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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.

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Presentation on theme: "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."— Presentation transcript:

1 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.

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

3 Slide Slide 3 Section 10-4 Variation

4 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.

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

6 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.)

7 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)

8 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

9 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.

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

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

12 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

13 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.

14 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.

15 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 ^

16 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

17 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

18 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

19 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).

20 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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