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Slide Slide 1 Warm Up: #23, 24 on page 555 Answer each of the following questions for #23, 24: a)Is there a linear correlation? Use software then Table A-6 to prove it. Are your answers the same? b)Graph the points (don’t forget axis labels). If there is a correlation, graph the LSRL and continue to do c-h. c)Find the vital statistics (r, r-squared, a, b, y-hat – don’t forget to define x and y) d)Tell me what r and r-squared means in the context of the problem (r: form, direction, strength) (r-squared: how much of the variation in x can be explained by the variation in y) e)Find the residuals f)Draw the residual plot – is the regression line a good model for the data? Why? g)For # 23, predict the winning time when the temperature is 73 degrees Fahrenheit. h)For #24, predict the height of a daughter when her mother is 66 inches tall.

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Slide Slide 2 The SAT essay: longer is better? Words Score Words Score Words13573 Score31

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Slide Slide 3 Section 10-4 Variation

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Slide Slide 4 Key Concept In this section we proceed to consider a method for constructing a prediction interval, which is an interval estimate of a predicted value of y. Using paired data (x,y), we describe the variation that can be explained between x and y and the variation that is unexplained.

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Slide Slide 5 Figure 10-9 Unexplained, Explained, and Total Deviation

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Slide Slide 6 Definitions Total Deviation The total deviation of ( x, y ) is the vertical distance y – ybar, which is the distance between the point ( x, y ) and the horizontal line passing through the sample mean y-bar. Explained Deviation The explained deviation is the vertical distance y-hat - y-bar, which is the distance between the predicted y- value and the horizontal line passing through the sample mean y-bar. Unexplained Deviation The unexplained deviation is the vertical distance y – y-hat, which is the vertical distance between the point ( x, y ) and the regression line. (The distance y – y-hat is also called a residual, as defined in Section 10-3.)

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Slide Slide 7 Particulars We can explain the discrepancy between y-bar = 9 and y-hat=13 by noting that there is a linear relationship best described by the LSRL (y-y-hat). The discrepancy between y-hat = 13 and y=19 can’t be explained by the LSRL = residual or unexplained deviation (y-y-hat)

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Slide Slide 8 (total deviation) = (explained deviation) + (unexplained deviation) ( y - y ) = ( y - y ) + (y - y ) ^ ^ (total variation) = (explained variation) + (unexplained variation) ( y - y ) 2 = ( y - y ) 2 + (y - y) 2 ^^ Formula 10-4 Relationships

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Slide Slide 9 Definition r2 =r2 = explained variation. total variation The value of r 2 is the proportion of the variation in y that is explained by the linear relationship between x and y. Coefficient of determination is the amount of the variation in y that is explained by the regression line.

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Slide Slide 10 Warm Up: Day 2 Consider the following data set: Find: a)Total variation b)Explained variation c)Unexplained variation XY

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Slide Slide 11 Try again! Consider the following data set: Find: a)Total variation b)Explained variation c)Unexplained variation XY

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Slide Slide 12 Not Old Faithful again! In section 10-2 we used the duration/interval after eruption times in Table 10-1 to find that r =.926. find the coefficient of determination. Also, find the percentage of the total variation in y (time interval after eruption) that can be explained by the linear relationship between the duration of time and the time interval after an eruption. Duration Interval After

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Slide Slide 13 Interpretation/New Def 86% of the total variation in time intervals after eruptions (y) can be explained by the duration times (x) 14% of the total variation in time intervals after eruptions can be explained by factors other than duration times. Recall: y-hat = x (x = duration in seconds, y = predicted time interval). When x = 180, we predict a y-hat of ____? This single value is called a point estimate. It is our best predicted value. How accurate is it? We use prediction intervals to answer this question.

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Slide Slide 14 Definitions Prediction Interval: an interval estimate of a predicted value of y. The development of a prediction interval requires a measure of the spread of sample points about the regression line. The standard error of estimate, denoted by se is a measure of the differences (or distances) between the observed sample y-values and the predicted values y that are obtained using the regression equation. That is, it is a collective measure of the spread of the sample points about the regression line. Se = A measure of how sample points deviate from their regression line.

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Slide Slide 15 Standard Error of Estimate s e = or s e = y 2 – b 0 y – b 1 xy n – 2 Formula 10-5 ( y – y ) 2 n – 2 ^

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Slide Slide 16 Given the sample data in Table 10-1, find the standard error of estimate s e for the duration/interval data. Example: Old Faithful Duration Interval After

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Slide Slide 17 y - E < y < y + E ^ ^ Prediction Interval for an Individual y where E = t 2 s e n(x2)n(x2) – ( x) 2 n(x0 – x)2n(x0 – x) n x 0 represents the given value of x t 2 has n – 2 degrees of freedom

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Slide Slide 18 E = t 2 s e + n( x 2 ) – ( x) 2 n(x0 – x)2n(x0 – x) n Example: Old Faithful For the paired duration/interval after eruption times in Table 10-1, we have found that for a duration of 180 sec, the best predicted time interval after the eruption is 76.9 min. Construct a 95% prediction interval for the time interval after the eruption, given that the duration of the eruption is 180 sec (so that x = 180). Duration Interval After

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Slide Slide 19 y – E < y < y + E 76.9 – 13.4 < y < < y < 90.3 ^ ^ Example: Old Faithful - cont For the paired duration/interval after eruption times in Table 10-1, we have found that for a duration of 180 sec, the best predicted time interval after the eruption is 76.9 min. Construct a 95% prediction interval for the time interval after the eruption, given that the duration of the eruption is 180 sec (so that x = 180).

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Slide Slide 20 Same problem, different x For the paired duration/interval after eruption times, find: 1)For a duration of 150 sec, the best predicted time interval after the eruption is _____ min. 2)Construct a 95% prediction interval for the time interval after the eruption, given that the duration of the eruption is 150 sec (so that x = 150). Duration Interval After E = t 2 s e + n( x 2 ) – ( x) 2 n(x0 – x)2n(x0 – x) n y – E < y < y + E ^ ^

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