7. Do not estimate values for y using x-values outside the limits of the data given. This is called extrapolation and is not reliable.

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1 AP Statistics 15 Inference for Regression I. Regression Review a. r à correlation coefficient or Pearson s coefficient: indicates strength and direction of the relationship between the explanatory variables (x) and the response variable (y). Values are between 1 to 1 with 0 being no relationship, 1 being a perfect positive relationship, and -1 being a perfect negative relationship. b. r 2 à Coefficient of Determination: example: r 2 =.857 Explanation of r 2 : 85.7% of the variation in the response variable in context can be explained by the linear relationship with explanatory variable in context. c. LSRL à Least Squares Regression Line: 1. Sample: ŷ = a + bx a is y-intercept and b is slope, always contains the point (x, y) [Calculator: STATSà CALCà 8.LinReg(a + bx)] 2. Population: µ y = α + βx α is y-intercept and β is slope 3. Example: predicted _ IQ _ score = (cry _ count) OR IQ Score (hat on top) 4. Explanation of slope or LSRL: For every additional unit in Crying Intensity unit of x in context, the IQ Score y variable in context increases (or decreases depending on sign of slope) by slope with units. 5. Explanation of y-intercept: y-intercept is the value of the IQ Score y variable in context when the Cry Intensity x variable in context is 0. NOTE: It doesn t always make sense. 6. b = r Sy à slope = (correlation coefficient)(standard deviation of y values/standard deviation Sx of x values) [Sy and Sx found in 2-Var Stats] 7. Do not estimate values for y using x-values outside the limits of the data given. This is called extrapolation and is not reliable. 8. residual: y ŷ à observed y-value (from data) expected y-value (calculated from LSRL). A positive residual indicates the point is above the LSRL, and a negative residual indicates the point is below the LSRL. II. Conditions for Inference (Confidence Interval and t-test) a. R: Random the data are produced from a well-designed random sample or a randomized experiment b. L: Linear relationship check scatterplot and residual plot (no pattern indicates a linear relationship) c. I: Independence Repeated responses y are independent of each other. d. N: Normal Distribution the response y varies according to a Normal distribution (check modified boxplot of residuals look for outliers and strong skewness) e. E: Equal Variance the standard deviation of y is the same for all values of x (check residual plot to see if the spread of the residuals stays approximately the same as x increases) III. Confidence Interval for Slope (LinRegTInt à t interval for linear regression) a. Procedure: 1. Conditions see above 2. Calculations: b ± t *(SE b ) à slope =/- (critical t value)(standard Error for slope) df = n 2 having area C between t* and t*

2 SE b = s (x x) 2 à s = standard error about the line = residuals 2 n 2 NOTE: SE b rarely has to calculated by hand and is usually given by regression software. 3. Interpretation: We are 95% confident that the interval between low value and high value units for slope captures the slope of the true regression line relating the x variable in context and y variable in context. b. Examples: 1. For their second semester project, two AP Statistics students decided to investigate the effect of sugar on the life of cut flowers. They went to the local grocery store and randomly selected 12 carnations. All the carnations seemed equally healthy when they were selected. When the students got home, they prepared 12 identical vases with exactly the same amount of water in each vase. They put one tablespoon of sugar in 3 vases, two tablespoons of sugar in 3 vases, and three tablespoons of sugar in 3 vases. In the remaining 3 vases, they put no sugar. After the vases were prepared and placed in the same location, the students randomly assigned one flower to each vase and observed how many hours each flower continued to look fresh. Here are the data: Sugar (TBSP) Freshness (hours) a) Construct a 95% confidence interval for the slope of the population regression line. b) Would you feel confident predicting the hours of freshness for 10 tbsp of sugar? Explain.

3 2. Does Fidgeting Keep You Slim? We examined data from a study that investigated why some people don t gain weight even when they overeat. Perhaps fidgeting and other non-exercise activity (NEA) explains why some people may spontaneously increase non-exercise activity when fed more. Researchers deliberately overfed a random sample of 16 healthy adults for 8 weeks. They measured fat gain (kg) and change in energy use (calories) from activity other than deliberate exercise fidgeting, daily living, and the like for each subject. Here are the data: NEA change (cal): Fat gain (kg): NEA change (cal): Fat gain (kg): Construct and interpret a 90% confidence interval for the slope of the population regression line.

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