Statistics Flashcards: Fit Linear Models To Scatter Plots

What this deck covers

This deck focuses on Fit Linear Models To Scatter Plots, giving you a quick way to review the definitions, rules, and examples that matter most for Statistics.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

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

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

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

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

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

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

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

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

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

Answer: m=rac{3-7}{6-2}=-1. Apply slope formula with rise over run.

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

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

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

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

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

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

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

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

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

Answer: 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Answer: m=rac{10-2}{5-1}=2. Use slope formula m=rac{y_2-y_1}{x_2-x_1}.

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

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

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

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

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

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

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

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