IAS 3.9 Bivariate Data

Transcription

1 Year 13 Mathematics IAS 3.9 Bivariate Data Robert Lakeland & Carl Nugent Contents Achievement Standard Bivariate Data Scatter Plots Relationships Correlation Coefficient Regression Analsis Outliers Residuals Residual Plots Causation Non-Linear Regression Bivariate Investigation Practice Internal Assessment Answers Innovative Publisher of Mathematics Tets

2 2 IAS 3.9 Bivariate Data NCEA 3 Internal Achievement Standard 3.9 Bivariate Data This achievement standard involves students investigating bivariate measurement data. Achievement Achievement with Merit Achievement with Ecellence Investigate bivariate measurement data. Investigate bivariate measurement data, with justification Investigate bivariate measurement data, with statistical insight. This achievement standard is derived from Level 8 of The New Zealand Curriculum and is related to the achievement objectives Carr out investigations of phenomena, using the statistical enquir ccle using eisting data sets finding, using, and assessing appropriate models (including linear regression for bivariate data), seeking eplanations, and making predictions using informed contetual knowledge and statistical inference communicating findings and evaluating all stages of the ccle in the Statistics strand of the Mathematics and Statistics Learning Area. Investigate bivariate measurement data involves showing evidence of using each component of the statistical enquir ccle. Investigate bivariate measurement data, with justification involves linking components of the statistical enquir ccle to the contet, and referring to evidence such as statistics, data values, trends, or features of visual displas in support of statements made. Investigate bivariate measurement data, with statistical insight involves integrating statistical and contetual knowledge throughout the investigation process, and ma include reflecting about the process; considering other relevant variables; evaluating the adequac of an models, or showing a deeper understanding of the models. Using the statistical enquir ccle to investigate bivariate measurement data involves: posing an appropriate relationship question using a given multivariate data set selecting and using appropriate displa(s) identifing features in the data finding an appropriate model describing the nature and strength of the relationship and relating this to the contet using the model to make a prediction communicating findings in a conclusion. Measurement data can either be discrete or continuous in nature. In regression analsis the -variable, or response variable, must be a continuous variable. The variable or eplanator variable can be either a discrete or continuous variable. The relationship ma be non-linear. Use and interpretation of R 2 is not epected at this level. IAS 3.9 Year 13 Mathematics and Statistics Published b N New Zealand Robert Lakeland & Carl Nugent

3 IAS 3.9 Bivariate Data 3 Bivariate Data Introduction In statistics we are often interested in identifing relationships involving more than one variable. In this booklet we stud Bivariate Data which deals with the stud of two quantitative variables (pairs of variables) and identifing relationships between them. We aim to identif the relationship between the variables, graphicall with the use of a scatter plot and quantitativel b looking at correlation (the degree to which two or more quantities are linearl associated). In addition we use regression to enable us to make quantitative predictions (interpolation) of one variable from the other. Much of the emphasis in this Achievement Standard is on the visual interpretation of scatter plots as well as linking statistical knowledge to the contet of the question and using appropriate reasoning and reflection. To this end we are suggesting that students have access to inzight a simple data analsis sstem which encourages eploring what data is saing without the distractions of driving comple software (inzight website). INZight can be downloaded from auckland.ac.nz/~wild/inzight/dlw.html and can be installed on either a Mac or Windows computer. All datasets used in this booklet can be downloaded from our website at co.nz under the Downloads link and imported directl into inzight or Ecel. IAS 3.9 Year 13 Mathematics and Statistics Published b N New Zealand Robert Lakeland & Carl Nugent

4 IAS 3.9 Bivariate Data 7 Relationships Inspecting a Scatter Plot When we are given a scatter plot and asked to comment on the relationship between the variables our first approach is a visual one. The first step is to make sure we understand the precise meaning of the variables being used as well as the units applied to the variables. Sometimes it ma be necessar to research the meaning of the variables so that ou have a better understanding prior to investigating a possible relationship. Net look at the degree of scatter of the points. Are the points clumped together or spread out? Are the points in a number of distinct clusters or is there just a single cluster of points with one or two points at the etremes of the scatter plot? Are there an unusual observations that go against the general trend of the scatter plot? Are these points still realistic, i.e. can the be eplained? What is the variation of scatter in the scatter plot? Are the points close to each other or is there significant gaps between them? Do the points follow a discernible pattern or trend, e.g. following a straight line? Correlation When we draw a scatter plot the shape of the plot is usuall defined as linear or non-linear (curved). A correlation eists between two variables when one of them is related or can be influenced b the other in some wa. The strength of a relationship can be identified visuall, on a scatter plot, b how tightl or spread out the plotted points are and the direction of the relationship can be identified b the general slope of the pattern of the data. Stud the scatter plots below and note the visual description of shape, strength and direction of the relationship between and. linear, strong positive relationship linear, moderate positive relationship linear, weak positive relationship non-linear, strong positive relationship linear, strong negative relationship linear, moderate negative relationship linear, weak negative relationship non-linear, moderate negative relationship An outlier is one or more points that do not follow the trend. no relationship strong relationship with outliers When determining whether a relationship eists between two variables we also want to be able to measure the strength and direction of a relationship quantitativel, and to do this for linear relationships we can calculate the (linear) correlation coefficient (r). This is eplained in detail on page 12. IAS 3.9 Year 13 Mathematics and Statistics Published b N New Zealand Robert Lakeland & Carl Nugent

5 IAS 3.9 Bivariate Data 29 Eample A heater is turned on in a cold room and the temperature ( C) of the room is recorded ever five minutes. The results are given below. Find the least squares regression line using inzight. Time () C () Import the dataset Eample into inzight and then draw a scatter plot of time versus temperature. To find the regression line click on the Get Summar button. Both the correlation coefficient and the regression line are displaed in the window. Add a linear trend line (regression line) to the scatter plot b clicking on the button Add to Plot and then on the radio button Add trend curves. Click on the checkbo linear followed b Show Changes and Done. The regression line is written in terms of the variable names rather than and, i.e. Degrees.C () = * Time.mins () IAS 3.9 Year 13 Mathematics and Statistics Published b N New Zealand Robert Lakeland & Carl Nugent

6 IAS 3.9 Bivariate Data 37 Achievement Answer the following questions. 48. Download the dataset called Question 48.csv from under the Downloads link on our website ( Import the file into inzight. a) Using inzight produce a scatter plot of d) Would our chosen point in part b) be versus. better identified as an influential Visuall describe our scatter plot and point rather than an outlier or both? identif an features that stand out, e.g. Justif our answer. strength (degree of scatter), groupings (clusters), unusual observations, and the variation of scatter etc. Comment on the linear correlation coefficient for our scatter plot and find the least squares regression line. Predict the value of when = 5.0. e) Import the file Question 48e.csv which is the same dataset as above with some additional points included. Draw a scatter plot of the data. Is there an point that ou could consider a possible outlier? If so identif the point. f) What effect does our chosen point in e) have on the linear correlation coefficient and regression line? b) Import the file Question 48b.csv which is the same dataset as above with some additional points included. Draw a scatter plot of the data. Is there an point that ou could consider a possible outlier? If so identif the point. g) Would our chosen point in part e) be better identified as an influential point or an outlier or both or neither? Justif our answer. c) What effect does our chosen point in b) have on the linear correlation coefficient and regression line? IAS 3.9 Year 13 Mathematics and Statistics Published b N New Zealand Robert Lakeland & Carl Nugent

7 42 IAS 3.9 Bivariate Data Eample Open the dataset Question 16 in Ecel, which comprises the mean lifespan (in ears), metabolic rate (cm 3 /g/h), gestation period (das) and brain weight (g) of 27 mammals. B either using the Analsis Toolpak which can be installed using Add-Ins for Ecel 2010 or StatPlus mac LE which is a separate program that can be downloaded from en/products/statplusmacle/ for Mac, produce a plot of the residuals for gestation period versus lifespan of the 27 mammals and comment on what the plot tells us about using a linear model for the association between these two variables. Instructions for StatPlus mac LE are given below. See the following page for information on the instructions for using the Analsis ToolPak. cont... the independent variable of our data pair, that is, Gestation Period. When ou return to StatPlus [Workbook1]Sheet1!$B$1:$B$28 should appear. Now click the checkboes Plot residuals vs fitted and Plot line fit followed b OK. Generated are the residuals, a plot of the residuals and a scatter plot in an Ecel spreadsheet. The spreadsheet includes a large amount of information including the least squares regression line, (Lifespan = * Gestation period), the linear correlation coefficient (r = ) as well as a table of residuals (see below and the net page). If ou are using StatPlus mac LE on a Mac begin b booting the application StatPlus. When StatPlus boots it will also boot Ecel. Open a new worksheet in Ecel. Regression equation Using Ecel open the dataset Question 16.csv. Cop the two columns C and D, i.e. Lifespan and Gestation Period from the Question 16.csv spreadsheet into columns A and B of our blank Correlation coefficient worksheet. Now close the Question 16.csv spreadsheet. In the StatPlus application choose the menu option Statistics, then Regression, then Multiple Linear Regression. The following window appears. Click on the button to the right of the Dependent variable field. Remember the dependent variable is the response or variable. The Ecel window will be activated so ou can select the cells representing the dependent variable of our data pair, i.e. Lifespan. When ou return to StatPlus [Workbook1] Sheet1!$S$1:$A$28 should appear. Click on the button to the right of the Independent variable field. The independent variable is the eplanator or variable. The Ecel window will be activated so ou can select the cells representing The predicted values are those generated b substituting the appropriate gestation period into the least squares regression line. For observation 1 the gestation period is 14. Substituting this into Lifespan = * Gestation period gives us a predicted value, i.e. Lifespan () of The observed point for a gestation period of 14 das is a lifespan of 50 ears so the difference between 50 and , i.e is our residual. IAS 3.9 Year 13 Mathematics and Statistics Published b N New Zealand Robert Lakeland & Carl Nugent

8 IAS 3.9 Bivariate Data 49 Non-Linear Regression cont... An alternative to fitting a logarithmic model could be to investigate using a negative parabola, especiall if ou are using inzight where ou have limited options (see below). inzight at present onl allows quadratic and cubic models for non-linear functions. If ou require eponential, logarithmic or power models the authors suggest ou use Ecel. You can start from the scatter plots produced b the Analsis Toolpak or the StatPlus mac LE. The equation for the quadratic trend is Strength = Da 19.35Da kpa which gives a strength of kPa for 2 das curing and a strength of kpa for 10 das curing. Comparing the three trends (linear, logarithmic and quadratic) the linear in the long term (> 25 das) will become less reliable as it will continue to increase over time. The same will be true of the quadratic because it will begin to decrease once the maimum point of the quadratic is reached. In the long term the logarithmic model is likel to be the best fit as it approaches an asmptote. Another option is to adopt a piecewise function which is a combination of the different functions over different subsets of. IAS 3.9 Year 13 Mathematics and Statistics Published b N New Zealand Robert Lakeland & Carl Nugent

9 58 IAS 3.9 Bivariate Data Page 10 Q16. cont... g) There appears to be a moderate positive linear association between the two variables. Mammals with a longer gestation period have a longer lifespan. Two unusual observations are the Echnida which has a gestation period of onl 14 das but a long lifespan of 50 ears and man with a gestation period of 267 das and a ver long lifespan of 86 ears. Most points fall within the gestation range of 0 to <200 das and lifespan of 0 to 20 ears. h) Metabolic rate (eplanator) and gestation period (response). i) Investigating how metabolic rate predicts gestation period not the other wa round n) Gestation period is a r = reasonable predictor of 19. r = lifespan although man and the E. American Page 17 (Answers ma var) mole go against the trend. A linear model could be Constant difference. suitable for these two 21. Close to 1. The older a car the variables. less its value. Page 11 Q16. cont... j) The have a low gestation period. k) Not an obvious linear one although greater metabolic rate does result in a lower gestation period. More likel a negative non-linear relationship. Page 11 Q16. cont... l) There appears to be weak negative non-linear association between the two variables. Mammals with a greater metabolic rate have a shorter gestation period. Most points are clustered in the metabolic range 0 to < 1.0 and gestation period of 0 to < 300 das. Etreme points are that of the Asian elephant, little brown bat and E. American mole although the would still fit a non-linear model. m) Metabolic rate versus 22. Somewhat negative. Less well lifespan would be educated people smoke. better suited to a non-linear model. 23. Close to 1. People with big feet Again the E. American are generall taller. mole goes against 24. Somewhat negative. More the trend. A suitable cigarettes smoked the lower the non-linear model would level of fitness. probabl be a good predictor of lifespan. With brain weight versus lifespan a linear model is affected b the Asian elephant and E. American mole. Page 11 Q16 n) cont... Page Overall gestation looks like the better predictor from a linear perspective but metabolic rate from a non-linear perspective. Brain weight is not as good as the other two r = The more ou spend the less ou save. 26. Close to 1. Tall men often choose tall wives. 27. Somewhat positive. More land area greater price. IAS 3.9 Year 13 Mathematics and Statistics Published b N New Zealand Robert Lakeland & Carl Nugent