Fitting Linear Functions to Data - Statistics and Probability

1. Choose the correct statement: correlation $r$ is unitless or depends on measurement units?

   $r$ is unitless. Correlation is a pure number, independent of measurement scales.

2. What does $r=-1$ mean for a scatterplot and linear fit?

   Perfect negative linear association. All points lie exactly on a downward-sloping line.

3. What is the range of possible values for the correlation coefficient $r$?

   $-1 \le r \le 1$. Correlation is bounded between perfect negative and perfect positive.

4. Identify the correlation coefficient if a linear fit reports $R^2=0.81$ and the slope is negative.

   $r=-0.9$. Since $r^2=0.81$ and slope is negative, $r=-\sqrt{0.81}$.

5. Identify the correlation coefficient if a linear fit reports $R^2=0.36$ and the slope is positive.

   $r=0.6$. Since $r^2=0.36$ and slope is positive, $r=+\sqrt{0.36}$.

6. Identify the stronger linear association: $r=-0.20$ or $r=0.05$.

   $r=-0.20$. Compare absolute values: $|-0.20|=0.20 > |0.05|=0.05$.

7. What does a correlation of $r=-0.93$ indicate about the linear fit quality?

   Very strong negative linear fit. Near -1 means points cluster tightly around downward line.

8. What does $r=1$ mean for a scatterplot and linear fit?

   Perfect positive linear association. All points lie exactly on an upward-sloping line.

9. Find $|r|$ if the correlation coefficient is $r=-0.34$.

   $|r|=0.34$. Absolute value removes the negative sign.

10. What does the sign of $r$ indicate about the direction of a linear relationship?

    $r>0$ positive; $r<0$ negative. Positive $r$ means upward trend; negative $r$ means downward trend.

11. Which value indicates a stronger linear association: $|r|=0.2$ or $|r|=0.8$?

    $|r|=0.8$. Closer to $|r|=1$ means stronger linear relationship.

12. What is a key limitation of using $r$ to describe a relationship?

    $r$ measures only linear association. Cannot detect curved or other nonlinear patterns.

13. What does $r=-1$ mean about the scatterplot and linear fit?

    Perfect negative linear association. All points lie exactly on a line with negative slope.

14. What does the sign of the correlation coefficient $r$ indicate about the association?

    Positive: increasing; negative: decreasing. Positive $r$ means $y$ increases with $x$; negative means $y$ decreases.

15. What is the range of the correlation coefficient $r$ for any data set?

    $-1 \le r \le 1$. Correlation is bounded between perfect negative and perfect positive.

16. What is the correlation coefficient $r$ if $R^2=0$ for a linear regression with an intercept?

    $r=0$. If $R^2=0$, then $r^2=0$, so $r=0$.

17. What happens to $r$ if all $x$-values are multiplied by $-3$?

    The sign of $r$ flips; magnitude unchanged. Negative scaling reverses direction but preserves strength.

18. Which statement is correct if $r=0.12$: the linear association is strong or weak?

    Weak. Values close to 0 indicate weak linear relationship.

19. Which option is correct: for data with a positive slope, can $r$ ever be negative?

    No, the sign of $r$ matches the slope. Correlation and slope always have the same sign in linear regression.

20. What does $r=0$ mean about linear association between two quantitative variables?

    No linear association. Variables have no linear relationship (may still have nonlinear).

21. Identify the main limitation: does a large $|r|$ guarantee a cause-and-effect relationship?

    No, correlation does not imply causation. High correlation shows association, not that one variable causes the other.

22. Which value shows a stronger linear association: $r=-0.81$ or $r=0.63$?

    $r=-0.81$. Compare absolute values: $|-0.81|=0.81 > |0.63|=0.63$.

23. What happens to $r$ if all $y$-values are multiplied by a positive constant $k$ (use $y'=ky$, $k>0$)?

    $r$ is unchanged. Scaling by positive constant preserves pattern and correlation.

24. What does a correlation of $r=0.97$ indicate about the linear fit quality?

    Very strong positive linear fit. Near 1 means points cluster tightly around upward line.

25. State the formula that relates $r$ and the coefficient of determination $r^2$.

    $r^2=(r)^2$. Square the correlation to get coefficient of determination.

26. Identify what an outlier typically does to correlation $r$ in a scatterplot with an otherwise linear trend.

    It can greatly change $r$. Outliers pull the line toward them, affecting correlation strength.

27. State the relationship between $r$ and $R^2$ for a linear regression with an intercept.

    $R^2=r^2$. Coefficient of determination equals correlation squared.

28. What does $r=0$ mean about linear association in the data?

    No linear association. Variables have no linear relationship, though other patterns may exist.

29. Which value shows a stronger linear association: $r=0.92$ or $r=0.48$?

    $r=0.92$. Compare absolute values: $|0.92|=0.92 > |0.48|=0.48$.

30. Which statement is correct if $r=-0.76$: the association is positive or negative?

    Negative. Negative $r$ indicates downward linear trend.

31. Find $r^2$ if the correlation coefficient is $r=-0.8$.

    $r^2=0.64$. Square the correlation: $(-0.8)^2 = 0.64$.

32. What happens to $r$ if all $x$-values are shifted by a constant $c$ (use $x'=x+c$)?

    $r$ is unchanged. Shifting doesn't change the pattern or strength of association.

33. What is the symbol for the correlation coefficient of a linear fit?

    $r$. Standard notation for Pearson's correlation coefficient.

34. What does $r=1$ mean about the scatterplot and linear fit?

    Perfect positive linear association. All points lie exactly on a line with positive slope.

35. Find the correct interpretation: if $r=-0.85$, is the linear association strong or weak, and positive or negative?

    Strong negative linear association. $|r|=0.85$ is close to 1, and negative sign shows decreasing trend.

36. Which situation can make $|r|$ misleadingly small even if a strong pattern exists: linear or curved trend?

    Curved (nonlinear) trend. Correlation only measures linear relationships, not curves.

37. What happens to $r$ if all $y$-values are multiplied by a negative constant $k$ (use $y'=ky$, $k<0$)?

    $r$ changes sign. Negative scaling reverses the direction of association.

38. Find the correct interpretation: if $r=0.12$, is the linear association strong or weak, and positive or negative?

    Weak positive linear association. $|r|=0.12$ is close to 0, and positive sign shows increasing trend.

39. What is a common effect of an outlier on the correlation coefficient $r$?

    It can greatly increase or decrease $r$. Outliers pull the line toward them, changing $r$ substantially.

40. What happens to $r$ if a constant $5$ is added to every $y$-value?

    $r$ is unchanged. Adding constants doesn't affect correlation strength or direction.

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