Fit Linear Models to Scatter Plots - Statistics
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What does a negative residual $y-\hat{y}<0$ indicate about the point?
What does a negative residual $y-\hat{y}<0$ indicate about the point?
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The point lies below the fitted line. Actual y-value is less than predicted value.
The point lies below the fitted line. Actual y-value is less than predicted value.
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What does a positive residual $y-\hat{y}>0$ indicate about the point?
What does a positive residual $y-\hat{y}>0$ indicate about the point?
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The point lies above the fitted line. Actual y-value exceeds predicted value.
The point lies above the fitted line. Actual y-value exceeds predicted value.
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What is the slope formula $m$ between two points $(x_1,y_1)$ and $(x_2,y_2)$?
What is the slope formula $m$ between two points $(x_1,y_1)$ and $(x_2,y_2)$?
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$m=\frac{y_2-y_1}{x_2-x_1}$. Calculates rise over run between two points.
$m=\frac{y_2-y_1}{x_2-x_1}$. Calculates rise over run between two points.
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What is the point-slope form of a line through $(x_1,y_1)$ with slope $m$?
What is the point-slope form of a line through $(x_1,y_1)$ with slope $m$?
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$y-y_1=m(x-x_1)$. Rearranges slope formula to express line equation.
$y-y_1=m(x-x_1)$. Rearranges slope formula to express line equation.
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Find the linear model $\hat{y}=mx+b$ through points $(1,4)$ and $(3,10)$.
Find the linear model $\hat{y}=mx+b$ through points $(1,4)$ and $(3,10)$.
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$\hat{y}=3x+1$. Slope $m=3$, then $b=4-3(1)=1$.
$\hat{y}=3x+1$. Slope $m=3$, then $b=4-3(1)=1$.
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Given $\hat{y}=2x+5$, what is the predicted value $\hat{y}$ when $x=7$?
Given $\hat{y}=2x+5$, what is the predicted value $\hat{y}$ when $x=7$?
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$\hat{y}=19$. Substitute: $\hat{y}=2(7)+5=14+5=19$.
$\hat{y}=19$. Substitute: $\hat{y}=2(7)+5=14+5=19$.
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Given $\hat{y}=1.5x+2$ and observed $(4,7)$, what is the residual $y-\hat{y}$?
Given $\hat{y}=1.5x+2$ and observed $(4,7)$, what is the residual $y-\hat{y}$?
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$-1$. $\hat{y}=1.5(4)+2=8$, so residual $=7-8=-1$.
$-1$. $\hat{y}=1.5(4)+2=8$, so residual $=7-8=-1$.
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What is the slope-intercept form of a linear model used to fit data?
What is the slope-intercept form of a linear model used to fit data?
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$\hat{y}=mx+b$. Standard form where $m$ is slope and $b$ is y-intercept.
$\hat{y}=mx+b$. Standard form where $m$ is slope and $b$ is y-intercept.
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What is the formula for the intercept $b$ given slope $m$ and a point $(x_1,y_1)$?
What is the formula for the intercept $b$ given slope $m$ and a point $(x_1,y_1)$?
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$b=y_1-mx_1$. Rearranges point-slope form to solve for y-intercept.
$b=y_1-mx_1$. Rearranges point-slope form to solve for y-intercept.
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What does a positive slope $m>0$ mean in the context of a fitted linear model?
What does a positive slope $m>0$ mean in the context of a fitted linear model?
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As $x$ increases, $\hat{y}$ tends to increase. Positive slope means upward trend from left to right.
As $x$ increases, $\hat{y}$ tends to increase. Positive slope means upward trend from left to right.
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What does a negative slope $m<0$ mean in the context of a fitted linear model?
What does a negative slope $m<0$ mean in the context of a fitted linear model?
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As $x$ increases, $\hat{y}$ tends to decrease. Negative slope means downward trend from left to right.
As $x$ increases, $\hat{y}$ tends to decrease. Negative slope means downward trend from left to right.
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What is the meaning of the intercept $b$ in the model $\hat{y}=mx+b$?
What is the meaning of the intercept $b$ in the model $\hat{y}=mx+b$?
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Predicted value when $x=0$. Y-intercept is where the line crosses the y-axis.
Predicted value when $x=0$. Y-intercept is where the line crosses the y-axis.
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Identify the predicted value for input $x$ using a fitted line $\hat{y}=mx+b$.
Identify the predicted value for input $x$ using a fitted line $\hat{y}=mx+b$.
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Substitute $x$ to get $\hat{y}=mx+b$. Plug in $x$ value to calculate predicted $y$ value.
Substitute $x$ to get $\hat{y}=mx+b$. Plug in $x$ value to calculate predicted $y$ value.
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Identify the line that best matches a positive linear trend: $\hat{y}=-2x+3$ or $\hat{y}=2x+3$?
Identify the line that best matches a positive linear trend: $\hat{y}=-2x+3$ or $\hat{y}=2x+3$?
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$\hat{y}=2x+3$. Positive slope ($m=2$) indicates positive trend.
$\hat{y}=2x+3$. Positive slope ($m=2$) indicates positive trend.
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Find $b$ if a fitted line has slope $m=4$ and passes through $(2,11)$.
Find $b$ if a fitted line has slope $m=4$ and passes through $(2,11)$.
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$b=3$. Using $11=4(2)+b$, so $b=11-8=3$.
$b=3$. Using $11=4(2)+b$, so $b=11-8=3$.
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Choose the better fit: residuals mostly positive or residuals balanced above and below $0$?
Choose the better fit: residuals mostly positive or residuals balanced above and below $0$?
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Residuals balanced above and below $0$. Balanced residuals indicate unbiased fit.
Residuals balanced above and below $0$. Balanced residuals indicate unbiased fit.
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Find the slope $m$ of the line through $(2,5)$ and $(6,13)$.
Find the slope $m$ of the line through $(2,5)$ and $(6,13)$.
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$m=2$. Using $m=\frac{13-5}{6-2}=\frac{8}{4}=2$.
$m=2$. Using $m=\frac{13-5}{6-2}=\frac{8}{4}=2$.
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What is a residual for a data point $(x,y)$ relative to a fitted line $\hat{y}$?
What is a residual for a data point $(x,y)$ relative to a fitted line $\hat{y}$?
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$y-\hat{y}$. Difference between actual and predicted y-values.
$y-\hat{y}$. Difference between actual and predicted y-values.
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What does it mean to fit a linear function to a scatter plot with a linear association?
What does it mean to fit a linear function to a scatter plot with a linear association?
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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.
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.
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What is the general form of a linear function used to model a scatter plot?
What is the general form of a linear function used to model a scatter plot?
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$y=mx+b$. Standard form where $m$ is slope and $b$ is y-intercept.
$y=mx+b$. Standard form where $m$ is slope and $b$ is y-intercept.
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What does the slope $m$ represent in a fitted line $y=mx+b$ for data?
What does the slope $m$ represent in a fitted line $y=mx+b$ for data?
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Average change in $y$ for each $1$-unit increase in $x$. Slope measures the rate of change between variables.
Average change in $y$ for each $1$-unit increase in $x$. Slope measures the rate of change between variables.
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Find the line through $(4,1)$ with slope $m=-2$ in the form $y=mx+b$.
Find the line through $(4,1)$ with slope $m=-2$ in the form $y=mx+b$.
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$y=-2x+9$. Substitute point into $y=mx+b$ to find $b$.
$y=-2x+9$. Substitute point into $y=mx+b$ to find $b$.
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Find the line through $(2,5)$ with slope $m=3$ in the form $y=mx+b$.
Find the line through $(2,5)$ with slope $m=3$ in the form $y=mx+b$.
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$y=3x-1$. Use point-slope form: $y-5=3(x-2)$, then simplify.
$y=3x-1$. Use point-slope form: $y-5=3(x-2)$, then simplify.
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Find the slope between points $(2,7)$ and $(6,3)$ to model the trend line.
Find the slope between points $(2,7)$ and $(6,3)$ to model the trend line.
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$m= rac{3-7}{6-2}=-1$. Apply slope formula with rise over run.
$m= rac{3-7}{6-2}=-1$. Apply slope formula with rise over run.
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Find the slope between points $(1,2)$ and $(5,10)$ to use in a linear model.
Find the slope between points $(1,2)$ and $(5,10)$ to use in a linear model.
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$m= rac{10-2}{5-1}=2$. Use slope formula $m= rac{y_2-y_1}{x_2-x_1}$.
$m= rac{10-2}{5-1}=2$. Use slope formula $m= rac{y_2-y_1}{x_2-x_1}$.
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Which option describes a good linear fit based on residuals plotted against $x$?
Which option describes a good linear fit based on residuals plotted against $x$?
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Residuals are randomly scattered around $0$ with no pattern. Random residuals indicate the linear model fits well.
Residuals are randomly scattered around $0$ with no pattern. Random residuals indicate the linear model fits well.
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What does a residual of $0$ mean for a point relative to the fitted line?
What does a residual of $0$ mean for a point relative to the fitted line?
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The point lies exactly on the fitted line. Zero residual means observed equals predicted value.
The point lies exactly on the fitted line. Zero residual means observed equals predicted value.
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What is an outlier in a scatter plot when fitting a linear model?
What is an outlier in a scatter plot when fitting a linear model?
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A point far from the overall linear pattern of the data. Outliers deviate significantly from the general trend.
A point far from the overall linear pattern of the data. Outliers deviate significantly from the general trend.
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Which option best describes a negative linear association in a scatter plot?
Which option best describes a negative linear association in a scatter plot?
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As $x$ increases, $y$ tends to decrease. Negative slope means variables move in opposite directions.
As $x$ increases, $y$ tends to decrease. Negative slope means variables move in opposite directions.
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Which option best describes a positive linear association in a scatter plot?
Which option best describes a positive linear association in a scatter plot?
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As $x$ increases, $y$ tends to increase. Positive slope means variables move in the same direction.
As $x$ increases, $y$ tends to increase. Positive slope means variables move in the same direction.
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