Assessing Model Fit with Residuals - Statistics
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What does a slope of $0$ mean for a linear model relating $y$ to $x$?
What does a slope of $0$ mean for a linear model relating $y$ to $x$?
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$y$ is predicted to stay constant as $x$ changes. Zero slope means a horizontal line with no change in $y$.
$y$ is predicted to stay constant as $x$ changes. Zero slope means a horizontal line with no change in $y$.
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What does a negative slope indicate about the relationship between $x$ and $y$?
What does a negative slope indicate about the relationship between $x$ and $y$?
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As $x$ increases, predicted $y$ decreases. Negative slope indicates an inverse relationship between variables.
As $x$ increases, predicted $y$ decreases. Negative slope indicates an inverse relationship between variables.
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What does a positive intercept $b>0$ mean in context (when $x=0$ is meaningful)?
What does a positive intercept $b>0$ mean in context (when $x=0$ is meaningful)?
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The predicted starting value of $y$ is above $0$. Positive intercept means the line crosses above the origin.
The predicted starting value of $y$ is above $0$. Positive intercept means the line crosses above the origin.
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What should you check before interpreting the intercept $b$ as a real starting value?
What should you check before interpreting the intercept $b$ as a real starting value?
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Whether $x=0$ is within the data range and meaningful in context. Intercept interpretation requires $x=0$ to make practical sense.
Whether $x=0$ is within the data range and meaningful in context. Intercept interpretation requires $x=0$ to make practical sense.
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Identify the slope and intercept in the model $y=4.5x-18$.
Identify the slope and intercept in the model $y=4.5x-18$.
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Slope $=4.5$, intercept $=-18$. Read coefficients directly from standard form $y=mx+b$.
Slope $=4.5$, intercept $=-18$. Read coefficients directly from standard form $y=mx+b$.
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What is the slope between points $(2,7)$ and $(5,19)$, and what does it represent?
What is the slope between points $(2,7)$ and $(5,19)$, and what does it represent?
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$m= rac{19-7}{5-2}=4$; $y$ increases $4$ per $1$ increase in $x$. Use slope formula $m= rac{y_2-y_1}{x_2-x_1}$ between two points.
$m= rac{19-7}{5-2}=4$; $y$ increases $4$ per $1$ increase in $x$. Use slope formula $m= rac{y_2-y_1}{x_2-x_1}$ between two points.
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What is the intercept of the line with slope $3$ passing through $(2,11)$?
What is the intercept of the line with slope $3$ passing through $(2,11)$?
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$b=11-3(2)=5$. Use point-slope form: $b=y-mx$ with known slope and point.
$b=11-3(2)=5$. Use point-slope form: $b=y-mx$ with known slope and point.
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Which quantity does the slope represent: fixed fee or per-unit fee in $y=mx+b$?
Which quantity does the slope represent: fixed fee or per-unit fee in $y=mx+b$?
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Per-unit fee (change in $y$ per $1$ unit of $x$). Slope is the variable rate that depends on usage or quantity.
Per-unit fee (change in $y$ per $1$ unit of $x$). Slope is the variable rate that depends on usage or quantity.
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Which quantity does the intercept represent: fixed fee or per-unit fee in $y=mx+b$?
Which quantity does the intercept represent: fixed fee or per-unit fee in $y=mx+b$?
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Fixed fee (the value when $x=0$). Intercept is the constant base amount regardless of usage.
Fixed fee (the value when $x=0$). Intercept is the constant base amount regardless of usage.
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Interpret $m=2.5$ in context if $x$ is gallons and $y$ is dollars.
Interpret $m=2.5$ in context if $x$ is gallons and $y$ is dollars.
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Cost increases by $2.5$ dollars per gallon: $ rac{ ext{dollars}}{ ext{gallon}}$. Slope gives the price rate with appropriate unit ratio.
Cost increases by $2.5$ dollars per gallon: $ rac{ ext{dollars}}{ ext{gallon}}$. Slope gives the price rate with appropriate unit ratio.
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Interpret the intercept in $y=4.5x-18$ when $x=0$ represents the start of the situation.
Interpret the intercept in $y=4.5x-18$ when $x=0$ represents the start of the situation.
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Predicted $y$ at $x=0$ is $-18$ (a deficit or below-zero baseline). Negative intercept indicates a starting deficit or debt.
Predicted $y$ at $x=0$ is $-18$ (a deficit or below-zero baseline). Negative intercept indicates a starting deficit or debt.
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What is the meaning of the slope $m$ in a linear model $y=mx+b$ in context?
What is the meaning of the slope $m$ in a linear model $y=mx+b$ in context?
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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.
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.
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What is the meaning of the intercept $b$ in a linear model $y=mx+b$ in context?
What is the meaning of the intercept $b$ in a linear model $y=mx+b$ in context?
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Predicted value of $y$ when $x=0$ (the baseline or starting value). The intercept is where the line crosses the $y$-axis.
Predicted value of $y$ when $x=0$ (the baseline or starting value). The intercept is where the line crosses the $y$-axis.
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What are the units of the slope $m$ if $x$ is in hours and $y$ is in dollars?
What are the units of the slope $m$ if $x$ is in hours and $y$ is in dollars?
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Dollars per hour: $ rac{ ext{dollars}}{ ext{hour}}$. Units of slope are always $ rac{ ext{units of }y}{ ext{units of }x}$.
Dollars per hour: $ rac{ ext{dollars}}{ ext{hour}}$. Units of slope are always $ rac{ ext{units of }y}{ ext{units of }x}$.
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What are the units of the intercept $b$ if $y$ is measured in miles traveled?
What are the units of the intercept $b$ if $y$ is measured in miles traveled?
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Miles (same units as $y$). The intercept always has the same units as the dependent variable.
Miles (same units as $y$). The intercept always has the same units as the dependent variable.
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Identify the slope and intercept in the model $C=12t+35$.
Identify the slope and intercept in the model $C=12t+35$.
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Slope $=12$, intercept $=35$. In $y=mx+b$ form, $m$ is the coefficient of $x$ and $b$ is the constant.
Slope $=12$, intercept $=35$. In $y=mx+b$ form, $m$ is the coefficient of $x$ and $b$ is the constant.
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What does the slope mean in $C=12t+35$ if $t$ is hours and $C$ is dollars?
What does the slope mean in $C=12t+35$ if $t$ is hours and $C$ is dollars?
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$C$ increases by $12$ dollars for each additional hour. The slope tells us the hourly rate of cost increase.
$C$ increases by $12$ dollars for each additional hour. The slope tells us the hourly rate of cost increase.
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What does the intercept mean in $C=12t+35$ if $t$ is hours and $C$ is dollars?
What does the intercept mean in $C=12t+35$ if $t$ is hours and $C$ is dollars?
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Predicted cost at $t=0$ is $35$ dollars (a fixed fee). The intercept represents the initial or base cost when no time has passed.
Predicted cost at $t=0$ is $35$ dollars (a fixed fee). The intercept represents the initial or base cost when no time has passed.
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Interpret the slope in $T=-0.8d+22$ where $d$ is days and $T$ is temperature in $^ $.
Interpret the slope in $T=-0.8d+22$ where $d$ is days and $T$ is temperature in $^ $.
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Temperature decreases by $0.8^ $ per day. Negative slope means temperature drops as days increase.
Temperature decreases by $0.8^ $ per day. Negative slope means temperature drops as days increase.
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Interpret the intercept in $T=-0.8d+22$ where $d$ is days and $T$ is temperature in $^ $.
Interpret the intercept in $T=-0.8d+22$ where $d$ is days and $T$ is temperature in $^ $.
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Predicted temperature at $d=0$ is $22^ $. The intercept is the initial temperature on day zero.
Predicted temperature at $d=0$ is $22^ $. The intercept is the initial temperature on day zero.
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Identify the meaning of $m = 0$ in a linear model $y = mx + b$.
Identify the meaning of $m = 0$ in a linear model $y = mx + b$.
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$y$ is constant; no linear change with $x$. Zero slope means horizontal line with no change.
$y$ is constant; no linear change with $x$. Zero slope means horizontal line with no change.
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Which units does the slope $m$ have if $x$ is in hours and $y$ is in miles?
Which units does the slope $m$ have if $x$ is in hours and $y$ is in miles?
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Miles per hour. Units are $\frac{\text{miles}}{\text{hour}}$ from slope definition.
Miles per hour. Units are $\frac{\text{miles}}{\text{hour}}$ from slope definition.
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What does the intercept $b$ represent in a linear model $y = mx + b$ in context?
What does the intercept $b$ represent in a linear model $y = mx + b$ in context?
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Predicted value of $y$ when $x = 0$. The $y$-value where the line crosses the $y$-axis.
Predicted value of $y$ when $x = 0$. The $y$-value where the line crosses the $y$-axis.
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What is the $x$-intercept in terms of $m$ and $b$ for $y = mx + b$ with $m \ne 0$?
What is the $x$-intercept in terms of $m$ and $b$ for $y = mx + b$ with $m \ne 0$?
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$x = -\frac{b}{m}$. Set $y = 0$ and solve for $x$.
$x = -\frac{b}{m}$. Set $y = 0$ and solve for $x$.
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Identify the meaning of slope $m = 0.5$ when $x$ is miles driven and $y$ is gallons used.
Identify the meaning of slope $m = 0.5$ when $x$ is miles driven and $y$ is gallons used.
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Gallons used increase by $0.5$ per additional mile. Each mile driven uses $0.5$ more gallons.
Gallons used increase by $0.5$ per additional mile. Each mile driven uses $0.5$ more gallons.
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Find the predicted value of $y$ at $x = 5$ for the model $y = 2x - 3$.
Find the predicted value of $y$ at $x = 5$ for the model $y = 2x - 3$.
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$y = 7$. Substitute: $y = 2(5) - 3 = 10 - 3 = 7$.
$y = 7$. Substitute: $y = 2(5) - 3 = 10 - 3 = 7$.
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What does the slope $m$ represent in a linear model $y = mx + b$ for real data?
What does the slope $m$ represent in a linear model $y = mx + b$ for real data?
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Rate of change in $y$ for a $1$-unit increase in $x$. Slope measures how much $y$ changes per unit change in $x$.
Rate of change in $y$ for a $1$-unit increase in $x$. Slope measures how much $y$ changes per unit change in $x$.
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Find the predicted change in $y$ when $x$ increases by $3$ if the slope is $m = -1.2$.
Find the predicted change in $y$ when $x$ increases by $3$ if the slope is $m = -1.2$.
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$\Delta y = -3.6$. $\Delta y = m \cdot \Delta x = -1.2 \times 3 = -3.6$.
$\Delta y = -3.6$. $\Delta y = m \cdot \Delta x = -1.2 \times 3 = -3.6$.
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Choose the word that best describes using $y = mx + b$ to predict far beyond the data range.
Choose the word that best describes using $y = mx + b$ to predict far beyond the data range.
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Extrapolation. Predicting beyond observed data is extrapolation.
Extrapolation. Predicting beyond observed data is extrapolation.
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What does it mean in context if $x = 0$ is outside the observed data range?
What does it mean in context if $x = 0$ is outside the observed data range?
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The intercept may be an extrapolation and not meaningful. Predictions outside data range may be unreliable.
The intercept may be an extrapolation and not meaningful. Predictions outside data range may be unreliable.
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