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# Maths A - Chapter 11

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### Maths A - Chapter 11

1. 1. 11syllabussyllabusrrefefererenceence Strand: Statistics and probability Core topic: Data collection and presentation In thisIn this chachapterpter 11A Scatterplots 11B Regression lines 11C Time series and trend lines Scatterplots and time series MQ Maths A Yr 11 - 11 Page 451 Thursday, July 5, 2001 11:03 AM
2. 2. 452 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Introduction The manager of a small ski resort has a problem. He wants to be able to predict the number of skiers using his resort each weekend in advance, so that he can organise additional resort stafﬁng and catering if needed. He knows that good deep snow will attract skiers in big numbers but scant covering is unlikely to attract a crowd. To investigate the situation further, he collects the following data over twelve consecutive weekends at his resort. As there are two types of data in this example, they are called bivariate data. For each item (weekend), two variables are considered (depth of snow and number of skiers). When analysing bivariate data, we are interested in examining the relationship between the two variables. In the case of the ski resort data, the manager might be interested in answering the following questions. 1. Are visitor numbers related to depth of snow? 2. If there is a relationship between visitor numbers and depth of snow, is it always true or is it just a guide? In other words, how strong is the relationship? 3. How much conﬁdence could be placed in the prediction? In this chapter we shall examine how questions such as these can be answered by the appropriate presentation of data. Depth of snow (m) Number of skiers 0.5 120 0.8 250 2.1 500 3.6 780 1.4 300 1.5 280 1.8 410 2.7 320 3.2 640 2.4 540 2.6 530 1.7 200 MQ Maths A Yr 11 - 11 Page 452 Thursday, July 5, 2001 11:03 AM
3. 3. C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 453 1 When graphing pairs of variables, we commonly have an ‘independent variable’ and a ‘dependent variable’. Explain these terms. 2 By convention, which of the above two variable types is graphed on the horizontal axis? 3 For the following pairs of variables, state which variable would be classiﬁed as the dependent variable. a age and height b distance travelled and time taken (at a ﬁxed speed) c temperature and elevation d blood alcohol level and reaction time e IQ and results on an academic test f overtime pay and hours worked g value of a car and its age h time taken to travel a given distance and speed of travel 4 For each of the pairs of variables in question 3, state what happens to the value of the second variable when there is an increase in the value of the ﬁrst variable. 5 If we collected data for each of the pairs of variables in question 3 and plotted the points, we would obtain a set of points scattered over the Cartesian plane. Consider a straight line which could be drawn indicating the general trend in each case. a Draw up a set of labelled axes (no scale) for each case in question 3. b Sketch a straight line which would show the general trend of the relationship between the two variables. Scatterplots We shall now look more closely at the data collected by the ski resort manager, and at the three questions he posed. To help answer these questions, the data can be arranged on a scatterplot. Each of the data points is represented by a single visible point on the graph. When drawing a scatterplot, it is important to choose the correct variable to assign to each of the axes. The convention is to place the independent variable on the x-axis and the dependent variable on the y-axis. The independent variable in an experiment or investigation is the variable that is deliberately controlled or adjusted by the investigator. The dependent variable is the variable that responds to changes in the independent variable. Neither of the variables involved in the ski resort data was controlled directly by the investigator but ‘Number of skiers’ would be considered the dependent variable because it is likely to change depending on depth of snow. (The snow 0 200 400 600 800 0 2 31 4 Numberofskiers Depth of snow (m) MQ Maths A Yr 11 - 11 Page 453 Thursday, July 5, 2001 11:03 AM
4. 4. 454 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d depth does not depend on numbers of skiers.) As ‘Number of skiers’ is the dependent variable, we graph it on the y-axis and the ‘Depth of snow’ on the x-axis. Notice how the scatterplot for the ski resort data shows a general upward trend. It is not a perfectly straight line but it is still clear that a general trend or relationship has formed: as the depth of snow increases, so too does the number of skiers. Once the scatterplot has been drawn, we can determine if any pattern is evident. Worked example 1 shows how, as a general rule, as height increases so does mass. We can also look to see if the pattern is linear. In worked example 1, although the points are not in a perfect straight line, they approximate a straight line. The ﬁgures below show examples of linear and non-linear relationships. Linear relationships Positive relationship between variables; Negative relationship between variables; that is, as one variable increases, the that is, as one variable increases, the other variable also increases. other variable decreases. The table below shows the height and mass of ten Year 11 students. Display the data on a scatterplot. Height (cm) 120 124 130 135 142 148 160 164 170 175 Mass (kg) 45 50 54 59 60 65 70 78 75 80 THINK WRITE Show the height on the x-axis and the mass on the y-axis. Plot the point given by each pair. 1 40 50 60 70 80 100 140 160120 180 Mass(kg) Height (cm) 2 1WORKEDExample 0 y x 0 y x MQ Maths A Yr 11 - 11 Page 454 Thursday, July 5, 2001 11:03 AM
5. 5. C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 455 Non-linear relationships In other cases it may be that there is no relationship at all between the two variables. Such a scatterplot would look like the one shown on the right. 0 y x 0 y x 0 y x 0 y x 0 y x The table below shows the length and mass of a dozen eggs. a Display this information in a scatterplot. b Determine if there is any relationship between the length and mass of the eggs and state if the relationship is linear. Length (cm) 6.2 3.9 4.5 5.8 7.2 7.6 6.1 6.7 7.3 5.1 6.0 7.3 Mass (g) 60 15 25 50 95 110 55 75 95 35 54 96 THINK WRITE a Display length on the x-axis and mass on the y-axis. a Plot the point given by each pair. b Study the scatterplot to see if mass increases as length increases. b As length increases, so does the mass of the egg. Study the scatterplot to see if the points seem to approximate a straight line. The points do not approximate a straight line and so the relationship is not linear. 1 0 20 40 60 80 100 120 0 81 2 3 4 5 6 7 Mass(g) Length (cm) 2 1 2 2WORKEDExample MQ Maths A Yr 11 - 11 Page 455 Thursday, July 5, 2001 11:03 AM
6. 6. 456 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d An understanding of the relationship between two variables offers a tool for future planning by businesses. Umbrellas can be stocked in preparation for the wet season, heaters can be stocked for an on-coming winter, the stocks of ice-cream are much lower during winter. An appreciation of these relationships provides an opportunity for a busi- ness to gain an edge over its competitors. The graphics calculator and statistical data A graphics calculator is a great labour-saving device when it comes to statistical calculations and display of data. If you have one readily available, its use in this chapter will greatly reduce your manual calculations. Let us repeat worked example 2 using a graphics calculator. (Instructions here apply to the Texas Instrument TI–83 calculator. Other brands offer similar functions.) 1 Clear the Y= editor and turn off any existing plots by pressing [STAT PLOT] and choosing 4: PlotsOff. 2 Press and select 1:Edit. Enter the x-data (length) in L1 and y-data (mass) in L2. 3 Press [STAT PLOT] then press to select the options for Plot 1. 4 Turn the plot ON; select the type as a scatterplot; the Xlist as L1; and the Ylist as L2. 5 Press and choose 9:ZoomStat. 6 The scatterplot shown in worked example 2 will be shown on the screen. Press , then use the arrow keys to examine the plot. 7 To examine the summary statistics of the x- and y-variables, press , then use the right arrow key to highlight CALC. Option 1 displays statistics of the variable x (mean, standard deviation, quartiles, minimum and maximum values etc.) while option 2 displays statistics for both variables. Note the LinReg(ax + b) (linear regression function) available. We will use this function later to determine the equation of the best straight line which ﬁts the data in the scatterplot. investigat ioninv estigat ion 2nd STAT 2nd ENTER ZOOM TRACE STAT remember 1. A scatterplot is a graph that is used to compare two variables. 2. One variable (the independent variable) is on the horizontal axis and the other variable (the dependent variable) is on the vertical axis. 3. Points are plotted by the pair formed by each variable. 4. A relationship between the variables exists if one increases as the other increases or if one decreases as the other increases. 5. If the points on the scatterplot seem to approximate a straight line, the relationship can be said to be linear. remember MQ Maths A Yr 11 - 11 Page 456 Thursday, July 5, 2001 11:03 AM
7. 7. C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 457 Scatterplots Use a graphics calculator if you have one available. 1 The table below shows the marks obtained by a group of 10 students in history and geography. Display this information on a scatterplot. 2 The table below shows the maximum temperature each day together with the number of people who attend the cinema that day. Display the information on a scatterplot. 3 The table below shows the wages of 20 people and the amount of money they spend each week on entertainment. Display this information on a scatterplot. 4 The table below shows the marks obtained by nine students in English and History. a Display the information on a scatterplot. b Is there any relationship between the marks obtained in English and in History? If there does appear to be a relationship, is the relationship linear? 5 The table below shows the daily temperature and the number of hot pies sold at the school canteen. a Display the information on a scatterplot. b Determine if there appears to be any relationship between the two variables and if the relationship appears to be linear. History 36 65 82 72 58 39 58 74 82 66 Geography 45 78 66 72 50 51 61 70 60 88 Temperature (°C) 25 33 30 22 15 18 27 22 28 20 No. at cinema 256 184 190 312 458 401 200 357 312 423 Wages (\$) 370 380 500 510 395 430 535 490 495 550 Amount spent on entertainment (\$) 55 85 150 75 145 100 130 115 70 150 Wages (\$) 810 460 475 520 530 475 610 780 350 460 Amount spent on entertainment (\$) 220 50 100 150 140 160 90 130 40 50 English 55 20 27 33 73 18 37 51 79 History 72 37 53 74 73 44 59 55 84 Temperature (°C) 24 32 28 23 16 14 26 20 29 21 No. of pies sold 56 20 24 60 84 120 70 95 36 63 11A WORKED Example 1 E XCEL Spread sheet Scatterplot WORKED Example 2 MQ Maths A Yr 11 - 11 Page 457 Thursday, July 5, 2001 11:03 AM
8. 8. 458 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 6 Container ships arriving on a wharf are unloaded by work teams. The table below shows the number of people in the work team and the time taken to unload the container ship. a Display the information on a scatterplot. b Determine if there appears to be a relationship between the number of people in the work team and the time taken to unload the container ship. If there is a relationship, does the relationship appear to be linear? 7 Which of the following scatterplots does not display a linear relationship? A B C D 8 In which of the following is no relationship evident between the variables? A B C D No. in work team 15 18 12 19 22 21 17 16 18 20 Hours taken 20 16 25 15 14 13 18 20 17 14 mmultiple choiceultiple choice y x y x y x y x mmultiple choiceultiple choice y x y x y x y x MQ Maths A Yr 11 - 11 Page 458 Thursday, July 5, 2001 11:03 AM
9. 9. C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 459 9 Give an example of a situation where the scatterplot may look like the ones below. a b Fitting straight lines to bivariate data The process of ‘ﬁtting’ straight lines to bivariate data enables us to analyse relation- ships between the data and possibly make predictions based on the given data set. Fitting a straight line by eye (line of best ﬁt) Consider the set of bivariate data points shown at right. In this case the x-values could be the hand lengths of a group of people, while y-values could be the lengths of their feet. We wish to determine a linear relationship between these two random variables. Of course, there is no single straight line which would go through all the points, so we can only estimate such a line. Furthermore, the more closely the points appear to be on or near a straight line, the more conﬁdent we are that such a linear relationship may exist and the more accurate our ﬁtted line should be. Consider the estimate, drawn ‘by eye’ in the ﬁgure below. It is clear that most of the points are on or very close to this straight line. This line was easily drawn since the points are very much part of an apparent linear relationship. However, note that some points are below the line and some are above it. Furthermore, if x is the hand length and y is the foot length, it seems that people’s feet are generally longer than their hands. Regression analysis is concerned with ﬁnding these straight lines of best ﬁt using various methods so that the number of points above and below the lines are ‘balanced’. Collecting bivariate data 1 Choose one of the following and collect data from within your class. a Each person’s hand span and height. b Each person’s resting heart rate and the time it takes for them to run 400 m. c Each person’s mark in mathematics and in science. 2 Display the results on a scatterplot. 3 Discuss any relationship that may be evident between the two variables. Work SHEET 11.1 0 y x 0 y x inv estigat ioninv estigat ion y x y x MQ Maths A Yr 11 - 11 Page 459 Thursday, July 5, 2001 11:03 AM
10. 10. 460 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Methods of drawing lines of best ﬁt There are many different methods of ﬁtting a straight line by eye. They may appear logical or even obvious but ﬁtting by eye involves a considerable margin of error. We are going to consider only one method: ﬁtting the line by balancing the number of points The technique of balancing the number of points involves ﬁtting a line so that there is an equal number of points above and below the line. For example, if there are 12 points in the data set, 6 should be above the line and 6 below it. As mentioned above, estimating the position for the line of best ﬁt by eye involves a considerable margin of error. For this reason, when making predictions of one variable from another using this graphical technique, the forecasts obtained can be considered rough estimates at best. A graphics calculator allows much more reliable predictions to be made. Draw the line of best ﬁt for the data in the ﬁgure using the equal-number-of-points method. THINK DRAW Note that the number of points (n) is 8. Fit a line where 4 points are above the line. Using a clear plastic ruler, try to ﬁt the best line. The ﬁrst attempt has only 3 points below the line where there should be 4. Make reﬁnements. The second attempt is an improvement, but the line is too close to the points above it. Improve the position of the line until a better ‘balance’ between upper and lower points is achieved. y x 1 2 y x 3 y x 4 y x 3WORKEDExample MQ Maths A Yr 11 - 11 Page 460 Thursday, July 5, 2001 11:03 AM
11. 11. C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 461 Regression lines After drawing a scatterplot, we look for a relationship between the variables. If there is a relationship, in many cases it appears to be linear. Once it has been established that a linear relationship exists, we can introduce a regression line that will enable us to make predictions about the two variables. Consider the example at the beginning of the chapter where the manager of the ski resort wants to predict the number of skiers at the resort on the weekend. Based on past data, we drew a scatterplot. On the scatterplot, we can draw the line of best ﬁt. This line of best ﬁt can be extended and then used to make predictions about the data. When this is done, the line is called a regression line. In the above example the depth of snow can then be used to predict the number of visitors to the resort. 0 200 400 600 800 0 2 31 4 Numberofskiers Depth of snow (m) 0 200 400 600 800 0 2 31 4 Numberofskiers Depth of snow (m) MQ Maths A Yr 11 - 11 Page 461 Thursday, July 5, 2001 11:03 AM
12. 12. 462 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Making predictions If a linear relationship exists between a pair of variables, it is useful to be able to make predictions of one variable from the other. There are two methods available: 1. Algebraically — we could ﬁnd an equation for this regression line (or line of best ﬁt), substitute the value of the known variable and so calculate the value of the unknown variable. A graphics calculator will generate the equation of the regression line from data entered in lists. 2. Graphically — we can draw lines horizontally or vertically from the known value to the line, then read off the corresponding value of the unknown variable. Using a graphics calculator, we can use the ‘trace’ function to determine variable values on the line. Algebraic predictions Using a graphics calculator If you have access to a graphics calculator, the equation of the regression line can readily be produced from the two sets of entered data. Manually To calculate the line of best ﬁt, we must determine the gradient and y-intercept of the drawn line. Equation of a line of best ﬁt or regression line To have the graphics calculator automatically ﬁnd the equation of a line of best ﬁt (also called a regression line), follow these steps. The example below uses the skiers and snow depth data given at the beginning of this chapter. 1. Enter the depth data in L1 and number of skiers in L2 (ﬁrst press , and select EDIT and 1:Edit...). 2. Press , select CALC and choose 4:LinReg(ax + b). 3. Enter L1, L2, Y1 (that is, press [L1] [L2] ; Y1 is found under VARS/Y-VARS/Function). 4. Press . 5. Ensure a STATPLOT is set up by entering [STATPLOT], setting Plot 1 to ON, setting the Type for a scatterplot, Xlist as L1 and Ylist as L2. 6. Adjust WINDOW settings to cope with extremes of the data, then press GRAPH. 7. Use [CALC] and select 1:Value or press to ﬁnd values on the line. 8. The graphics calculator shows the line of best ﬁt to have the equation y = 186.4x + 28.3. inv estigat ioninv estigat ion STAT STAT 2nd , 2nd , ENTER 2nd 2nd TRACE MQ Maths A Yr 11 - 11 Page 462 Thursday, July 5, 2001 11:03 AM
13. 13. C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 463 Once the regression line has been drawn, to make predictions, it is useful to ﬁnd its equation. The equation of a straight line has the form y = mx + c, where m is the gradient and c is the y-intercept. The gradient of a regression line is best found with a ruler and pencil by measuring the rise and the run from two points on the line of regression. The gradient is then found using: The y-intercept can then be seen by noting the point where the line crosses the y-axis. Consider worked example 3. From this ﬁgure we can see that the equation of the regression line is approximately y = x + 3. That is, chemistry = × physics + 3. A teacher may then be able to estimate a student’s chemistry mark by using their result in physics. For example a student who got a mark of 5 in physics may get: y = × 5 + 3 = 6 The teacher would estimate a mark of 6 for this student in chemistry. Use a graphics calculator to determine the equation of the regression line for the relationship between the chemistry and physics marks. Your resulting equation should be y = 0.6x + 2.8 (close to the one determined here). The table below shows the marks of 10 students in a physics and chemistry quiz. Draw a scatterplot and on the graph show the regression line. Physics 2 5 6 7 7 8 8 9 9 10 Chemistry 4 7 5 8 6 6 9 7 10 9 THINK WRITE Put physics on the x-axis and chemistry on the y-axis. Plot the point corresponding to each pair of marks. Add a line of regression. 1 0 2 4 6 8 10 0 4 62 8 10 Chemistry Physics 2 3 4WORKEDExample SkillS HEET 11.1Gradient m( ) rise run --------= 5 8 --- 0 2 4 6 8 10 0 4 62 8 run = 8 rise = 5 y-intercept = approx. 3 10 Chemistry Physics 5 8 --- Cabri Geo metry Gradient Cabri Geo metry Graphs and y-intercept 5 8 --- 1 8 --- MQ Maths A Yr 11 - 11 Page 463 Thursday, July 5, 2001 11:03 AM
14. 14. 464 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d The following table shows the bus fare charged by a bus company for different distances. a Represent the data using a scatterplot and add the regression line. b Find the gradient and y-intercept of the regression line; hence, ﬁnd its equation, manually. c Enter the data as lists into a graphics calculator and determine the equation of the regression line. Distance (km) Fare (\$) 1.5 2.10 0.5 2.00 7.5 4.50 6 4.00 6 4.50 2.5 2.60 0.5 2.10 8 4.50 4 3.50 3 3.00 THINK WRITE a Show the distance on the x-axis and the fare on the y-axis. a Plot the points given by each pair. Add the regression line to the graph. b Take two points on or closest to the regression line and ﬁnd the gradient between them by substituting in the formula and simplifying. b rise = 2, run = 5.5 m = ≈ 0.36 1 0.00 1.00 2.00 3.00 4.00 5.00 0 81 2 3 4 5 6 7 Fare(\$) Distance (km) 2 3 1 m rise run --------= 0.00 1.00 2.00 3.00 4.00 5.00 0 8 run = 5.5 rise = 2 1 2 3 4 5 6 7 Fare(\$) Distance (km) 2 5.5 ------- 5WORKEDExample EXCE L Spreadshe et Gradient MQ Maths A Yr 11 - 11 Page 464 Thursday, July 5, 2001 11:03 AM
15. 15. C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 465 Once the equation of the regression line has been found, it is possible to use this equation to make predictions by substituting the value of the given variable into it. Graphical predictions As mentioned earlier, once the regression line has been drawn, horizontal or vertical lines can be added from the axes to the line to read off the value of the unknown vari- able. Remember that the values so determined are only estimates, and not exact values. THINK WRITE Note the y-intercept. y-intercept = 1.8 Substitute into the formula y = mx + c to ﬁnd the equation. y = 0.36x + 1.8 c Enter distances into L1 and fares into L2. c The equation of the regression line is y = 0.37x + 1.8 Follow the procedures outlined previously to determine the equation of the regression line. 2 3 1 2 A casino records the number of people, N, playing a jackpot game and the prize money, p, for that game and plots the results on a scatterplot. The regression line is found to have the equation N = 0.07p + 220. a Find the number of people playing when the prize money is \$2500. b Find the likely prize on offer when there are 500 people playing. THINK WRITE a Write the equation of the regression line. a N = 0.07p + 220 Substitute 2500 for p. N = 0.07 × 2500 + 220 Calculate N. = 395 Give a written answer. There would be approximately 395 people playing. b Write the equation of the regression line. b N = 0.07p + 220 Substitute 500 for N. 500 = 0.07p + 220 Solve the equation. 280 = 0.07p p = p = 4000 Give a written answer. The prize would be approximately \$4000. 1 2 3 4 1 2 3 280 0.07 ---------- 4 6WORKEDExample MQ Maths A Yr 11 - 11 Page 465 Thursday, July 5, 2001 11:03 AM
16. 16. 466 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Causality The term causality refers to one variable causing another. For example, there is a strong relationship between a person’s shoe size and shirt size (as one tends to increase, the other also tends to increase). However, this does not mean that one causes the other. There also appears to be a strong relationship between the number of cigarettes smoked and the chance of contracting lung cancer. In this case, we are advised that smoking causes lung cancer. We cannot draw the conclusion that one variable in a relationship causes another just because there is a strong relationship between the two. In fact, sometimes the strong relationship between two variables is caused by their relationship to a third variable. In the example of shoe and shirt sizes, above, it is more likely that they are both dependent on age. Consider the graph drawn in worked example 5 representing bus fares for travelling various distances. Graphically estimate: a the fare for a journey of 5 km b how far you could expect to travel for a cost of \$3.25. THINK WRITE a Find the value of 5 km on the x-axis. a Draw a vertical line from this point to meet the regression line. Draw a horizontal line from this point to the y-axis. Read this value on the y-axis. A distance of 5 km of bus travel would cost about \$3.60. b Locate \$3.25 on the y-axis. b Draw a horizontal line from this point to the regression line. From this point, draw a vertical line to the x-axis. Read off the x-value at this point. For \$3.25 you could expect to travel about 4 km. 1 0.00 1.00 2.00 3.00 4.00 5.00 0 1 2 3 4 5 6 7 Fare(\$) Distance (km) \$3.60 \$3.25 5 km 4 km 2 3 4 1 2 3 4 7WORKEDExampleinv estigat ioninv estigat ion MQ Maths A Yr 11 - 11 Page 466 Thursday, July 5, 2001 11:03 AM
17. 17. C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 467 Interpolation and extrapolation We use the term interpolation for the making of predictions from a graph’s regression line from within the bounds of the original experimental data. Data may be interpolated either algebraically or graphically as shown by the pre- vious examples. We use the term extrapolation for the making of predictions from a graph’s regression line from outside the bounds of the original experimental data. Suppose, in the ski resort problem, that we wished to ﬁnd the number of skiers if the depth of snow was 4.5 m. The data could be extrapolated graphically by ﬁrst extending the trend line to make the prediction possible. The graph shows that about 900 skiers would be attracted to the resort if the snow depth was 4.5 m. How reliable is this prediction? Reliability of predictions Results predicted (whether algebraically or graphically) from the line of best ﬁt of a scatterplot can be considered reliable only if: 1. a reasonably large number of points are used to draw the scatterplot, 2. the plotted points lie reasonably close to the line of best ﬁt, 3. the predictions are made using interpolation and not extrapolation. Extrapolated results can never be considered to be reliable because when extrapolation is used we are assuming that the relationship holds true for untested values. In the case of the ski resort data we would be assuming that the relationship continues to hold true even when the snow depth is extreme. This may or may not be the case. It might be that there In a small group, for each of the following topics, discuss: a whether a positive or negative relationship exists between the variables b how strong the relationship might be c whether one variable causes the other. 1 Hours of study and exam marks. 2 Hours of exercise and resting pulse rate. 3 Weight and shirt size. 4 The number of hotels and churches in country towns. 5 The number of motels in a town and the number of ﬂights landing at the nearest airport. 6 Age and the time taken to run 100 metres. 0 0 200 400 600 800 1000 1 2 3 4 5 Snow depth (m) Numberofskiers 0 0 200 400 600 800 1000 1 2 3 4 5 Snow depth (m) Numberofskiers MQ Maths A Yr 11 - 11 Page 467 Thursday, July 5, 2001 11:03 AM
18. 18. 468 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d are only a certain number of skiers in the population — which places a limit on the maximum number visiting the resort. In this situation the trend line, if continued, may look like that in the graph above. Alternatively, it might be that when the snow depth is extreme then skiers cannot even get to the slopes. So — take care when extrapolating data! Crayﬁsh A graphics calculator is recommended for this activity. Rock lobsters (crayﬁsh) are sized according to the length of their carapace (main body shell). The table below gives the age and carapace length of 15 male rock lobsters. inv estigat ioninv estigat ion Age (years) Length of carapace (mm) Age (years) Length of carapace (mm) 3 65 14 210 2.5 59 4.5 82 4.5 80 3.5 74 3.25 68 2.25 51 7.75 130 1.76 48 8 150 10 171 6.5 112 9.5 160 12 200 MQ Maths A Yr 11 - 11 Page 468 Thursday, July 5, 2001 11:03 AM
19. 19. C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 469 1 Draw a scatterplot of the data. 2 Draw in the line of best ﬁt. 3 Suggest reasons why the points on the scatterplot are not perfectly linear. 4 Find an equation that represents the relationship between the length of the rock lobster, L, and its age, a. 5 Find the likely size of a 5-year-old male rock lobster. 6 Find the likely size of a 16-year-old male rock lobster. 7 Rock lobsters reach sexual maturity when their carapace length is approximately 65 mm. Find the age of the rock lobster at this stage. 8 The Fisheries Department wishes to set minimum size restrictions so that the rock lobsters have three full years from the time of sexual maturity in which to breed before they can be legally caught. What size should govern the taking of male crayﬁsh? 9 Which of your answers to parts 5 to 9 do you consider reliable? Why? World population The table below shows world population during the years 1955 to 1985. 1 Use a graphics calculator to create a scatterplot of the data. 2 Does the scatterplot of the Population-versus- Year data appear to be linear? 3 Use the graphics calculator to ﬁt a regression line to the data; hence ﬁnd the equation of the line and graph it on the scatterplot. 4 Use the equation to predict the world population in 1950. 5 Use the equation to predict the world population in 1982. 6 Use the equation to predict the world population in 1997. 7 Which of your answers to parts 4 to 6 do you consider reliable? Why? 8 In fact the world population in 1997 was 5840 million. Account for the discrepancy between this and your answer to part 6. 9 What would you estimate the world population to be in 2010? inv estigat ioninv estigat ion Year 1955 1960 1965 1970 1975 1980 1985 World population (millions) 2757 3037 3354 3696 4066 4432 4828 MQ Maths A Yr 11 - 11 Page 469 Thursday, July 5, 2001 11:03 AM
20. 20. 470 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Regression lines Note: For many of these questions your answers may differ somewhat from those in the back of the book. The answers are provided as a guide but there are likely to be individual differences when ﬁtting straight lines by eye. Where appropriate, the use of a graphics calculator is recommended if one is available. 1 Fit a straight line to the data in the scatterplots using the equal-number-of-points method. a b c d e remember 1. To ﬁt a straight line to bivariate data by eye, the line is drawn by balancing the number of points, ensuring an equal number of points above and below the ﬁtted line. This line is called the line of best ﬁt or regression line. 2. The equation of the regression line can be found by measuring the gradient and the y-intercept of the regression line and using the formula y = mx + c. Sometimes the gradient of the regression line will be negative. This occurs if one variable increases while the other variable decreases. 3. The equation of the regression line can also be determined using a graphics calculator. 4. Once the equation of the regression line has been found, it can then be used to make predictions about the variables. 5. Predictions can also be made graphically by drawing lines from the axes to the regression line. 6. Interpolation involves making predictions within the bounds of the original data. 7. Extrapolation involves making predictions beyond the bounds of the original data. 8. Predictions can be considered reliable if they are obtained using interpolation from a large number of points which lie reasonably close to the line of best ﬁt. remember 11B WORKED Example 3 y x y x y x y x y x MQ Maths A Yr 11 - 11 Page 470 Thursday, July 5, 2001 11:03 AM
21. 21. C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 471 2 The table below shows the marks achieved by a class of students in English and Mathematics. Show these data on a scatterplot and on the graph show the regression line. 3 Position the straight line of best ﬁt through each of the following graphs and ﬁnd the equation of each. a b c 4 In an experiment, a student measures the length of a spring (L) when different masses (M) are attached to it. Her results are shown below. a Draw a scatterplot of the data and on it draw the line of regression. b Find the gradient and y-intercept of the regression line and, hence, ﬁnd the equation of the regression line. Write your equation in terms of the variables L and M. English 64 75 81 63 32 56 47 59 73 64 Maths 76 62 89 56 49 57 53 72 80 50 Mass (g) Length of spring (mm) 0 220 100 225 200 231 300 235 400 242 500 246 600 250 700 254 800 259 900 264 WORKED Example 4 Math cad Gradient and y-intercept 0 10 20 30 40 50 60 70 y x0 4 62 8 10 0 10 20 30 40 50 60 70 0 40 6020 80 100 120 x y 0 500 1000 1500 2000 2500 3000 0 10 155 20 25 y x WORKED Example 5 MQ Maths A Yr 11 - 11 Page 471 Thursday, July 5, 2001 11:03 AM
22. 22. 472 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 5 A scientist who measures the volume of a gas at different temperatures provides the following table of values. a Draw a scatterplot of the data and on it draw the line of regression. b Give the equation of the line of best ﬁt. Write your equation in terms of the variables: volume of gas, V, and its temperature, T. 6 A sports scientist is interested in the importance of muscle bulk to strength. He measures the biceps circumference of ten people and tests their strength by asking them to complete a lift test. His results are given in the following table. a Draw a scatterplot of the data and draw the line of regression. b Find a rule for determining the ability of a person to complete a lift test, S, from the circumference of their biceps, B. Temperature (°C) Volume (L) −40 1.2 −30 1.9 −20 2.4 0 3.1 10 3.6 20 4.1 30 4.8 40 5.3 50 6.1 60 6.7 Circumference of biceps (cm) Lift test (kg) 25 50 25 52 27 58 28 51 30 60 30 62 31 53 33 62 34 61 36 66 MQ Maths A Yr 11 - 11 Page 472 Thursday, July 5, 2001 11:03 AM
23. 23. C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 473 7 A taxi company adjusts its meters so that the fare is charged according to the following equation: F = 1.2d + 3 where F is the fare, in dollars, and d is the distance travelled, in km. a Find the fare charged for a distance of 12 km. b Find the fare charged for a distance of 4.5 km. c Find the distance that could be covered on a fare of \$27. d Find the distance that could be covered on a fare of \$13.20. 8 Detectives can use the equation H = 6.1f − 5 to estimate the height of a burglar who leaves footprints behind. (H is the height of the burglar, in cm, and f is the length of the footprint.) a Find the height of a burglar whose footprint is 27 cm in length. b Find the height of a burglar whose footprint is 30 cm in length. c Find the footprint length of a burglar of height 185 cm. (Give your answer correct to 2 decimal places.) d Find the footprint length of a burglar of height 152 cm. (Give your answer correct to 2 decimal places.) 9 A football match pie seller ﬁnds that the number of pies sold is related to the temperature of the day. The situation could be modelled by the equation N = 870 − 23t, where N is the number of pies sold and t is the temperature of the day. a Find the number of pies sold if the temperature was 5 degrees. b Find the number of pies sold if the temperature was 25 degrees. c Find the likely temperature if 400 pies were sold. d How hot would the day have to be before the pie seller sold no pies at all? 10 The following table shows the average annual costs of running a car. It includes all ﬁxed costs (registration, insurance etc.) as well as running costs (petrol, repairs etc.). a Draw a scatterplot of the data. b Draw in the line of best ﬁt. c Find an equation which represents the relationship between the cost of running a vehicle, C, and the distance travelled, d. d Use your graph and its equation to ﬁnd: i the annual cost of running a car if it is driven 15 000 km ii the annual cost of running a car if it is driven 1000 km iii the likely number of kilometres driven if the annual costs were \$8000 iv the likely number of kilometres driven if the annual costs were \$16 000. Distance (km) Annual cost (\$) 5 000 4 000 10 000 6 400 15 000 8 400 20 000 10 400 25 000 12 400 30 000 14 400 WORKED Example 6 SkillS HEET 11.4 SkillS HEET 11.3 SkillS HEET 11.2 WORKED Example 7 MQ Maths A Yr 11 - 11 Page 473 Thursday, July 5, 2001 11:03 AM
24. 24. 474 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 11 A market researcher ﬁnds that the number of people who would purchase ‘Wise-up’ (the thinking man’s deodorant) is related to its price. He provides the following table of values. a Draw a scatterplot of the data. b Draw in the line of best ﬁt. c Find an equation that represents the relationship between the number of cans of ‘Wise-up’ sold, N (in thousands), and its price, p. d Use the equation to predict the number of cans sold each week if: i the price was \$3.10 ii the price was \$4.60. e At what price should ‘Wise-up’be sold if the manufacturers wished to sell 80 000 cans? f Given that the manufacturers of ‘Wise-up’ can produce only 100 000 cans each week, at what price should it be sold to maximise production? 12 The following table gives the adult return air fares between some Australian cities. a Draw a scatterplot of the data and on it draw the regression line. b Find an equation that represents the relationship between the air fare, A, and the distance travelled, d. c Use the equation to predict the likely air fare (to the nearest dollar) from: i Sydney to the Gold Coast (671 km) ii Perth to Adelaide (2125 km) iii Hobart to Sydney (1024 km) iv Perth to Sydney (3295 km). Price (\$) Weekly sales (× 1000) 1.40 105 1.60 101 1.80 97 2.00 93 2.20 89 2.40 85 2.60 81 2.80 77 3.00 73 3.20 69 3.40 65 City Distance (km) Price (\$) Melbourne–Sydney 713 580 Perth–Melbourne 2728 1490 Adelaide–Sydney 1172 790 Brisbane–Melbourne 1370 890 Hobart–Melbourne 559 520 Hobart–Adelaide 1144 820 Adelaide–Melbourne 669 570 EXCE L Spreadshe et Interpolation/ extrapolation MQ Maths A Yr 11 - 11 Page 474 Thursday, July 5, 2001 11:03 AM
25. 25. C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 475 An electrical repair business charges its customers using the formula C = 40h + 35, where C is the cost of the repairs and h is the time taken for the repairs, in hours. Find the cost of a repair job that took: 1 2 hours 2 5 hours 3 1 hour and 15 minutes. Estimate the time taken (in hours and minutes) for repairs if the cost of the repairs was: 4 \$175 5 \$275 6 \$145. The information below is to be used for questions 7–10. A survey relating exam marks to the amount of television watched ﬁnds that the regression line has the equation M = 95 − 15t, where M is the mark obtained and t is the average number of hours of television watched each night by the students. 7 Estimate the mark of a person who averages one hour of television per night. 8 Estimate the mark of a person who watches an average 4 hours of television per night. 9 Estimate the amount of television watched per night by a person who scores a mark of 65. 10 Jodie scored 27.5 on the exam. Estimate the average amount of television that Jodie watches each night. Time series and trend lines We have looked at bivariate, or (x, y), data where both x and y could vary independ- ently. We shall now consider cases where the x-variable is time and, generally, where time goes up in even increments such as hours, days, weeks or even years. In these cases we have what is called a time series. The main purpose of a time series is to see how some quantity varies with time. For example, a company may wish to record its daily sales ﬁgures over a 10-day period. We could also make a graph of this time series as shown in the ﬁgure. As can be seen from this graph, there seems to be a trend upwards — clearly, this company is increasing its revenues! Time Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8 Day 9 Day 10 Sales (\$) 5200 5600 6100 6200 7000 7100 7500 7700 7700 8000 1 Days Sales(\$) 4000 5000 6000 7000 8000 9000 10000 0 1 2 3 4 5 6 7 8 9 10 t MQ Maths A Yr 11 - 11 Page 475 Friday, July 6, 2001 2:37 PM
26. 26. 476 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Types of trend Many types of trend exist. They can be classiﬁed as secular, seasonal, cyclic and random. Secular trends If over a reasonably long period of time a trend appears to be either increasing or decreasing steadily, with no major changes of direction, then it is called a secular trend. It is important to look at the data over a long period. If the trend in the ﬁgure on the previous page continued for, say, 30 days, then we could safely conclude that the company was indeed becoming more proﬁtable. What appears to be a steady increase over a short term — say, stock market share prices — can turn out to be something quite different over the long run. Seasonal trends Certain data seem to ﬂuctuate during the year, as the seasons change. Consequently, this is termed a seasonal trend. The most obvious example of a seasonal trend would be the average monthly maximum temperatures over a year. A sample of this type of trend is shown in the ﬁgure. Note that the months have been coded, so that 1 = Jan., 2 = Feb., and so on. From which hemisphere of the world would these data come? Cyclic trends Like seasonal trends, cyclic trends show ﬂuctuations upwards and downwards, but not according to season. Businesses often have cycles where at times proﬁts increase, then decline, then increase again. A good example of this would be the sales of a new major software product, such as a word processor. At ﬁrst, sales are slow; then they pick up as the product becomes popular. When enough people have bought the product, sales may fall off until a new version of the product comes on the market, causing sales to increase again. This cycle can be repeated many times, which is why there are many versions of some software products. Random trends Trends may seem to occur at random. This can be caused by external events such as ﬂoods, wars, new technologies or inventions, or anything else that results from random causes. There is no obvious way to predict the direction of the trend or even when it changes direction. Months Temp.(°C) 14 18 22 26 30 0 1 2 3 4 5 6 7 8 9 10 11 12 t MQ Maths A Yr 11 - 11 Page 476 Thursday, July 5, 2001 11:03 AM
27. 27. C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 477 In the ﬁgure, there are a couple of minor ﬂuctuations at t = 4 and t = 8, and a major one at t = 13. The major ﬂuctuation could have been caused by a change in government which positively affected sales. The trend line If we want to predict the future values of a trend, it is important to be able to ﬁt a straight line to the data that we already have. There are a number of techniques which can be used to determine the trend line. We will use the method of ﬁtting the line ‘by eye’ employing the ‘equal number of points’ technique that was used previously. Trend lines in ABS census data This activity can be undertaken using a spreadsheet or graphics calculator. The Australian Bureau of Statistics has published data from the 1996 Australian Census on their web site, http://www.abs.gov.au. (It will take some time for the 2001 Census material to be available. Download the updated data when it is published.) An extract from their material shows the age breakdown year by year for males and females to age 79 (groupings occur after this date). Profits 14 18 22 26 30 0 2 4 6 8 10 12 14 16 t Fit a straight line to the following time series data, which represent the body temperature of a patient with appendicitis, taken every hour. THINK WRITE Attempt to ﬁt a line using your eye. By trial and error, a line such as the one at right could be the trend line. Try to balance the number of points either side of the line. Evaluate the trend. It is unlikely that the temperature will continue to rise indeﬁnitely, but the line may be signiﬁcant over the short term. Hours Temp.(°C) 30.0 30.5 31.0 31.5 32.0 32.5 33.0 0 1 2 3 4 5 6 7 8 9 10 t 1 Hours Temp.(°C) 30.0 30.5 31.0 31.5 32.0 32.5 33.0 0 1 2 3 4 5 6 7 8 9 10 t 2 8WORKEDExample inv estigat ioninv estigat ion MQ Maths A Yr 11 - 11 Page 477 Thursday, July 5, 2001 11:03 AM
28. 28. 478 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d AUSTRALIANBUREAUOFSTATISTICS1996CensusofPopulationandHousing Australia7688965.464sqkms B03AGEBYSEX AllPersons MaleFemalePersons 0years123101116506239607 1126232119980246212 2133747126933260680 3132414126111258525 4133393126489259882 0–46488876160191264906 5133908127701261609 6133759127144260903 7130505124238254743 8129595123650253245 9129424123389252813 5–96571916261221283313 10131934125566257500 11131046126007257053 12132320126563258883 13134123127098261221 14131117123939255056 10–146605406291731289713 15130655123635254290 16127911121142249053 17126772120884247656 18126441120841247282 19127615123792251407 15–196393946102941249688 20127481124020251501 21130978128427259405 22131793129213261006 23135793133924269717 24140685140143280828 20–246667306557271322457 25144157145410289567 26134645136527271172 27133443136049269492 28130488132804263292 29128474131264259738 25–296712076820541353261 MaleFemalePersons 30132867136256269123 31130269133606263875 32139492142313281805 33142209145143287352 34142150145761287911 30–346869877030791390066 35145277148560293837 36141012146205287217 37137969141896279865 38137076141660278736 39134507137849272356 35–396958417161701412011 40137481140360277841 41127118130373257491 42131417134898266315 43130649132966263615 44124503125515250018 40–446511686641121315280 45127342128336255678 46124358125646250004 47120734121165241899 48125216122796248012 49132224127795260019 45–496298746257381255612 50111979109136221115 5110151998391200082 5210248999391201880 539209589216181311 549028887136177424 50–54498370483442981812 558662483913170537 568389280821164713 578022578436158661 587796777312155279 597617974680150859 55–59404887395162800049 MaleFemalePersons 607332974559147888 616614366340132483 626786868018135886 636656068291134851 646628866370132658 60–64340188343578683766 656894772040140987 666598268925134907 676550568270133775 686373768776132513 696163966933128572 65–69325810344944670754 706062168794129415 715506062832117892 725412164498118619 735071762169112886 744818860441108629 70–74268707318734587441 754552358811104334 76405645390894472 77315584329874856 78297024288472586 79281224134569467 75–79175469240246415715 80–84103256174476277732 85–894371694151137867 90–94122863561747903 95–982331840810739 99yearsandover58721572744 Overseasvisitor6579873796139594 Total8849224904319917892423 MQ Maths A Yr 11 - 11 Page 478 Thursday, July 5, 2001 11:03 AM
29. 29. C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 479 Time series and trend lines Graphics calculators or spreadsheets can be used for the following where appropriate. 1 Identify whether the following trends are likely to be secular, seasonal, cyclic or random: a the amount of rainfall, per month, in north Queensland b the number of soldiers in the United States army, measured annually c the number of people living in Australia, measured annually d the share price of BHP, measured monthly e the number of seats held by the Liberal Party in Federal Parliament. 2 Fit a trend line to the data in the graph at right. Consider this data set as a time series of age groupings 0–4, 5–9 etc. (Note that the totals for these groupings occur within the data.) 1 Graph the trend of male numbers over time. 2 On the same graph, plot the female numbers over time. 3 Examine the differences in these two trends. a Compare the numbers of males and females in Australia in each age category. b What has happened to the birth rate over time? c Compare the age of death of males and females. d Which age group represents the greatest proportion of the population? 4 Write a report of the age composition of Australians. Present your ﬁndings, backed by reference to tables and graphs. remember 1. Time series are a set of measurements taken over (usually) equally spaced time intervals, such as hourly, daily, weekly, monthly or annually. 2. There are 4 basic types of trend: (a) secular: increasing or decreasing steadily (b) seasonal: varying from season to season (c) cyclic: similar to seasonal but not tied to a calendar cycle (d) random: varying from external causes happening at random. remember 11C Days Temp.(°C) 10 15 20 25 30 35 40 0 1 2 3 4 5 6 7 8 9 10 t MQ Maths A Yr 11 - 11 Page 479 Thursday, July 5, 2001 11:03 AM
30. 30. 480 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 3 Fit a trend line to the following data. What type of trend is best reﬂected by these data? 4 The monthly share prices of a recently privatised telephone company were recorded as follows. Graph the data (let 1 = Jan. 2001, 2 = Feb. 2001 . . . and so on) and ﬁt a trend line to the data. Use this line to predict the share price in January 2002. Comment on the feas- ibility of the predicted share price. 5 Plot the following monthly sales data for umbrellas. Fit a trend line. Discuss the type of trend best reﬂected by the data and the limitations of your trend line. 6 Consider the data in the ﬁgure, which represent the price of oranges over a 19-week period. a Fit a straight trend line to the data. b Predict the price in week 25. 7 The following table represents the quarterly sales ﬁgures (in thousands) of a popular software product. Plot the data and ﬁt a trend line using the ‘equal number of points’ method. Discuss the type of trend best reﬂected by these data. 8 The number of employees at the Comnatpac Bank was recorded over a 10-month period. Plot and ﬁt a trend line to the data. What would you say about the trend? t 1 2 3 4 5 6 7 8 9 10 11 12 y 6 9 13 8 9 14 15 17 14 11 15 19 Date Jan. 01 Feb. 01 Mar. 01 Apr. 01 May 01 Jun. 01 Jul. 01 Aug. 01 Price (\$) 2.50 2.70 3.00 3.20 3.60 3.70 3.90 4.20 Month Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec. Sales 5 10 15 40 70 95 100 90 60 35 20 10 Quarter Q1-99 Q2-99 Q3-99 Q4-99 Q1-00 Q2-00 Q3-00 Q4-00 Q1-01 Q2-01 Q3-01 Q4-01 Sales 120 135 150 145 140 120 100 110 120 140 190 220 Month Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec. Employees 6100 5700 5400 5200 4800 4400 4200 4000 3700 3300 WORKED Example 8 Weeks Price(cents) 20 40 60 80 100 0 5 10 15 20 25 t Work SHEET 11.2 MQ Maths A Yr 11 - 11 Page 480 Thursday, July 5, 2001 11:03 AM
31. 31. C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 481 Weather report for snowﬁelds The Bureau of Meteorology publishes statistical data on their web site (http://www.bom.gov.au). A visit to this site reveals climate averages for many places throughout Australia. The following data set is the result of measurements collected at Mount Buffalo Chalet over a number of years. inv estigat ioninv estigat ion CLIMATEAVERAGES—longtermmeanvaluesofweatherdata 083073MOUNTBUFFALOCHALETCommenced:1910Lastrecord:2000 Latitude:36.72SLongitude:146.82EElevation:1372.0mState:VIC JANFEBMARAPRMAYJUNJULAUGSEPOCTNOVDECANNNo.%age Yrscomp MeanDailyMaxTemp(degC)19.518.916.411.67.94.93.74.68.010.814.217.511.643.368 Meanno.Days,Max>=40.0degC0.00.00.00.00.00.00.00.00.00.00.00.00.02.892 Meanno.Days,Max>=35.0degC0.00.00.00.00.00.00.00.00.00.00.00.00.02.892 Meanno.Days,Max>=30.0degC0.00.00.00.00.00.00.00.00.00.00.00.00.02.892 HighestMaxTemp(degC)29.023.921.724.418.912.214.012.814.818.521.529.529.52.894 MeanDailyMinTemp(degC)10.811.09.05.32.60.2–0.7–0.31.83.86.49.25.043.669 Meanno.Days,Min=<2.0degC0.70.00.74.317.322.327.326.015.511.34.71.7131.82.997 Meanno.Days,Min=<0.0degC0.00.00.31.37.013.720.019.09.55.71.30.378.22.997 LowestMinTemp(degC)1.73.30.0–0.6–1.9–4.5–5.3–4.4–4.4–3.6–1.1–0.6–5.32.986 Mean9amAirTemp(degC)14.915.012.98.75.62.71.72.44.97.410.313.28.440.889 Mean9amWet-bulbTemp(degC)10.611.29.76.43.81.40.61.23.05.07.29.65.940.688 Mean9amRelativeHumidity(%)5660647073788178716863616840.588 Mean9amWindSpeed(km/hr)6.96.48.17.79.08.410.19.39.87.97.27.58.210.165 Mean3pmWindSpeed(km/hr)4.93.44.35.36.67.98.98.17.96.86.36.36.52.175 MeanRainfall(mm)88.483.2100.8131.0197.3212.7230.8227.0196.8191.4130.1112.61902.085.596 Median(Decile5)Rainfall(mm)64.257.981.8108.3168.2181.4214.3225.1190.5159.5112.794.81884.675 Decile9Rainfall(mm)189.6184.2256.9278.5375.9392.2386.2356.2332.1399.3253.8256.82493.875 Decile1Rainfall(mm)12.19.215.122.456.656.385.186.465.545.149.727.61209.975 Meanno.ofRaindays6.66.57.49.012.413.414.615.513.512.810.38.9131.084.395 HighestMonthlyRainfall(mm)364.7412.9359.2764.1614.0607.4717.4557.7496.0525.3323.1374.685.596 LowestMonthlyRainfall(mm)1.30.04.80.00.032.02.321.431.74.010.45.185.596 HighestRecordedDailyRain(mm)193.0134.4174.0189.7173.0216.7157.0145.4195.0212.0108.6109.7216.785.196 Meanno.ofClearDays0.00.00.00.00.00.00.00.00.00.00.00.00.03.0100 Meanno.ofCloudyDays0.00.00.00.00.00.00.00.00.00.00.00.00.03.0100 MQ Maths A Yr 11 - 11 Page 481 Thursday, July 5, 2001 11:03 AM
32. 32. 482 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d The following article explains the ideal conditions for heavy snowfalls, and graphically displays the recorded snow depth in the Snowy Mountains over the years 1963 to 1995. 1 Carefully examine all the information presented. 2 Visit the web site for more information, if necessary. 3 Your school is planning a visit for your year level to the snowﬁelds some time during the year. Analyse the data presented (together with any other you have collected) and write a report to the organisers identifying: a the general weather conditions in the snowﬁelds throughout the year (refer to as many factors as possible) b the best time of the year for the excursion (in your opinion) c the type of weather you would hope for, prior to your visit, to produce good snowfalls in the area. Supply tables and graphs to support your recommendations. Abundant snow seasons The prime conditions for heavy snowfalls in theAustralianAlps are persistent strong westerlies through the winter, which produce abundant precipitation and are generally accompanied by relatively low temperatures. There is no particular relationship with El Nino or La Nina, though the tendency for less precipitation in El Nino years usually leads to a poor season. The favoured conditions are generally different to those which produce snow at low levels, for which a strong cold outbreak (with generally south to southwesterly winds) is required. On the Snowy Mountains the greatest snow depth in the 46 years from 1954 to 1999 was just over 3.5 metres, achieved in both 1961 and 1981. The winter of 1981 was also exceptionally wet in SouthAustralia, Victoria and southern New South Wales: a severe storm struckAdelaide on 1 June, and this heralded three months of almost uninterrupted inﬂuence by depressions in the westerlies, which brought copious rain and snow. At Perisher Valley the snow depth rose steadily to 3.6 metres in early September before beginning to decline. Snow didn’t disappear completely until mid-November. More recently, 1992 was a very long season with deep snow. At Spencer Creek and Perisher Valley in NSW the snow depth reached the high level of three metres at regular four-year intervals: 1956, 1960, 1964 and 1968. These were also years of heavy snow cover on the Victorian Alps. In each case weather patterns were characterised by strong, persistent westerlies and abundant rain at lower levels. In each case the snow cover persisted until late Novem- ber, and in 1956, into De- cember. In 1964 a particularly stormy third week of July dumped pro- digious quantities of snow, stranding people at Falls Creek and Mt Hotham. In Australia, the annual spring melting of the mountain snow is not in it- self sufﬁcient to cause ﬂooding. However, in years when the snow depth is great, a combination of heavy rain and mild to warm conditions can aug- ment spring ﬂoods. Winter snow depth (cm) at Spencer Creek in the Snowy Mountains, 1964 through 1995 (data courtesy of Snowy Mountains Hydro-electric Authority MQ Maths A Yr 11 - 11 Page 482 Thursday, July 5, 2001 11:03 AM
33. 33. C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 483 Scatterplots • When looking for a relationship between two variables, data can be represented on a scatterplot. • One variable (the independent variable) is on the x-axis and the other variable (the dependent variable) is on the y-axis. • Points are plotted by the coordinates formed by each piece of data. • If the dependent variable consistently increases or decreases as the independent variable increases, a relationship exists. • If all points on the scatterplot form a straight line, the relationship is said to be linear. • The pattern of the scatterplot gives an indication of the strength of the relationship or level of association between the variables. • A strong relationship between variables does not imply that one variable causes the other to occur. Regression lines • The regression line is the line of best ﬁt on a scatterplot. • By measuring the gradient and the y-intercept on the regression line, we can use the formula y = mx + c to ﬁnd the equation. A graphics calculator also can be used to determine the equation of the regression line. • When the equation of the regression line has been found, it can then be used to make predictions about the data. Predictions • Predictions can be made algebraically or graphically. • Reliable predictions are produced by interpolation using a large number of points that lie reasonably close to the regression line. • Be careful about extrapolating data. Time series • A time series is a set of measurements taken over (usually) equally spaced time intervals, such as hourly, daily, weekly, monthly or annually. Trend lines • There are 4 basic types of trend: 1. secular: increasing or decreasing steadily 2. seasonal: varying from season to season 3. cyclic: similar to seasonal but not tied to a calendar cycle 4. random: variations caused by external triggers happening at random. Fitting trend lines • The trend line is a straight line that can be used to represent the entire time series. Trend lines can be used for predicting the future values of the time series. The line can be found by eye using the ‘equal number of points’ technique. summary MQ Maths A Yr 11 - 11 Page 483 Thursday, July 5, 2001 11:03 AM
34. 34. 484 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Answers to some of these questions may vary due to different positions of the regression line. The use of a graphics calculator is recommended, where appropriate. 1 The table below shows the maximum and minimum temperature on 10 days chosen at random throughout the year. Display this information on a scatterplot. 2 The table below shows the number of sick days taken by ten employees and relates this to the number of children that they have. a Show this information on a scatterplot. b Does a relationship appear to exist between the number of sick days taken and the number of children they have? If so, is the relationship linear? 3 The table below shows the number of cars and number of televisions in each household. a Show this information on a scatterplot. b Does a relationship appear to exist between the number of televisions in each household and the number of cars they have? If so, is the relationship linear? 4 A researcher administers different amounts of fertiliser to a number of trial plots of potato crop. She then measures the total mass of potatoes harvested from each plot. When drawing the scatterplot, the researcher should graph: A mass of harvest on the x-axis because it is the independent variable, and amount of ferti- liser on the y-axis because it is the dependent variable B mass of harvest on the y-axis because it is the independent variable, and amount of ferti- liser on the x-axis because it is the dependent variable Maximum temperature (°C) 25 36 21 40 24 26 30 18 20 25 Minimum temperature (°C) 12 21 11 23 12 15 19 10 8 13 No. of children 1 0 3 2 2 4 6 0 1 2 No. of sick days 5 3 10 8 4 12 12 0 1 5 No. of cars 1 1 2 2 2 3 1 0 1 2 No. of televisions 2 1 1 2 0 1 4 3 1 1 11A CHAPTER review 11A 11A 11A mmultiple choiceultiple choice MQ Maths A Yr 11 - 11 Page 484 Thursday, July 5, 2001 11:03 AM
35. 35. C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 485 C mass of harvest on the x-axis because it is the dependent variable, and amount of ferti- liser on the y-axis because it is the independent variable D mass of harvest on the y-axis because it is the dependent variable, and amount of ferti- liser on the x-axis because it is the independent variable. 5 Which of the following graphs best depicts a strong negative relationship between the two variables? A B C D 6 What type of relationship is shown by the graph on the right? A Strong positive relationship B Moderate positive relationship C Moderate negative relationship D Strong negative relationship 7 The table below shows the relationship between two variables, x and y. a Prepare a scatterplot of the data. b On the scatterplot, draw the regression line. c By measuring the gradient and the y-intercept of the regression line, ﬁnd its equation. 8 A survey is conducted comparing household income, I, with house value, V. A scatterplot is drawn and the regression line is found to have the equation V = 3.7I + 50 000. Use the equation to ﬁnd: a the likely value of a house owned by a family with an income of \$52 000 b the likely income (to the nearest \$1000) of a family living in a house valued at \$320 000. 9 An entomologist conducted an experiment in which small amounts of insecticide were introduced to a container of 100 blowﬂies. The results are detailed below. a Display the above information on a scatterplot and draw the line of regression. b Find the equation of the regression line. c Use the equation to predict the number of blowﬂies that would remain after two hours if 4.25 micrograms of insecticide was introduced. d Estimate the amount of insecticide needed to remove all blowﬂies. x 2 4 18 7 9 12 2 7 11 10 16 y 103 75 20 66 70 50 95 40 27 42 30 Insecticide (I) (micrograms) 1 2 3 4 5 6 7 8 9 10 No. remaining after 2 h (F) 99 92 81 74 62 68 52 45 38 24 11Ammultiple choiceultiple choice y x y x y x y x 11Ammultiple choiceultiple choice y x 11B 11B 11B MQ Maths A Yr 11 - 11 Page 485 Thursday, July 5, 2001 11:03 AM
36. 36. 486 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 10 The price of oranges over a 16-month period is recorded in the ﬁgure. The trend can be described as: A Cyclic B Seasonal C Random D Secular E There is no trend. 11 The number of uniforms sold in a school uniform shop is reported in the table. Fit a trend line to these data. What type of trend is best reﬂected by these data? Can you explain these trends? Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec. 118 92 53 20 47 102 90 42 35 26 12 58 11C mmultiple choiceultiple choice testtest CHAPTER yyourselfourself testyyourselfourself 11 t 50 40 30 20 10 0 0 2 4 6 8 Months 10 12 14 Priceoforanges 16 11C MQ Maths A Yr 11 - 11 Page 486 Thursday, July 5, 2001 11:03 AM