Fit Lines to Scatter Plots - 8th Grade Math

1. What is a residual for a data point relative to a fitted line?

   Residual $=$ actual $y$ minus predicted $y$. Measures vertical distance from point to line.

2. What is the predicted $y$ if the fitted line is $y=3x+1$ and $x=4$?

   $y=13$. Substitute $x=4$ into equation: $y=3(4)+1=12+1=13$.

3. What does it mean if a scatter plot shows no linear association?

   There is no clear upward or downward linear trend. Points are scattered randomly without a linear pattern.

4. If a fitted line is $y = -3x + 10$, what is the predicted $y$ when $x = 2$?

   $y = 4$. Substitute: $y = -3(2) + 10 = -6 + 10 = 4$.

5. What is a line of best fit in the context of a scatter plot?

   A line that models the overall trend so points are roughly balanced above and below. Represents the general linear trend in the data.

6. What does a weak linear association look like in a scatter plot?

   Points are widely scattered around any line. Large scatter means the linear model fits poorly.

7. Which model fit is better: points tightly clustered or widely scattered around a line?

   Tightly clustered points indicate a better fit. Less scatter means more reliable predictions.

8. If a data point is at $(3,9)$ and the model predicts $y = 7$ at $x = 3$, what is the residual?

   Residual $= 2$. Residual = actual - predicted = $9 - 7 = 2$.

9. Identify the term for using a fitted line to predict $y$ for an $x$ value inside the data range.

   Interpolation. Making predictions within the observed data range.

10. What is another key rule for informally drawing a line of best fit?

    Draw the line through the middle of the point cluster. The line should represent the overall trend of all points.

11. What is one key rule for informally drawing a line of best fit?

    Place about half the points above and half below the line. This balances the line through the data's center.

12. What does a residual of $0$ mean for a point and the fitted line?

    The point lies on the line (actual $y$ equals predicted $y$). Zero residual means perfect prediction for that point.

13. Find the slope of the line through $(1,8)$ and $(5,2)$.

    $m = -\frac{3}{2}$. $m = \frac{2-8}{5-1} = \frac{-6}{4} = -\frac{3}{2}$.

14. What is a line of best fit used for in a scatter plot?

    To model the relationship and make predictions. The line helps estimate values and understand trends.

15. Identify the slope sign if a fitted line goes down from left to right.

    Negative slope, $m < 0$. Falling lines have negative slopes.

16. What is a line of best fit (informal fit) on a scatter plot?

    A straight line drawn to represent the overall trend of points. This line approximates the general direction of the data points.

17. What does the slope of a fitted line represent in a real-world context?

    The rate of change: the change in $y$ for each $1$ unit increase in $x$. Shows how much $y$ changes per unit of $x$.

18. What does it mean if points are very close to a fitted line?

    The linear model is a good fit (strong linear association). Points cluster tightly around the line.

19. If a fitted line has equation $y = 2x + 1$, what is the predicted $y$ when $x = 4$?

    $y = 9$. Substitute: $y = 2(4) + 1 = 8 + 1 = 9$.

20. What does a strong linear association look like in a scatter plot?

    Points lie close to a straight line. Little scatter means the linear model fits well.

21. Identify the slope sign if a fitted line goes up from left to right.

    Positive slope, $m > 0$. Rising lines have positive slopes.

22. What is the slope between points $(1,9)$ and $(5,1)$?

    $\frac{1-9}{5-1}=-2$. Use slope formula: $\frac{y_2-y_1}{x_2-x_1}$.

23. What is a positive linear association in a scatter plot?

    As $x$ increases, $y$ tends to increase. Both variables move in the same direction together.

24. What does a negative linear association look like in a scatter plot?

    Points trend downward from left to right as $x$ increases. Shows a negative correlation between the variables.

25. What is one goal when informally drawing a line of best fit?

    Balance points so about half are above and half are below. Equal distribution minimizes overall prediction error.

26. What does it mean when a scatter plot shows a linear association?

    Points trend around a straight line as $x$ changes. Linear association means data follows a straight-line pattern.

27. What is the residual if the line predicts $y=20$ but the actual value is $y=16$?

    $16-20=-4$. Residual = actual - predicted value.

28. What does a positive linear association look like in a scatter plot?

    Points trend upward from left to right as $x$ increases. Shows a positive correlation between the variables.

29. Find the $y$ value on the line $y=-2x+10$ when $x=6$.

    $y=-2$. Substitute $x=6$: $y=-2(6)+10=-12+10$.

30. Which direction indicates a positive association in a scatter plot?

    An upward trend: as $x$ increases, $y$ tends to increase. Both variables move in the same direction together.

31. What does the $y$-intercept of a fitted line represent in context?

    The predicted value of $y$ when $x=0$ (if $x=0$ is meaningful). The starting value when the independent variable is zero.

32. Which fitted line is steeper: $y=0.5x+2$ or $y=2x-1$?

    $y=2x-1$. Compare slopes: $2 > 0.5$, so larger slope means steeper line.

33. Which option is usually less reliable for a fitted line: interpolation or extrapolation?

    Extrapolation. Predictions beyond data range are less certain.

34. What does it mean if points are widely scattered from a fitted line?

    The linear model is a weak fit (weak linear association). Large scatter suggests the line poorly represents the data pattern.

35. What is the slope between points $(2,3)$ and $(6,11)$?

    $\frac{11-3}{6-2}=2$. Use slope formula: $\frac{y_2-y_1}{x_2-x_1}$.

36. Identify the slope of the fitted line through $ (2,9) $ and $ (6,1) $.

    $m=\frac{1-9}{6-2}=-2$. Apply slope formula: rise over run between the two points.

37. What is a scatter plot used for when studying two quantitative variables?

    A graph that plots $(x,y)$ pairs to show the relationship between two variables. Visualizes how two quantitative variables relate to each other.

38. Identify the slope of the fitted line through $(0,2)$ and $(4,10)$.

    $m=\frac{10-2}{4-0}=2$. Use slope formula $m=\frac{y_2-y_1}{x_2-x_1}$ with given points.

39. What does it mean to assess model fit informally on a scatter plot?

    Judge how close the points are to the fitted line. Closer points mean better model accuracy.

40. What does it mean if points are tightly clustered around a fitted line?

    The linear model is a good fit (strong linear association). Close clustering indicates the line accurately represents the data.

41. What is the residual if actual $y=14$ and predicted $y=12$?

    Residual $=14-12=2$. Subtract predicted from actual to find vertical distance.

42. What is a scatter plot used to display for two quantitative variables?

    A set of plotted $(x,y)$ pairs showing their relationship. Scatter plots visualize how two variables relate to each other.

43. What is an outlier in a scatter plot?

    A point far from the overall pattern of the data. Outliers can significantly affect the line of best fit.

44. What is the residual if actual $y=7$ and predicted $y=10$?

    Residual $=7-10=-3$. Actual minus predicted gives negative when point is below line.

45. What is the residual if the line predicts $y=12$ but the actual value is $y=15$?

    $15-12=3$. Residual = actual - predicted value.

46. Find the $y$ value on the line $y=3x+2$ when $x=4$.

    $y=14$. Substitute $x=4$: $y=3(4)+2=12+2$.

47. Identify the term for predicting $y$ for an $x$ value outside the data range using a fitted line.

    Extrapolation. Making predictions beyond the observed data range.

48. What word describes a scatter plot with no clear upward or downward trend?

    No association. The variables show no consistent relationship pattern.

49. Which direction indicates a negative association in a scatter plot?

    A downward trend: as $x$ increases, $y$ tends to decrease. The variables move in opposite directions.

50. What does it mean if a scatter plot shows a linear association?

    The points follow an approximately straight-line pattern. Linear association means the data forms a roughly straight pattern.

51. What does it mean if points are widely scattered around a fitted line?

    The linear model is a poor fit (weak linear association). Points deviate significantly from the line.

52. What does it mean if a residual is negative?

    The point is below the line (actual $y$ is less than predicted). Negative residual means the actual value is less than predicted.

53. What is an outlier in a scatter plot?

    A point far from the overall pattern of the other points. Outliers deviate significantly from the general data trend.

54. What does it mean if a residual is positive?

    The point is above the line (actual $y$ is greater than predicted). Positive residual means the actual value exceeds prediction.

55. Which option best describes a good informal line of fit for a linear scatter plot?

    A line through the middle of the data with about equal points above and below. Balances the data to minimize overall distance from points.

56. What is a negative linear association in a scatter plot?

    As $x$ increases, $y$ tends to decrease. Variables move in opposite directions.

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