation of the least squares regression line for this data and write it below using correct variables. Write the coefficients correct to two decimal places.

(4 marks)

Remember that:b=r s ysx

a= y−b x y=a+bx

mean age of car = 16.5 years standard deviation of car ages = 5.3mean price of car = $2534.44 standard deviation of car prices =$1121.91

13


14


(3F) Explain the meaning in real terms of the gradient in this equation.(1 mark)

15


QUESTION FOURAndrew’s sister Andrea wants to further explore how the way that the age of a Corolla is related to its price. She looks at the prices of various age Corollas if they are purchased from carsguide.com.au and she gathers the data shown below:

Age of Corolla (years)

Price $(AUS)1 15,0003 14,3005 10,1007 7,7859 6,00011 3,90013 3,50015 3,10017 2,80019 2,80021 2,30023 1,75525 1,65527 1,50029 1,45531 1,455

(4A) Calculate the equation of the least squares regression line for this data and write it below correct to 2 decimal places and using the correct variables. (2 marks)

16


(4B) Calculate the value of Pearson’s correlation coefficient and write it correct to 2 decimal places. Interpret the meaning of this value.

(2 marks)

(4C) Calculate the value of the coefficient of determination correct to 2 decimal places and interpret the meaning of this value.

(2 marks)

(4D) Use the least squares regression equation to calculate the price of a 10 year old

Corolla. (2 marks)

(4E) Andrea is looking at a 10 year old Corolla which is advertised at $5000. Calculate the residual of the price for this vehicle.

(1 mark)

17


18


(4F) Add the least squares regression line to the scatterplot provided below for the Corolla data.

(2 marks)

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 320

100020003000400050006000700080009000

10000110001200013000140001500016000


Pric

e ($

)

(4G) Complete the table of residuals correct to the nearest dollar for the Corolla data shown below. (2 marks)


Residual of price ($)13579 – 1,845

11 – 3,12213 – 2,69815 – 2,27517 – 1,75119 – 92721 – 60423 – 32525 39827 1,06729 1,846

19


31 2,669(4H) The residual plot for the Corolla data is shown below. It is also incomplete. Plot the missing values from the table in part (4H) on this residual plot. (2 marks)

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

-3500-3000-2500-2000-1500-1000

-5000

5001000150020002500300035004000

(4J) What does the residual plot indicate about the relationship between the age of a Corolla and its price?

(2 marks)

20


QUESTION FIVEAndrea has come to the conclusion that a transformation may enable her to make a more accurate prediction of the price of the Corolla given its age. Use the table at the start of question 4 to help you answer the following questions.

(5A) Perform each of the transformations listed below and write the value of r2 correct to 2 decimal places and describe the residual plot of each transformation. (6 marks)

Relationship r2

valueDescribe Residual Plot

Price vs age

Price vs log(age)

1/price vs age

(5B) State which relationship you consider best and give reasons for your answer.

(2 marks)

(5C) Show that the equation of the regression line for your transformation is:

Price= 10.000023× Age−0.0000144

(2 marks)

21


(5D) Predict the price of a 10-year-old Corolla using the relationship that you consider best correct to the nearest dollar. (2 marks)

(5E) Comment on the reliability of your prediction in part (5D). (2 marks)

(5F) Calculate the residual for the 10-year-old Corolla advertised at $5000.

(2 marks)

(5G) When predicting from the linear relationship for the same vehicle a residual of −$2433.58 was obtained. Compare this value to the one obtained in part (5F), and explain, referring to the result of the transformation, why any difference was observed.

(2 marks)

22


End of SAC

23

further mathematics unit 3 sac 1mcmullenmaths.weebly.com/.../0/8/10083938/car_sac_final.docx · web...

Documents