Fitting Linear Functions to Data - Statistics
Card 1 of 30
What does $r=0$ mean about linear association between two quantitative variables?
What does $r=0$ mean about linear association between two quantitative variables?
Tap to reveal answer
No linear association. Variables have no linear relationship (may still have nonlinear).
No linear association. Variables have no linear relationship (may still have nonlinear).
← Didn't Know|Knew It →
What does $r=1$ mean for a scatterplot and linear fit?
What does $r=1$ mean for a scatterplot and linear fit?
Tap to reveal answer
Perfect positive linear association. All points lie exactly on an upward-sloping line.
Perfect positive linear association. All points lie exactly on an upward-sloping line.
← Didn't Know|Knew It →
What does $r=-1$ mean for a scatterplot and linear fit?
What does $r=-1$ mean for a scatterplot and linear fit?
Tap to reveal answer
Perfect negative linear association. All points lie exactly on a downward-sloping line.
Perfect negative linear association. All points lie exactly on a downward-sloping line.
← Didn't Know|Knew It →
Which value shows a stronger linear association: $r=0.92$ or $r=0.48$?
Which value shows a stronger linear association: $r=0.92$ or $r=0.48$?
Tap to reveal answer
$r=0.92$. Compare absolute values: $|0.92|=0.92 > |0.48|=0.48$.
$r=0.92$. Compare absolute values: $|0.92|=0.92 > |0.48|=0.48$.
← Didn't Know|Knew It →
Which value shows a stronger linear association: $r=-0.81$ or $r=0.63$?
Which value shows a stronger linear association: $r=-0.81$ or $r=0.63$?
Tap to reveal answer
$r=-0.81$. Compare absolute values: $|-0.81|=0.81 > |0.63|=0.63$.
$r=-0.81$. Compare absolute values: $|-0.81|=0.81 > |0.63|=0.63$.
← Didn't Know|Knew It →
Identify the stronger linear association: $r=-0.20$ or $r=0.05$.
Identify the stronger linear association: $r=-0.20$ or $r=0.05$.
Tap to reveal answer
$r=-0.20$. Compare absolute values: $|-0.20|=0.20 > |0.05|=0.05$.
$r=-0.20$. Compare absolute values: $|-0.20|=0.20 > |0.05|=0.05$.
← Didn't Know|Knew It →
Which statement is correct if $r=-0.76$: the association is positive or negative?
Which statement is correct if $r=-0.76$: the association is positive or negative?
Tap to reveal answer
Negative. Negative $r$ indicates downward linear trend.
Negative. Negative $r$ indicates downward linear trend.
← Didn't Know|Knew It →
Which statement is correct if $r=0.12$: the linear association is strong or weak?
Which statement is correct if $r=0.12$: the linear association is strong or weak?
Tap to reveal answer
Weak. Values close to 0 indicate weak linear relationship.
Weak. Values close to 0 indicate weak linear relationship.
← Didn't Know|Knew It →
What does a correlation of $r=0.97$ indicate about the linear fit quality?
What does a correlation of $r=0.97$ indicate about the linear fit quality?
Tap to reveal answer
Very strong positive linear fit. Near 1 means points cluster tightly around upward line.
Very strong positive linear fit. Near 1 means points cluster tightly around upward line.
← Didn't Know|Knew It →
What does a correlation of $r=-0.93$ indicate about the linear fit quality?
What does a correlation of $r=-0.93$ indicate about the linear fit quality?
Tap to reveal answer
Very strong negative linear fit. Near -1 means points cluster tightly around downward line.
Very strong negative linear fit. Near -1 means points cluster tightly around downward line.
← Didn't Know|Knew It →
State the formula that relates $r$ and the coefficient of determination $r^2$.
State the formula that relates $r$ and the coefficient of determination $r^2$.
Tap to reveal answer
$r^2=(r)^2$. Square the correlation to get coefficient of determination.
$r^2=(r)^2$. Square the correlation to get coefficient of determination.
← Didn't Know|Knew It →
Find $r^2$ if the correlation coefficient is $r=-0.8$.
Find $r^2$ if the correlation coefficient is $r=-0.8$.
Tap to reveal answer
$r^2=0.64$. Square the correlation: $(-0.8)^2 = 0.64$.
$r^2=0.64$. Square the correlation: $(-0.8)^2 = 0.64$.
← Didn't Know|Knew It →
Find $|r|$ if the correlation coefficient is $r=-0.34$.
Find $|r|$ if the correlation coefficient is $r=-0.34$.
Tap to reveal answer
$|r|=0.34$. Absolute value removes the negative sign.
$|r|=0.34$. Absolute value removes the negative sign.
← Didn't Know|Knew It →
What happens to $r$ if all $x$-values are multiplied by $-3$?
What happens to $r$ if all $x$-values are multiplied by $-3$?
Tap to reveal answer
The sign of $r$ flips; magnitude unchanged. Negative scaling reverses direction but preserves strength.
The sign of $r$ flips; magnitude unchanged. Negative scaling reverses direction but preserves strength.
← Didn't Know|Knew It →
What happens to $r$ if a constant $5$ is added to every $y$-value?
What happens to $r$ if a constant $5$ is added to every $y$-value?
Tap to reveal answer
$r$ is unchanged. Adding constants doesn't affect correlation strength or direction.
$r$ is unchanged. Adding constants doesn't affect correlation strength or direction.
← Didn't Know|Knew It →
What is a key limitation of using $r$ to describe a relationship?
What is a key limitation of using $r$ to describe a relationship?
Tap to reveal answer
$r$ measures only linear association. Cannot detect curved or other nonlinear patterns.
$r$ measures only linear association. Cannot detect curved or other nonlinear patterns.
← Didn't Know|Knew It →
What is a common effect of an outlier on the correlation coefficient $r$?
What is a common effect of an outlier on the correlation coefficient $r$?
Tap to reveal answer
It can greatly increase or decrease $r$. Outliers pull the line toward them, changing $r$ substantially.
It can greatly increase or decrease $r$. Outliers pull the line toward them, changing $r$ substantially.
← Didn't Know|Knew It →
What is the symbol for the correlation coefficient of a linear fit?
What is the symbol for the correlation coefficient of a linear fit?
Tap to reveal answer
$r$. Standard notation for Pearson's correlation coefficient.
$r$. Standard notation for Pearson's correlation coefficient.
← Didn't Know|Knew It →
What is the range of possible values for the correlation coefficient $r$?
What is the range of possible values for the correlation coefficient $r$?
Tap to reveal answer
$-1 \le r \le 1$. Correlation is bounded between perfect negative and perfect positive.
$-1 \le r \le 1$. Correlation is bounded between perfect negative and perfect positive.
← Didn't Know|Knew It →
What does the sign of $r$ indicate about the direction of a linear relationship?
What does the sign of $r$ indicate about the direction of a linear relationship?
Tap to reveal answer
$r>0$ positive; $r<0$ negative. Positive $r$ means upward trend; negative $r$ means downward trend.
$r>0$ positive; $r<0$ negative. Positive $r$ means upward trend; negative $r$ means downward trend.
← Didn't Know|Knew It →
Choose the correct statement: correlation $r$ is unitless or depends on measurement units?
Choose the correct statement: correlation $r$ is unitless or depends on measurement units?
Tap to reveal answer
$r$ is unitless. Correlation is a pure number, independent of measurement scales.
$r$ is unitless. Correlation is a pure number, independent of measurement scales.
← Didn't Know|Knew It →
Identify the correlation coefficient if a linear fit reports $R^2=0.36$ and the slope is positive.
Identify the correlation coefficient if a linear fit reports $R^2=0.36$ and the slope is positive.
Tap to reveal answer
$r=0.6$. Since $r^2=0.36$ and slope is positive, $r=+\sqrt{0.36}$.
$r=0.6$. Since $r^2=0.36$ and slope is positive, $r=+\sqrt{0.36}$.
← Didn't Know|Knew It →
Identify what an outlier typically does to correlation $r$ in a scatterplot with an otherwise linear trend.
Identify what an outlier typically does to correlation $r$ in a scatterplot with an otherwise linear trend.
Tap to reveal answer
It can greatly change $r$. Outliers pull the line toward them, affecting correlation strength.
It can greatly change $r$. Outliers pull the line toward them, affecting correlation strength.
← Didn't Know|Knew It →
Identify the correlation coefficient if a linear fit reports $R^2=0.81$ and the slope is negative.
Identify the correlation coefficient if a linear fit reports $R^2=0.81$ and the slope is negative.
Tap to reveal answer
$r=-0.9$. Since $r^2=0.81$ and slope is negative, $r=-\sqrt{0.81}$.
$r=-0.9$. Since $r^2=0.81$ and slope is negative, $r=-\sqrt{0.81}$.
← Didn't Know|Knew It →
Which option is correct: for data with a positive slope, can $r$ ever be negative?
Which option is correct: for data with a positive slope, can $r$ ever be negative?
Tap to reveal answer
No, the sign of $r$ matches the slope. Correlation and slope always have the same sign in linear regression.
No, the sign of $r$ matches the slope. Correlation and slope always have the same sign in linear regression.
← Didn't Know|Knew It →
State the relationship between $r$ and $R^2$ for a linear regression with an intercept.
State the relationship between $r$ and $R^2$ for a linear regression with an intercept.
Tap to reveal answer
$R^2=r^2$. Coefficient of determination equals correlation squared.
$R^2=r^2$. Coefficient of determination equals correlation squared.
← Didn't Know|Knew It →
What happens to $r$ if all $x$-values are shifted by a constant $c$ (use $x'=x+c$)?
What happens to $r$ if all $x$-values are shifted by a constant $c$ (use $x'=x+c$)?
Tap to reveal answer
$r$ is unchanged. Shifting doesn't change the pattern or strength of association.
$r$ is unchanged. Shifting doesn't change the pattern or strength of association.
← Didn't Know|Knew It →
What happens to $r$ if all $y$-values are multiplied by a positive constant $k$ (use $y'=ky$, $k>0$)?
What happens to $r$ if all $y$-values are multiplied by a positive constant $k$ (use $y'=ky$, $k>0$)?
Tap to reveal answer
$r$ is unchanged. Scaling by positive constant preserves pattern and correlation.
$r$ is unchanged. Scaling by positive constant preserves pattern and correlation.
← Didn't Know|Knew It →
What happens to $r$ if all $y$-values are multiplied by a negative constant $k$ (use $y'=ky$, $k<0$)?
What happens to $r$ if all $y$-values are multiplied by a negative constant $k$ (use $y'=ky$, $k<0$)?
Tap to reveal answer
$r$ changes sign. Negative scaling reverses the direction of association.
$r$ changes sign. Negative scaling reverses the direction of association.
← Didn't Know|Knew It →
What is the correlation coefficient $r$ if $R^2=0$ for a linear regression with an intercept?
What is the correlation coefficient $r$ if $R^2=0$ for a linear regression with an intercept?
Tap to reveal answer
$r=0$. If $R^2=0$, then $r^2=0$, so $r=0$.
$r=0$. If $R^2=0$, then $r^2=0$, so $r=0$.
← Didn't Know|Knew It →