Tables and graphs taken from Glencoe, Advanced Mathematical Concepts

R 2 near 1.0 indicates that a regression line is a “good fit” R 2 closer to 0 indicates a regression line is not a “good fit” Indicates the proportion of variability that can be attributed to the statistical model Indicates how accurate the model should be when used for predicting future outcomes Models are typically most accurate for finding values within the range of the given data – called interpolation. When extrapolating, finding measures outside the range of the given data, use extra caution in considering reliability.

Let’s look back at the data set we used previously, where x = 3 represents the school year For this data we found the following linear regression: Find each and decide which is most representative – or best-fit model for the data Quadratic Regression (5:QuadReg) Cubic Regression (6:CubicReg) Quartic Regression (7:QuartReg) Exponential Regression (0:ExpReg) Power Regression (A:PwrReg)

Let’s look back at the data set we used previously, where x = 3 represents the school year For this data we found the following linear regression: Find each and decide which is most representative – or best-fit model for the data Quadratic Regression (5:QuadReg) Cubic Regression (6:CubicReg) Quartic Regression (7:QuartReg) Exponential Regression (0:ExpReg) Power Regression (A:PwrReg) Notice that two of these have an r 2 of.99 or better. You should ALWAYS check the graphs of the regression models against the actual data to know for sure. On the next slide we will learn how to do this.

Find each and decide which is most representative – or best-fit model for the data Quartic Regression (7:QuartReg) Power Regression (A:PwrReg) To verify the correct model we need to graph the data (STATPLOT) along with the equations that seem to model it most closely. MODE – change from sequential to simultaneous (SIMUL) STATPLOT (2 nd y=) – turn on for scatter plot – first choice QUIT Recalculate the Quartic Regression Equation (STAT, CALC, 7:QuartReg), L1, L2 Y = Y1=VARS, 5:STATISTICS,<EQ,1:RegEQ,enter [this will put the actual equation in Y1 for you – must use actual equation when graphing] ZOOM, 9:ZOOMSTAT, then ZOOM, 3:ZOOMOUT SKETCH what you see on paper REPEAT the process – QUIT (2 nd MODE) clear Y1 the QUIT (2 nd MODE) Recalculate the Power Regression (STAT, CALC, A:PwrReg) Y= Y1=VARS, 5:STATISTICS,<EQ,1:RegEQ,enter [this will put the actual equation in Y1 for you – must use actual equation when graphing] ZOOM, 9:ZOOMSTAT, then ZOOM, 3:ZOOMOUT SKETCH what you see on paper Higher r vs BEST FIT MODEL

Let’s look back at the data set we used previously. For this data we found the following linear regression: Find each and decide which is most representative – or best-fit model for the data Quadratic Regression (5:QuadReg) Cubic Regression (6:CubicReg) Quartic Regression (7:QuartReg) Exponential Regression (0:ExpReg) Power Regression (A:PwrReg)

Let’s look back at the data set we used previously, where x = 3 represents the school year For this data we found the following linear regression: Find each and decide which is most representative – or best-fit model for the data Quadratic Regression (5:QuadReg) Cubic Regression (6:CubicReg) Quartic Regression (7:QuartReg) Exponential Regression (0:ExpReg) Power Regression (A:PwrReg) Notice that one of these has an r 2 much better than the other models. You should ALWAYS check the graphs of the regression models against the actual data to know for sure.

Find each and decide which is most representative – or best-fit model for the data Cubic Regression (6:CubicReg) To verify the correct model we need to graph the data (STATPLOT) along with the equations that seem to model it most closely. MODE – change from sequential to simultaneous (SIMUL) STATPLOT (2 nd y=) – turn on for scatter plot – first choice QUIT Recalculate the Cubic Regression (STAT, CALC, 6:CubicReg) Y= Y1=VARS, 5:STATISTICS,<EQ,1:RegEQ,enter [this will put the actual equation in Y1 for you – must use actual equation when graphing] ZOOM, 9:ZOOMSTAT, then ZOOM, 3:ZOOMOUT SKETCH what you see on paper BEST FIT MODEL Remember, cubic models change direction twice. It is difficult to see on this graph but we can examine the table to learn more. TBLSET (2 nd Window) TblStart = 0 ∆Tbl = 100 Look at the table and you see that this model increases for approximately from x = 0 through x = 600, and then decreases from x > 600 through x = 1800, and then begins increasing again for x < 1800.

Find each and decide which is most representative – or best-fit model for the data Linear Regression (4:LinReg) Quadratic Regression (5:QuadReg) Cubic Regression (6:CubicReg) Quartic Regression (7:QuartReg) Exponential Regression (0:ExpReg) Power Regression (A:PwrReg)

Find each and decide which is most representative – or best-fit model for the data Linear Regression (4:LinReg) Quadratic Regression (5:QuadReg) Cubic Regression (6:CubicReg) Quartic Regression (7:QuartReg) Exponential Regression (0:ExpReg) Power Regression (A:PwrReg) When looking at the graph you see that the data only changes direction one time, indicative of a quadratic function. Because quadratic, cubic, and quartic all have such high r 2 values, it is critical that you see which one most closely mimics the graph.

Find each and decide which is most representative – or best-fit model for the data Linear Regression (4:LinReg) Quadratic Regression (5:QuadReg) Cubic Regression (6:CubicReg) Quartic Regression (7:QuartReg) Exponential Regression (0:ExpReg) Power Regression (A:PwrReg) When looking at the graph the data does not appear to change direction at all, i.e. there is no curve to the data, indicative of a linear function. Since all have such high r 2 values, it is critical that you see which one most closely mimics the graph.

Find each and decide which is most representative – or best-fit model for the data Linear Regression (4:LinReg) Quadratic Regression (5:QuadReg) Cubic Regression (6:CubicReg) Quartic Regression (7:QuartReg) Exponential Regression (0:ExpReg) Power Regression (A:PwrReg) When looking at the graph the data changes direction one time, indicative of a quadratic function. Since quadratic, cubic, and quartic all have such high r 2 values, it is critical that you see which one most closely mimics the graph.

Find each and decide which is most representative – or best-fit model for the data Linear Regression (4:LinReg) Quadratic Regression (5:QuadReg) Cubic Regression (6:CubicReg) Quartic Regression (7:QuartReg) Exponential Regression (0:ExpReg) Power Regression (A:PwrReg) Notice that all 4 regressions are essentially the same – they are equivalent. When we found the linear regression equation, the r 2 = 1 indicated that it was a perfect linear relationship. We could have stopped there.

Find each and decide which is most representative – or best-fit model for the data Linear Regression (4:LinReg) Quadratic Regression (5:QuadReg) Cubic Regression (6:CubicReg) Quartic Regression (7:QuartReg) Exponential Regression (0:ExpReg) Power Regression (A:PwrReg) Notice that the quadratic, cubic, and quartic regressions are essentially the same – they are equivalent. When we found the quadratic regression equation, the r 2 = 1 indicated that it was a perfect quadratic relationship. We could have stopped there.

Find each and decide which is most representative – or best-fit model for the data Linear Regression (4:LinReg) Quadratic Regression (5:QuadReg) Cubic Regression (6:CubicReg) Quartic Regression (7:QuartReg) Exponential Regression (0:ExpReg) Power Regression (A:PwrReg) Notice that the quartic regression has the only r 2 value over.90. When we check the graph we also see that the data changes direction 3 times, indicative of a quartic function. We can feel comfortable that this is the most accurate model for this data.

Find each and decide which is most representative – or best-fit model for the data Linear Regression (4:LinReg) Quadratic Regression (5:QuadReg) Cubic Regression (6:CubicReg) Quartic Regression (7:QuartReg) Exponential Regression (0:ExpReg) Power Regression (A:PwrReg) Notice how close the r 2 values for the cubic and quartic regression equations are. When we check the graph we also see that the data changes direction 2 times, indicative of a cubic function.

Find each and decide which is most representative – or best-fit model for the data Linear Regression (4:LinReg) Quadratic Regression (5:QuadReg) Cubic Regression (6:CubicReg) Quartic Regression (7:QuartReg) Exponential Regression (0:ExpReg) Power Regression (A:PwrReg) The r 2 value for the quartic regression equation is far greater than any of the others. When we check the graph we also see that the data changes direction 3 times, indicative of a quartic function.

Find each and decide which is most representative – or best-fit model for the data Linear Regression (4:LinReg) Quadratic Regression (5:QuadReg) Cubic Regression (6:CubicReg) Quartic Regression (7:QuartReg) Exponential Regression (0:ExpReg) Power Regression (A:PwrReg) None of the r 2 values are that high, however, the values for cubic and quartic are the highest. When we check the graphs we see that the data changes direction 2 times, indicative of a cubic function, so we would probably go with the cubic regression. As indicated by the r 2 values, neither model is accurate enough to predict values with much certainty.

Find each and decide which is most representative – or best-fit model for the data Linear Regression (4:LinReg) Quadratic Regression (5:QuadReg) Cubic Regression (6:CubicReg) Quartic Regression (7:QuartReg) Exponential Regression (0:ExpReg) Power Regression (A:PwrReg) Both cubic and quartic have r 2 values over When we check the graphs we see that the data changes direction 3 times, indicative of a quartic function.

Approximately 20% If % put in as decimals, 23% as 0.23 If % put in as whole numbers, 23% as 23

Find each two ways, including the data from 1993 and after deleting the data from 1993, and decide which is most representative – or best-fit model for the data. (5:QuadReg) (6:CubicReg) (7:QuartReg) Incl w/o Incl w/o Incl w/o. 1993

31.6 million To find this using the calculator, put regression equation in y1 and set y2 = 150. Then, graph by hitting ZOOM, ZOOMSTAT. Finally, use the CALC menu (2 nd Trace) to find the intersection (5:intersect) of the two equations. Approximately 18 years after 1983, so 2001.

c.In what year would the population have been 10 (in 100 millions)? d.In what year would the population have been 20 (in 100 millions)? e.In what year would the population be 30 (in 100 millions)? f.Find the expected population in g.Find the expected population in Approximately 47 years after 1950, or Approximately 79 years after 1950, or Approximately 98 years after 1950, or Approximately 48.7 in 100 millions, or 4.9 billions. Approximately 20.4 in 100 millions, or 2.04 billions.

a.How many hours will it take for there to be 50 bacteria (thousands/cc)? b.How many bacteria (thousands/cc) will there be after 15 hours? c.How many hours will it take for there to be 60 bacteria (thousands/cc)? d.How many bacteria (thousands/cc) will there be after 12.5 hours? Approximately 12.5 hrs. Approximately 13.6 hrs. Approximately (thousands/cc). Approximately (thousands/cc).

a.How long will it take the skater to slow down to a velocity of 3.75 m/s? b.What will the skater’s velocity be after 9 seconds? c.How long will it take the skater to slow down to a velocity of 1.25 m/s? d.What will the skater’s velocity be after 40 seconds? Approximately 2.45 seconds Approximately seconds Approximately 2.25 m/s Approximately 0.53 m/s

b.How many minutes will it take for the grams of U-239 to be present to have dropped to 2.3 grams? c.How many grams of U-239 will be present after 27 minutes? d.How many minutes will it take for the grams of U-239 to be present to have dropped to 1.5 grams? e.How many grams of U-239 will be present after 40 minutes? Approximately 4.44 gramsApproximately 3.00 grams Approximately 49 minutes Approximately 63 minutes

b.How many total hours will it take for the number of bacteria (in millions) to have doubled from that observed at 4 hours? c.How many bacteria (in millions) will be present after 2.5 hours? d.How many hours will it take for the number of bacteria to reach 200 (in millions)? e.How many bacteria (in millions) will be present after 5.6 hours? Approximately 20Approximately 117 Approximately 1.3 more hours, at t = 5.3 hours. Approximately 6.5 hours

b.How many minutes will it take for the number of number of grams of Pb-211 present to reach 0.20? c.How many grams of PB-211 will be present after 45 mins? Approximately 0.42 grams Approximately 82 minutes b.What is the half-life of PB-211 in minutes? c.How many grams of PB-211 will be present after 1 hour? Approximately 36 minutes Approximately 0.31 grams

b.How much would have been in the account on January 1, 1965? c.How much would have been in the account on January 1, 1975? d.How much would have been in the account on January 1, 1985? e.How much would have been in the account on January 1, 1995? Approximately $ f.How much would have been in the account on January 1, 2005? g.How much would have been in the account on January 1, 2015? h.How many years will it take in all for the balance to reach$75,000? i.How many years will it take in all for the balance to reach $100,000? Approximately $ Approximately $10,172 Approximately $17,110 Approximately $28,779 Approximately $48,407 Approximately 68.4 years Approximately 74 years