Quantitative Bivariate Data

Transcription

1 Statistics 211 (L02) - Linear Regression Quantitative Bivariate Data Consider two quantitative variables, defined in the following way: X i - the observed value of Variable X from subject i, i = 1, 2,, n. Y i - the observed value of Variable Y from subject i, i = 1, 2,, n. Some examples of data that are bivariate in nature: 1. a Statistics 211 student s midterm exam mark and final exam mark. 2. the number of years of post-secondary education an individual has and their annual income. 3. A year s inflation and interest rate. 4. The average price of oil in a month and the average price of the Canadian dollar (relative to the U.S. dollar). When an experimental study or random sampling method produces data on two different quantitative variables, there are three research questions that are posed. 1. Is there a relationship between the two variables? Is there a relationship between a Statistics 211 student s midterm mark and their final exam mark? Is there a relationship between how much post-secondary education one has and their income? Is there a relationship between your parent s IQ and your own? If there is a relationship, what is the direction of the relationship? Is the relationship positive? negative (or inverse)? Is the relationship seem to be linear? non-linear? 2. If a relationship exists between X and Y, how strong is this relationship? Is the relationship between a student s midterm exam mark and their final exam mark strong or weak? What about the relationship, should one exist, between your parent s IQ and your own? Is such a relationship subtle, or strong? 3. If the relationship between X and Y is strong, can the existing relationship be used to predict what will happen in the future? That is, if after collecting some data on past student s 211 midterm exam mark (X) and their corresponding final exam mark (Y ), can we project this existing relationship to predict your final exam mark based solely on how it is related with your midterm exam mark? That is, can we create a mathematical function - y = f(x) - which will predict one s final exam mark once the midterm exam mark has been applied to this function? 1. Relationship between X and Y. In order to determine if a relationship exists between the two underlying variables once the data has been collected, one should graph the data prior to any consideration of the calculation of various statistics. A visual picture of the data is constructed as a primary diagnostic tool to see if a relationship does exist between the two variables under study. The plot used is called a scatter-plot/scatter-diagram. A scatterplot simply graphs the data, where each data-point is in the form of an ordered pair: (X 1 = x 1, Y 1 = y 1, X 2 = x 2, Y 2 = y 2,,X n = x n, Y n = y n ) Scatter-plots/scatter-diagrams can have one of the three forms:

2 Consider the example below: Example 1: The following data was produced by a controlled experiment where male volunteer subjects, over the age of 21, were to consume a certain type of beer over a three-hour period. At the end of the three-hours, each subject was to provide a blood sample, from which their blood alcohol content (BAL) was measured. The raw data is given below. X - No. of Beers Consumed Y - BAL A scatterplot of the data is given below. From this one can attempt to make generalizations: is there a relationship, and if there is, what is the nature of the relationship? Beer Consumption to BAL Content BAL Content No. of Beers Consumed

3 2. Strength of the Relationship in X and Y. The scatterplot will give an indication as to how strong (or weak) the relationship is between variable X and variable Y. This can be determined by the compactness of the scattering. The more compact the scattering of the data around some imaginary linear line, the stronger the two variables are. Using the scatterplot as a diagnostic tool to identify the strength of the relationship can be somewhat subjective. To remove the subjectivity from the analysis, one can compute the correlation coefficient between X and Y. The correlation coefficient (r) measures the degree of linear association between X and Y. The correlation coefficient is calculated using the following formula below: ni=1 X i Y i (n Ave X Ave Y ) r = 1 r 1 (n 1)S X S Y where Ave X, SD X are the average and standard deviation of the data on variable X; Ave Y, SD Y are the average and standard deviation on the data on variable Y. In addition, n X i Y i i=1 = X 1 Y 1 + X 2 Y X n Y n Question 2. Calculate the correlation coefficient. Does this support the findings in from the scatterplot? Interpret the meaning of this statistic in the context of the data. Solution: The data is summarized: n = 10, Ave X = 5 S X = 2.36 Ave Y = S Y = i=1 X i Y i = The correlation coefficient is given by: ni=1 X i Y i (n Ave X Ave Y ) r = (n 1)S X S Y ( ) = (10 1)(2.36)(0.051) 0.96 = r = The value of the correlation coefficient is r = This supports the commentary about the relationship obtained from an inspection of the scatterplot: the relationship between one s blood alcohol content (Y ) and the number of beers consumed (X) is positive, as indicated by the non-negativity on the value of ; the numerical value of r is closer to 1 than zero, so perhaps one can say that the relationship between the amount of alcoholic beverage (beer in this case) and one s blood alcohol content is somewhat strong.

4 3. Simple Linear Regression Once one has determined that the linear relationship between the X variable and the Y variable is one that is significant - or somewhat strong - regardless of the direction of the relationship, an attempt can be made to construct a statistical crystal ball - a statisitcal/mathematical model which attempts to predict what will happen in the future based on what has happened in the past, the latter being what is happening in the bivariate data. Simple linear regression is a statistical technique which aims to construct a statistical crystal ball. Specifically, simple linear regression is a statistical method that attempts to build a stastistical prediction model, one that will attempt to predict the value of one variable based entirely on its historical relationship with another variable. The value of the variable that is being predicted is called the response variable, denoted by Y. The variable being used as the basis for the prediction is called the predictor variable, denoted by X. In some texts, the response variable is referred to as the dependent variable Y ; the predictor variable is commonly referred to as the independent variable X. Simple linear regression attempts to fit the data, or model the data, collected on both the predictor variable X and the response variable Y in the following form: PredictedY = a + (b V alueofx) where X - value of the predictor variable from subject i - or element i - randomly selected from the target population. PredictedY - value of the response variable from subject i - or element i - again, randomly selected from the target population. a - Y -intercept term. b - the slope term. The slope term represents the average change to the response variable Y that corresponds to a unit change in the predictor variable X. The values of a and b are calcuated through the following formulae. ( ) SY b = r S X a = Ave Y (b Ave X )

5 Question 3. Find the regression line which expresses the home furnishing store s monthly sales as a linear function of its monthly advertising expenditure. Solution: The regression line is: ( ) SY b = r S X = (0.8862) ( ) b = a = Ave Y (b Ave X ) = ( ) a = The regression line which attempts to predict one s blood alcohol content - Ŷ - for a given value number of beers consumed over a three-hour period - X is: PredictedY = a + (b V alueofx) = (0.019 V alueofx)

6 Question 4 - An Application of the Model. Using the model estimated in Question 3, predict the blood alcohol content of a male consuming 6 beers in a three-hour period. Interpret the meaning of this predicted value. Question 5: Interpret the meaning of the Y -intercept term (a) and the slope term (b) - in the context of the question.

7 Example 2: Is there a relationship, and if so, to what extent, between a person s age and a maximum safe heart rate? This may be of interest to those people who use heart-rate monitors when exercising in order to achieve target heart rates for optimal training. The maximum safe heart-rate is thought to be around 220 beats per minute minus one s age in years. Bivariate data about age and maximum heart rate was collected on 27 individuals The data is summarized below. Person Age - X Maximum Heart Rate - Y : : : The data is summarized: Ave X = 40 S X = 13.2 Ave Y = S Y = i=1 X i Y i = Age versus Maximum Heart Rate mhr age

8 From the data, answer the following questions: (a) From the scatter plot given, does there appear to be a relationship between the response variable (maximum heart rate) and the predictor variable (age)? Provide a statement regarding the nature of this relationship. (b) Find the value of the correlation coefficient. (c) Find the regression line which expresses a person s maximum heart rate as a linear function of the person s age. (d) Predict the maximum heart rate of a person who is (i) 37 years old. (ii) 75 years old. (e) As the a person s age increases by a year, how does this their maximum heart rate?

9 Example 3: What is the relationship between a student s midterm and final exam score in an introductory statistics course, such as Statistics 211. To investigate this, a statistics professor randomly selected 10 students who completed an introductory statistics course similar to Statistics 211 in the Fall Term of For each student chosen, their midterm exam mark and final exam mark (in percentage of marks earned) was observed. The raw data is provided below. Midterm Exam Score (%) Final Exam Score (%) Ave SD i=1 Midterm Final = Final Exam Mark Midterm Exam Mark (a) From the scatter plot, what can you say about the (i) direction of the relationship? (ii) the strength of the relationship? (iii) does the relationship seem to be linear?

10 (b) Find the value of the correlation coefficient. (c) Assuming one is trying to model his/her final exam mark as a linear function of his/her midterm exam mark. Find the regression line and draw this into the scatter-plot given. Interpret the meaning of the slope term in the context of this data. (d) From the regression line in (b), estimate your final exam mark.