Assessing Model Fit with Residuals - Statistics and Probability

1. Identify the meaning of $m < 0$ in a linear model relating $x$ to $y$.

   $y$ decreases as $x$ increases. Negative slope means inverse relationship between variables.

2. Identify the meaning of $m = 0$ in a linear model $y = mx + b$.

   $y$ is constant; no linear change with $x$. Zero slope means horizontal line with no change.

3. Choose the word that best describes using $y = mx + b$ to predict far beyond the data range.

   Extrapolation. Predicting beyond observed data is extrapolation.

4. What are the units of the intercept $b$ if $y$ is measured in miles traveled?

   Miles (same units as $y$). The intercept always has the same units as the dependent variable.

5. Identify the slope and intercept in the model $C=12t+35$.

   Slope $=12$, intercept $=35$. In $y=mx+b$ form, $m$ is the coefficient of $x$ and $b$ is the constant.

6. What does a negative slope indicate about the relationship between $x$ and $y$?

   As $x$ increases, predicted $y$ decreases. Negative slope indicates an inverse relationship between variables.

7. What is the intercept $b$ in terms of $m$ and a point $(x_1,y_1)$ on the line?

   $b = y_1 - mx_1$. Rearrange $y_1 = mx_1 + b$ to solve for $b$.

8. Which quantity does the slope represent: fixed fee or per-unit fee in $y=mx+b$?

   Per-unit fee (change in $y$ per $1$ unit of $x$). Slope is the variable rate that depends on usage or quantity.

9. Which units does the intercept $b$ have in $y = mx + b$ if $y$ is measured in dollars?

   Dollars. Intercept has same units as the dependent variable $y$.

10. Interpret $m=2.5$ in context if $x$ is gallons and $y$ is dollars.

    Cost increases by $2.5$ dollars per gallon: $\frac{\text{dollars}}{\text{gallon}}$. Slope gives the price rate with appropriate unit ratio.

11. What does it mean in context if $x = 0$ is outside the observed data range?

    The intercept may be an extrapolation and not meaningful. Predictions outside data range may be unreliable.

12. What is the intercept of the line with slope $3$ passing through $(2,11)$?

    $b=11-3(2)=5$. Use point-slope form: $b=y-mx$ with known slope and point.

13. What are the units of the slope $m$ if $x$ is in hours and $y$ is in dollars?

    Dollars per hour: $\frac{\text{dollars}}{\text{hour}}$. Units of slope are always $\frac{\text{units of }y}{\text{units of }x}$.

14. What is the $x$-intercept in terms of $m$ and $b$ for $y = mx + b$ with $m \ne 0$?

    $x = -\frac{b}{m}$. Set $y = 0$ and solve for $x$.

15. What does the intercept $b$ represent in a linear model $y = mx + b$ in context?

    Predicted value of $y$ when $x = 0$. The $y$-value where the line crosses the $y$-axis.

16. What does a positive intercept $b>0$ mean in context (when $x=0$ is meaningful)?

    The predicted starting value of $y$ is above $0$. Positive intercept means the line crosses above the origin.

17. What is the slope between points $(x_1,y_1)$ and $(x_2,y_2)$?

    $m = \frac{y_2 - y_1}{x_2 - x_1}$. Rise over run formula for slope between two points.

18. What does a positive intercept $b > 0$ indicate about the predicted value at $x = 0$?

    The model predicts $y$ is positive when $x = 0$. Positive $b$ means $y > 0$ when $x = 0$.

19. Interpret the intercept $b = 10$ in $y = 4x + 10$ where $x$ is weeks and $y$ is dollars.

    Predicted $y$ is $10 at $x = 0$ weeks. At week $0$, the model predicts $10.

20. What is the slope between points $(2,7)$ and $(5,19)$, and what does it represent?

    $m=\frac{19-7}{5-2}=4$; $y$ increases $4$ per $1$ increase in $x$. Use slope formula $m=\frac{y_2-y_1}{x_2-x_1}$ between two points.

21. Interpret the intercept in $T=-0.8d+22$ where $d$ is days and $T$ is temperature in $^ $.

    Predicted temperature at $d=0$ is $22^ $. The intercept is the initial temperature on day zero.

22. What should you check before interpreting the intercept $b$ as a real starting value?

    Whether $x=0$ is within the data range and meaningful in context. Intercept interpretation requires $x=0$ to make practical sense.

23. Find the slope $m$ of the model passing through $(2,5)$ and $(6,13)$.

    $m = 2$. $m = \frac{13-5}{6-2} = \frac{8}{4} = 2$.

24. What does the slope mean in $C=12t+35$ if $t$ is hours and $C$ is dollars?

    $C$ increases by $12$ dollars for each additional hour. The slope tells us the hourly rate of cost increase.

25. Which units does the slope $m$ have if $x$ is in hours and $y$ is in miles?

    Miles per hour. Units are $\frac{\text{miles}}{\text{hour}}$ from slope definition.

26. What does the slope $m$ represent in a linear model $y = mx + b$ for real data?

    Rate of change in $y$ for a $1$-unit increase in $x$. Slope measures how much $y$ changes per unit change in $x$.

27. Find the predicted change in $y$ when $x$ increases by $3$ if the slope is $m = -1.2$.

    $\Delta y = -3.6$. $\Delta y = m \cdot \Delta x = -1.2 \times 3 = -3.6$.

28. Interpret the slope $m = 4$ in the model $y = 4x + 10$ where $x$ is weeks and $y$ is dollars.

    $y$ increases by $4 per week. Slope of $4$ means $4 increase per week.

29. What is the meaning of the slope $m$ in a linear model $y=mx+b$ in context?

    Rate of change: change in $y$ for each $1$-unit increase in $x$. Slope represents how much $y$ changes when $x$ increases by one unit.

30. What is the meaning of the intercept $b$ in a linear model $y=mx+b$ in context?

    Predicted value of $y$ when $x=0$ (the baseline or starting value). The intercept is where the line crosses the $y$-axis.

31. What does a slope of $0$ mean for a linear model relating $y$ to $x$?

    $y$ is predicted to stay constant as $x$ changes. Zero slope means a horizontal line with no change in $y$.

32. Find the intercept $b$ for $y = 3x + b$ if the line passes through $(2,11)$.

    $b = 5$. Substitute: $11 = 3(2) + b$, so $b = 11 - 6 = 5$.

33. Find the predicted value of $y$ at $x = 5$ for the model $y = 2x - 3$.

    $y = 7$. Substitute: $y = 2(5) - 3 = 10 - 3 = 7$.

34. Interpret the slope in $T=-0.8d+22$ where $d$ is days and $T$ is temperature in $^ $.

    Temperature decreases by $0.8^ $ per day. Negative slope means temperature drops as days increase.

35. Which quantity does the intercept represent: fixed fee or per-unit fee in $y=mx+b$?

    Fixed fee (the value when $x=0$). Intercept is the constant base amount regardless of usage.

36. Interpret the intercept in $y=4.5x-18$ when $x=0$ represents the start of the situation.

    Predicted $y$ at $x=0$ is $-18$ (a deficit or below-zero baseline). Negative intercept indicates a starting deficit or debt.

37. Identify the meaning of slope $m = 0.5$ when $x$ is miles driven and $y$ is gallons used.

    Gallons used increase by $0.5$ per additional mile. Each mile driven uses $0.5$ more gallons.

38. What does the intercept mean in $C=12t+35$ if $t$ is hours and $C$ is dollars?

    Predicted cost at $t=0$ is $35$ dollars (a fixed fee). The intercept represents the initial or base cost when no time has passed.

39. Identify the slope and intercept in the model $y = -2.5x + 7$.

    $m = -2.5$, $b = 7$. Read coefficients directly from standard form.

40. Identify the slope and intercept in the model $y=4.5x-18$.

    Slope $=4.5$, intercept $=-18$. Read coefficients directly from standard form $y=mx+b$.

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