Fit Linear Models to Scatter Plots - Statistics and Probability

1. Identify the predicted value for input $x$ using a fitted line $\hat{y}=mx+b$.

   Substitute $x$ to get $\hat{y}=mx+b$. Plug in $x$ value to calculate predicted $y$ value.

2. What is an outlier in a scatter plot when fitting a linear model?

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

3. Which option describes a good linear fit based on residuals plotted against $x$?

   Residuals are randomly scattered around $0$ with no pattern. Random residuals indicate the linear model fits well.

4. Find the line through $(2,5)$ with slope $m=3$ in the form $y=mx+b$.

   $y=3x-1$. Use point-slope form: $y-5=3(x-2)$, then simplify.

5. Which option best describes a positive linear association in a scatter plot?

   As $x$ increases, $y$ tends to increase. Positive slope means variables move in the same direction.

6. What does a positive slope $m>0$ mean in the context of a fitted linear model?

   As $x$ increases, $\hat{y}$ tends to increase. Positive slope means upward trend from left to right.

7. Find the slope between points $(2,7)$ and $(6,3)$ to model the trend line.

   $m=\frac{3-7}{6-2}=-1$. Apply slope formula with rise over run.

8. Find the linear model $\hat{y}=mx+b$ through points $(1,4)$ and $(3,10)$.

   $\hat{y}=3x+1$. Slope $m=3$, then $b=4-3(1)=1$.

9. Given $\hat{y}=2x+5$, what is the predicted value $\hat{y}$ when $x=7$?

   $\hat{y}=19$. Substitute: $\hat{y}=2(7)+5=14+5=19$.

10. What does a residual of $0$ mean for a point relative to the fitted line?

    The point lies exactly on the fitted line. Zero residual means observed equals predicted value.

11. What is the point-slope form of a line through $(x_1,y_1)$ with slope $m$?

    $y-y_1=m(x-x_1)$. Rearranges slope formula to express line equation.

12. What does it mean to fit a linear function to a scatter plot with a linear association?

    Choose a line $y=mx+b$ that models the trend of the points. Find the line that best represents the overall pattern of the data points.

13. What is the general form of a linear function used to model a scatter plot?

    $y=mx+b$. Standard form where $m$ is slope and $b$ is y-intercept.

14. Choose the better fit: residuals mostly positive or residuals balanced above and below $0$?

    Residuals balanced above and below $0$. Balanced residuals indicate unbiased fit.

15. What is the slope-intercept form of a linear model used to fit data?

    $\hat{y}=mx+b$. Standard form where $m$ is slope and $b$ is y-intercept.

16. Which option best describes a negative linear association in a scatter plot?

    As $x$ increases, $y$ tends to decrease. Negative slope means variables move in opposite directions.

17. Find $b$ if a fitted line has slope $m=4$ and passes through $(2,11)$.

    $b=3$. Using $11=4(2)+b$, so $b=11-8=3$.

18. Find the line through $(4,1)$ with slope $m=-2$ in the form $y=mx+b$.

    $y=-2x+9$. Substitute point into $y=mx+b$ to find $b$.

19. What is a residual for a data point $(x,y)$ relative to a fitted line $\hat{y}$?

    $y-\hat{y}$. Difference between actual and predicted y-values.

20. What does a negative slope $m<0$ mean in the context of a fitted linear model?

    As $x$ increases, $\hat{y}$ tends to decrease. Negative slope means downward trend from left to right.

21. What is the meaning of the intercept $b$ in the model $\hat{y}=mx+b$?

    Predicted value when $x=0$. Y-intercept is where the line crosses the y-axis.

22. Given $\hat{y}=1.5x+2$ and observed $(4,7)$, what is the residual $y-\hat{y}$?

    $-1$. $\hat{y}=1.5(4)+2=8$, so residual $=7-8=-1$.

23. What is the formula for the intercept $b$ given slope $m$ and a point $(x_1,y_1)$?

    $b=y_1-mx_1$. Rearranges point-slope form to solve for y-intercept.

24. Identify the line that best matches a positive linear trend: $\hat{y}=-2x+3$ or $\hat{y}=2x+3$?

    $\hat{y}=2x+3$. Positive slope ($m=2$) indicates positive trend.

25. What does a positive residual $y-\hat{y}>0$ indicate about the point?

    The point lies above the fitted line. Actual y-value exceeds predicted value.

26. What is the slope formula $m$ between two points $(x_1,y_1)$ and $(x_2,y_2)$?

    $m=\frac{y_2-y_1}{x_2-x_1}$. Calculates rise over run between two points.

27. Find the slope between points $(1,2)$ and $(5,10)$ to use in a linear model.

    $m=\frac{10-2}{5-1}=2$. Use slope formula $m=\frac{y_2-y_1}{x_2-x_1}$.

28. What does a negative residual $y-\hat{y}<0$ indicate about the point?

    The point lies below the fitted line. Actual y-value is less than predicted value.

29. What does the slope $m$ represent in a fitted line $y=mx+b$ for data?

    Average change in $y$ for each $1$-unit increase in $x$. Slope measures the rate of change between variables.

30. What does the intercept $b$ represent in a fitted line $y=mx+b$ for data?

    Predicted value of $y$ when $x=0$ (if meaningful). Y-intercept shows where the line crosses the y-axis.

31. Find the slope $m$ of the line through $(2,5)$ and $(6,13)$.

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

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