In last class, we learned statistical inference for population mean. Meaning. The population mean. The sample mean

Transcription

1 RECALL: In last class, we learned statistical inference for population mean. Problem. Notation Populati on Notation X σ Meaning The population mean The sample mean The population standard deviation s The sample standard deviation n The sample size

2 RECALL: Point estimation. (sample mean X ) Distribution of X Confidence Interval One-sample z-interval (population SD is known) One-sample t-interval (only sample SD is known) X ± t Remark: 1. T-interval needs normal assumption. * 2. t n 1, which is related to n-1 and C%, can be obtained from t-table. * n 1 s n

3 RECALL: Hypothesis Testing about μ Null Hypothesis H 0 vs. Alternative Hypothesis H A H 0 : Z-Test (population SD is known) Test statistic: P-value: vs. Alternative Hypothesis H A H A : H A : H A : (two-sided) (one-sided) (one-sided) P-value formula H A : (two-sided) P-value=2P(Z> z ) H A : (one-sided) P-value=P(Z>z) H A : (one-sided) P-value=P(Z<z)

4 RECALL: Hypothesis Testing about μ Null Hypothesis H 0 vs. Alternative Hypothesis H A H A : (two-sided) H 0 : vs. H A : (one-sided) H A : (one-sided) T-Test (sample SD s is known) Test statistic: X µ 0 t = s n P-value:(df=n-1) Alternative Hypothesis H A P-value formula H A : (two-sided) Two-tail prob. of t H A : (one-sided) One-tail prob. of t H A : (one-sided) One-tail prob. of t

5 RECALL: TI commands (under STAT TESTS): T-interval: use 8: T Interval T-Test: use 2:T-Test Decisions: If p-value< alpha level, reject H 0, and we say the test is statistically significant at this alpha level); If p-value>alpha level, fail to reject H 0, and we say the test is not statistically significant at this alpha level); Errors: Type I error: decide to reject H 0, but actually H 0 is true; Type II error: decide to retain H 0, but actually H 0 is false; P(Type I error)=alpha level.

6 Exploring Relationship Between Variables Chapter 7: Scatterplots, Association, and Correlation Chapter 8: Linear Regression

7 WHERE ARE WE GOING? People might ask the following questions in the real life: 1. Is the price of sneakers related to how long they last? 2. Is smoking related to lung cancer? 3. Do baseball teams that score more runs sell more tickets to their games? Chapter 7 will look at relationships between two quantitative variables X and Y. Scatterplot Correlation

8 TERM 1: SCATTERPLOTS Is the price of sneakers related to how long they last? Following table shows some data collected for sneakers: Years Price($) Price This is an example of scatterplot. x-axis represents variable years and y-axis represents prices.

9 TERM 1: SCATTERPLOT Scatterplots may be the most common and most effective display for paired data. Scatterplots are the best way to start observing the relationship and the ideal way to picture associations between two quantitative variables Price X-axis: Years, Explanatory variable which explains or influences changes in the other variable. Y-axis: Price, Response variable which measures an outcome of a study.

10 TERM 1: SCATTERPLOTS How do we describe the scatterplot? Or, What information about the relationship of the two variables can we get by looking at the scatterplot? Please look at the scatterplot of the sneakers example, and think about what can you tell about the relationship of years and price Price We are going to describe the relationship from four different aspects. 1) Direction 2) Form 3) Strength 4) Unusual features

11 TERM 1: SCATTERPLOT Scatterplot Look for direction: What s my design positive, negative or neither? Negative A pattern like this that runs from the upper left to the lower right is said to be negative. Y variable decreases as the X variable increases. Positive A pattern running the other way is called positive. Y variable increases as X variable increases. Y Y X Scatterplot X

12 TERM 1: SCATTERPLOT The example in the text shows a negative association between central pressure and maximum wind speed As the central pressure increases, the maximum wind speed decreases

13 TERM 1: SCATTERPLOTS Look for Form: straight, curved or something exotic, or no pattern? Scatterplot Scatterplot Scatterplot Y Y Y X X X Straight line, linear Curved No pattern In this part, we are more interested in the linear pattern.

14 TERM 1: SCATTERPLOTS Look for strength: how much scatter? Or, how strong the relationship is? Strong: the points appear tightly clustered in a single stream. Scatterplot Scatterplot Scatterplot Y Y Y X X Weak: the X swarm of points seem to form a vague cloud through which we can barely discern any trend or pattern Scatterplot Y X

15 TERM 1: SCATTERPLOTS Look for the Unusual Features: Are there outliers or subgroups? Scatterplot Scatterplot Y Y The point circled is a potential outlier X There are two clusters. X

16 TERM 1: SCATTERPLOT-ROLES FOR VARIABLES It is important to determine which of the two quantitative variables goes on the x-axis and which on the y-axis. Slide 1-16 This determination is made based on the roles played by the variables. When the roles are clear, the explanatory or predictor variable goes on the x-axis, and the response variable goes on the y-axis.

17 TERM 1: SCATTERPLOTS Summary A Scatterplot shows the relationship between two quantitative variables measured on the same individual. The variable that is designated the X variable is called the explanatory variable The variable that is designated the Y variable is called the response variable Always plot the explanatory variable on the horizontal (x) axis Always plot the response variable on the vertical (y) axis In examining scatterplots, look for an overall pattern showing the form, direction and strength of the relationship Look also for outliers or other deviations from this pattern

18 TERM 1: SCATTERPLOT Example: Fast food is often considered unhealthy because much of it is high in fat. Are fat and calories related? Here are the fat and calories contents of several brands of burgers. Analyze the association between fat content and calories. Fat(g) Calories Calorie Fat Comment on the scatterplot: 1) Direction Positive 2) Form Roughly linear 3) Strength Moderately strong 4) Unusual features No.

19 TERM 2: CORRELATION From scatterplots, we can look for the relationship between two quantitative variables and whether the relationship is strong or weak. But how strong is it? Correlation coefficient (or simply correlation) is a quantitative measure of linear relationship (association) between two quantitative variables. Finding the correlation coefficient, denoted by r, by hand: ( x x)( y y) r = ( n 1) s x s y s s Where x and y are standard deviations for X and Y respectively. Remarks: Before you use correlation, you must check several conditions: Quantitative Variables Condition Straight Enough Condition Outlier Condition

20 TERM 2: CORRELATION (Revisit the calories example) Here are the fat and calories contents of several brands of burgers. X: Fat(g) Y: Calories What is the correlation coefficient of x (fat) and y (calories)? Solution: Deviations in x Deviations in y Product 20-35= =-180 (-15)*(-180)= = =-10 (-5)*(-10)= = = 0 0*0= = =-20 1*(-20)= = = 50 5*50= = = 90 5*90= = = 70 9*70=630 Add up the products: (-20) =4060 Correlation r=4060/{(7-1)*7.98*89.81}=0.9442

21 TERM 2: CORRELATION

22 CORRELATION PROPERTIES The sign of a correlation coefficient gives the direction of the linear association. Positive sign Positive linear association Negative sign Negative linear association Correlation is always between -1 and +1. Correlation can be exactly equal to -1 or +1, but these values are unusual in real data because they mean that all the data points fall exactly on a single straight line. A correlation near zero corresponds to a weak linear association. Example: The correlation between fat and calories as indicates a strong positive linear association between them. Slide 1-22

23 y y TERM 2: CORRELATION Cautions about correlation: Quantitative Variables Condition: Correlation applies only to quantitative variables. Straight Enough Condition: Correlation measures the strength only of the linear association r= x -2 r= x Outlier Condition: Outliers can distort the correlation dramatically. y With the outlier: r=0.795 Without the outlier: r= x

24 TERM 2: CORRELATION Correlation Causation Fast food is often considered unhealthy because much of it is high in fat. Are fat and calories related? Based on the fat and calories contents of several brands of burgers, the correlation between them is r= Which conclusion is most accurate? A. More fat in the burgers causes higher calories B. The burgers containing more fat tend to have higher calories Comment: Even though A sounds all right, it is not the conclusion can be derived/explained by the correlation. Correlation is an objective story teller of the linear association between two variables. It can t tell the causation.

25 CORRELATION PROPERTIES (CONT.) Correlation treats x and y symmetrically: The correlation of x with y is the same as the correlation of y with x. Slide 1-25 Correlation has no units. Correlation is not affected by shifting and rescaling of either variable. Correlation depends only on the z-scores, and they are unaffected by changes in center or scale. i.e. corr(ax+b,cy+d)=corr(x,y) where a,b,c,d are constants.

26 TERM 2: CORRELATION Example: Here are several scatterplots. The calculated correlations are , , and Which is which? (a) (b) Y Y X X (c) (d) Y Y X X

27 QUESTION: CAN WE DO MORE? Scatterplot and correlation are useful tolls helping us to learn the (linear) association between two quantitative variables. Can we answer the following question: Fast food is often considered unhealthy because much of it is high in fat. What is the calorie content of a kind of fast food with 28g fat? 700 If we want to estimate a unknown value based on the 650 known values, this is called a 600 prediction. Calorie Fat One way to do the prediction is by constructing a linear model.

28 TERM 3: LINEAR MODEL Let s look at the burger example again. Fat(g) Calories BURGERS CALORIES FAT The red line does not go through all the points, but it can summarize the general pattern with only a couple of parameters: Calories = a+b*fat. This model can be used to predict the Calories based on the fat contain. Explanatory Var: Fat Response Var: Calories

29 TERM 3: LINEAR MODEL BURGERS Predicted value: we call the estimate made from a model the predicted value, denoted as. Residual: The difference between the observed value and its associated predicted value is called the residual. The line of best fit is the line for which the sum of the squared residuals is smallest. And it s called the least squares line. ŷ CALORIES residual Prediction FAT

30 TERM 3: LINEAR MODEL

31 TERM 3: LINEAR MODEL X: Fat(g) Y: Calories Q1: Please construct a linear regression model to predict the calories based on fat. Fat: Calories: Correlation: r= Slope: Intercept: Linear model: Q2: What is the predicted calorie when the fat is 30g? When x=30, Q3: What is the residual for the burger with 30g fat? When x=30, the residual is CALORIES BURGERS = x FAT

32 TERM 3: LINEAR MODEL Remarks: Since regression and correlation are closely related, we need to check the same conditions for regressions as we did for correlations: Quantitative Variables Condition Straight Enough Condition Outlier Condition

33 TERM 3: LINEAR MODEL (PARAMETERS) We write a and b for the slope and intercept of the line. They are called the coefficients of the linear model. The coefficient b is the slope, which tells us how rapidly the predicted value ( ŷ ) changes with respect to x. As the value of x increases by 1 unit, the predicted value of y will be increased by b units. The coefficient a is the intercept, which tells where the line hits (intercepts) the y-axis. In other words, the intercept a is the predicted value of y when x=0

34 Intercept and Slope (examples) Fast food is often considered unhealthy because much of it is high in fat. Are fat and calories related? Here are the fat and calories contents of several brands of burgers. To analyze the association between fat content and calories, the equation of the regression model is: Predicted calories= *fat For this linear equation, slope=10.63, intercept= Q1: What does the slope mean? A1: An increase in fat of 1 gram is associated with an increase in calories of Q2: If the fat increases by 2 grams, how many more calories are expected to be contained in the burger? A2: 2*10.63=21.26 Q3: What does the intercept mean here? A3: Theoretically, it means: when the burger contains no fat at all, the amount of calories is

35 TERM 4: RESIDUAL PLOT After you construct the linear model, you have to check whether the linear model makes sense or not. Residual plot can be used to check the appropriateness of the linear model. Residual plot is the scatterplot of the residuals versus the x- values. If a linear model is appropriate, then the residual plot shouldn t have any interesting features, like a direction or shape. It should stretch horizontally, with about the same amount of scatter throughout. It should show no bends, and it should have no outliers. Residuals X

36 TERM 4: RESIDUAL SCATTERPLOT Now, let s try to diagnose the model for the calorie and fat example. Fat(g): x Calories: y Predicted calories: Residual: Residual plot residuals fat x

37 TERM 4: RESIDUAL PLOT Example: Tell what each of the residual plots below indicates about the appropriateness of the linear model that was fit to the data. (a) (b) (c) y y y x x x3

38 TI for correlation and regression equation The first time you do this: Press 2 nd, CATALOG (above 0) Scroll down to DiagnosticOn Press ENTER, ENTER Read Done Your calculator will remember this setting even when turned off Enter predictor (x) values in L1 Enter response (y) values in L2 Pairs must line up There must be the same number of predictor and response values Press STAT, > (to CALC) Scroll down to 8:LinReg(a+bx), press ENTER, ENTER Read intercept a, slope b and correlation r at the screen

39 IMPORTANT NOTES: Take-home quiz is due on Monday. No late submission will be accepted. Keep the ID assignment and bring it to class on Monday. Sample exam will be handed out on Monday. We will discuss the questions on Wednesday. Suggested Problem Set 4 will be collected on next Thursday. Final exam will be on next Thursday. 2 hours in class. Please prepare one page A4 size cheat sheet (one-sided) on your own. Formula sheet will not be provided in final exam. Cheat sheet will be collected together with the final exam.