1. Bring project data to enter into Fathom
Tomorrow
Bring project data to enter into Fathom

2. Section 3.2
Day 3

3. Find the slope of the line that passes through the points (1960, 0
Find the slope of the line that passes through the points (1960, 0.80) and (2000, 4.80)

4. Find the slope of the line that passes through the points (1960, 0
Find the slope of the line that passes through the points (1960, 0.80) and (2000, 4.80)
Slope =

5. Find the slope of the line that passes through the points (1960, 0
Find the slope of the line that passes through the points (1960, 0.80) and (2000, 4.80)
m = 0.10

6. Find the equation for that line in slope-intercept form, y = mx + b
through the points (1960, 0.80) and (2000, 4.80)

7. Find the equation for that line in slope-intercept form, y = mx + b
If we use (1960, 0.80):
0.80 = (0.10)(1960) + b
b =

8. Find the equation for that line in slope-intercept form, y = mx + b
If we use (1960, 0.80):
0.80 = (0.10)(1960) + b
b =
Equation is y = 0.10x –

9. Equation is y = 0.10x –
Recall, in statistics, the equation is written with y-intercept first.
So, y = x

10. The line on the plot is the ____________.

11. The line on the plot is the line of best fit.

12. What does the slope of 0.10 tell us in context?

13. Slope of 0. 10 tells us the minimum wage increased by about $0
Slope of 0.10 tells us the minimum wage increased by about $0.10 per year over the 45-year period 1960 – 2005.

14. Slope of 0. 10 tells us the minimum wage increased by about $0
Slope of 0.10 tells us the minimum wage increased by about $0.10 per year over the 45-year period 1960 – 2005.
“by about”: have to qualify interpretation of slope because although the scatterplot is linear, all the points do not fall on the same line.

15. To make a prediction of y for a given value of x:

16. To make a prediction:
If have equation for line, substitute value for x and solve for y

17. To make a prediction:
If have equation for line, substitute value for x and solve for y
If looking at graph, find y-value that corresponds with x-value you’re interested in

18. Example
Use the equation y = x to predict the minimum wage in the years 2003 and 1950.

19. Example
Use the equation y = x to predict the minimum wage in the years 2003 and 1950.
Easier to see if put equation in context.
Minimum wage = (year)

20. Example
Use the equation y = x to predict the minimum wage in the years 2003 and 1950.
Minimum wage = (year)
2003: min wage = (2003) = $5.10

21. Example
Use the equation y = x to predict the minimum wage in the years 2003 and 1950.
Minimum wage = (year)
2003: min wage = (2003) = $5.10
1950: min wage = (1950) =
-$0.20

22. Two types of predictions:

23. Two types of predictions:
1) interpolation – making prediction when value of x falls within range of the actual data; usually safe

24. Two types of predictions:
1) interpolation – making prediction when value of x falls within range of the actual data; usually safe
2003: min wage = (2003) = $5.10

25. 2) extrapolation – making prediction when value of x falls outside range of actual data; may be very risky, risk increases the further away you get from the range of actual data

26. 2) extrapolation – making prediction when value of x falls outside range of actual data; may be very risky, risk increases the further away you get from the range of actual data
1950: min wage = (1950) =
-$0.20

27. Prediction Error
Prediction error: difference between the actual value of y and value of y predicted from a regression line

28. Prediction Error
Prediction error: difference between the actual value of y and value of y predicted from a regression line
Usually unknown except for the points used to construct the regression line, whose prediction errors are called residuals

29. Residual = observed value of y – predicted value of y, or y , (read “y-hat”)
Residual = y - y

30. Residual is the signed vertical distance from an observed data point to the regression line.
Positive if point above the line
Negative if point below the line

31. Questions 1) What does it mean if the residual is 0?
2) If the residual is large and negative where is the observed point located with respect to the line?
3) If someone said they had fit a line to a set of data points and all their residuals were positive, what would you say to them?

32. Questions 1) What does it mean if the residual is 0?
The observed point is on the line of best fit
2) If the residual is large and negative where is the observed point located with respect to the line?

33. Questions
2) If the residual is large and negative where is the observed point located with respect to the line? Far below the line of best fit
3) If someone said they had fit a line to a set of data points and all their residuals were positive, what would you say to them?

34. Questions
3) If someone said they had fit a line to a set of data points and all their residuals were positive, what would you say to them?
The fitted line is too low. Move the line up so there are both positive and negative residuals.

35. Line of Best Fit
The line of best fit lies with about one-half the points above and one-half the points below.
Thus, we want to make the sum of the residuals equal to 0.

36. Line of Best Fit
The line of best fit lies with about one-half the points above and one-half the points below.
Thus, we want to make the sum of the residuals equal to 0.
However, there are many lines that could also satisfy this condition so we must use a different method to determine the line of best fit.

37. Method of Least Squares
Sum of squared errors is known as SSE.

38. LSRL
Least squares regression line, also called least squares line or regression line, is the line for which the sum of the squared errors or SSE is as small as possible.

39. LSRL
Least squares regression line, also called least squares line or regression line, is the line for which the sum of the squared errors or SSE is as small as possible.
SSE = (residuals)2

40. Demo SSE and LSRL

41. Find the least squares line for this passenger jets data.

42. Put explanatory values in L1 and response values in L2

43. Put explanatory values in L1 and response values in L2
STAT
CALC
8. LinReg (a + bx)
LinReg (a + bx) L1, L2, Y1

44. Put explanatory values in L1 and response values in L2
STAT
CALC
8. LinReg (a + bx)
LinReg (a + bx) L1, L2, Y1
To get Y1, go to VARS, Y-VARS,
1: Function, ENTER, 1: Y1, ENTER

45. So, what is equation for LSRL?
LinReg
y = a + bx
a =
b = 16
r2 =
r =
So, what is equation for LSRL?
Note: if r and r2 don’t show on your calculator, you must turn on diagnostics by:
1. 2nd Catalog [the blue “Catalog” button is the same as 2nd 0 (i.e. the zero button)]
2. Scroll down the alphabetical list to “DiagnosticOn”
3. <Enter> <Enter>
4. Re-enter the LinReg function

46. y = 367 + 16x Is this it? LinReg y = a + bx a = 366.6666667 b = 16
So, what is equation for LSRL?
y = x
Is this it?

47. y = 367 + 16x Is this it? Cost = 367 + 16(seats)
So, what is equation for LSRL?
y = x
Is this it?
Cost = (seats)

48. Cost = (seats)
Interpret the slope and y-intercept.

49. Cost = (seats)
Interpret the slope and y-intercept.
Slope: For each additional seat, the cost increases by about $16 per hour

50. Cost = (seats)
Interpret the slope and y-intercept.
Slope: For each additional seat, the cost increases by about $16 per hour
Y-intercept: If a passenger jet had 0 seats, it would cost about $367 per hour to operate.

51. Page 134, E14

52. Page 134, E14 a. Fuel consumption rate is the explanatory variable.
Operating cost is the response variable.

53. Page 134, E14 b. The slope is approximately 2.5.
Can not just count squares as scales are different.

54. Page 134, E14 b. The slope is approximately 2.5.
For each additional gallon of fuel used per
hour, the operating cost tends to be about
$2.50 per hour more.
This could be the cost of 1 gallon of fuel.

55. Page 134, E14 c. This value means that if an aircraft used no fuel,
the cost per hour would be $470 per hour.
While it doesn’t make sense for an aircraft to use
no fuel, it does make sense that there are costs
besides fuel.
The y-intercept would represent the cost per hour
of running an aircraft in addition to fuel costs.

56. Page 134, E14 d. The cost per hour for a plane that
consumes 1500 gallons per hour of fuel is
approximately (1500) or $4220.

57. Questions?
Bring project data for class tomorrow