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3. Regression Analysis (Chapters 5 and 14) pg 183 Study relationship between 2 quantitative variables One variable is _____________________ variable, denoted by y. Measures the outcome of the study. Also called the _____________________ variable. Other variable is ____________________ variable, denoted by x. Thought to explain changes in the response. Also called the _____________________ variable.

4. Regression Analysis First Step … Examine data graphically … scatterplot displays the relationship. Look for overall pattern and then if any departures. If linear relationship reasonable, find a model (equation of line) to summarize relationship.  Resulting equation used for prediction.  If assumptions hold, can assess significance of linear relationship and make CIs for estimations/predictions.

5. Regression Example: Exam 2 and Final Exam Scores How well does the exam 2 score for a Stats 350 student predict their final exam score? Below are the scores for a random sample of n = 6 students from a previous term. Which is response and which is explanatory? Response (dependent) variable y = _____________________ Explanatory (independent) variable x = __________________ Exam 2336544646040 Final538078938858

6. Step 1: Examine data graphically w/ scatterplot Exam 2336544646040 Final538078938858

7. Interpretation of the Scatterplot overall form (is the average pattern a straight line or curved?) direction of association (positive or negative) strength of association (how much do the points vary around the average pattern?) any deviations from the overall form?

8. Describing a Linear Relationship with a Regression Line pg 185 Simplest kind of relationship between two variables is a straight line – the analysis in this case is called Linear Regression Regression Line for Exam 2 vs Final Remember equation of a line? Regression line for a sample: where b 0 b 1

9. Goal: Find the Best Fitting Line pg 185 How? What do we mean by “best”? Look at “observed errors” in prediction. Observed errors = _____________________ are called ________________________ Choose line for which sum of squares of observed errors is least. Line called: _______________________________________

10. Least Squares Regression Line Equations for estimated slope and intercept for LS regression line are given by: b 1 b 0 Least squares regression line (estimated regression function):

11. Finding the Least Squares Regression Line

12. Slope Estimate: y-intercept Estimate: Estimated Regression Line: Predict final exam score for student who scored 60 on exam 2:

13. Residuals Predicted final exam score is 84.43 Note: 5 th observation had exam 2 of 60 and final was 88 pts Find corresponding residual…  Notation: The residuals … You could compute residual for each observation. Exam 2336544646040 Final538078938858

14. Residuals pg 187 SSE = sum of squared errors (residuals) 

15. Where are we? Regression Example Exam 2 vs Final Least Squares Regression Line: SSTO = 1300 SSE = 272.1 Exam 2336544646040 Final538078938858

16. Measuring Strength and Direction of a Linear Relationship with Correlation Properties about r 1.r ranges from... correlation coefficient r = measure of strength of the linear relationship between y and x.

17. Properties about Correlation r 2.Sign of r indicates... 3.Magnitude of r indicates... (what is strong? discipline specific) 4.r ONLY measures the strength of the _____________________ relationship.

18. Some Pictures: The Formula:

19. Exam Scores Example pg 188 r = _______________ Interpretation:

20. 1. Statistically Significant – In a statistical test of hypotheses, we say the data are statistically significant at the level a (select all that apply) A) if a is 0.05 B) if a is small C) if the p-value is larger than a D) if the p-value is less than a E) if the p-value is equal to a

21. 2.Skipping Breakfast – Gallup poll reported a 90% CI for the proportion of all young adults skip breakfast regularly to be: (0.20, 0.28). True (T) or False (F). a.The above confidence interval was constructed with a margin of error of 8 percentage points.TF b.One of the assumptions for the interval to be valid is that the population has a normal distribution. TF

22. 2.Skipping Breakfast – Gallup poll reported a 90% CI for the proportion of all young adults skip breakfast regularly to be: (0.20, 0.28). True (T) or False (F). c.If the poll was repeated many times, 90% of the time the proportion of all young adults who skip breakfast regularly will be between 0.20 and 0.28.TF d.At 10% significance level, value of 26% is acceptable for the proportion of all young adults who skip breakfast regularly, but the value of 30% would be rejected. TF

23. 3.Average Service Time – Population of all weekday lunch customers at a local restaurant. Claim: mean service time 15 minutes and std dev 2 minutes. Random sample of n = 64 weekday lunch customers to be selected and sample mean service time computed. Detailed sketch of (approx) distribution of possible values for sample mean. Include labels and values.