You Are Wondering Whether the Relationship Between Being a Minority

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The Answer is a. (IV = seriousness; DV = arrest) and (0 = minor; 1 = serious).
b. H₀: There is no statistically significant difference between severity of the traffic violation and arrests after a traffic stop.
H₁: There is a statistically significant difference severity of the traffic violation and arrests after a traffic stop.
c. df = (c - 1) × (r - 1) = 1.
d. You reject the null hypothesis if χ² is greater than ±3.841.
e.
$a. (IV = seriousness; DV = arrest) and (0 = minor; 1 = serious). b. H0: There is no statistically significant difference between severity of the traffic violation and arrests after a traffic stop. H1: There is a statistically significant difference severity of the traffic violation and arrests after a traffic stop. c. df = (c - 1) × (r - 1) = 1. d. You reject the null hypothesis if χ2 is greater than ±3.841. e. \begin{array}{|l|c|c|c|c|c|} \hline ext { Cell } & O & E & O-E & (O- & (O- \ \hline \mathrm{A} & 20 & 16.46 & 3.54 & 12.5316 & 0.761337 \ \hline \mathrm{B} & 4 & 7.54 & -3.54 & 12.5316 & 1.662016 \ \hline \mathrm{C} & 4 & 7.54 & -3.54 & 12.5316 & 1.662016 \ \hline \mathrm{D} & 7 & 3.46 & 3.54 & 12.5316 & 3.62185 \ \hline & & & & \sum & 7.707218 \ \hline \end{array} \begin{aligned} \chi ^ { 2 } & = 7.7072 \ \phi & = \sqrt { \frac { 7.7072 } { 35 } } = 0.4693 \end{aligned} f. You reject the null hypothesis (alpha 0.05) because χ (7.7072) exceeds 3.841, meaning that there is a statistically significant moderate (phi = 0.4693) relationship between the severity of a traffic violation and arrest (χ2(1) = 7.7072, p < 0.05). Whereas 63.6% of individuals stopped for a severe traffic violation were subsequently arrested, only 16.7% of individuals stopped for a minor traffic violation were. The value of chi square also indicates that severity of the traffic violation has a greater influence on an arrest decision than being a minority driver. Arrest × Race (minor violation) \begin{array} { l c c c } & ext { Nonminority } & ext { Minority } & ext { Total } \ ext { No arrest } & ext { A } & ext { B } & \ & 15 & 5 & 20 \ ext { Arrest } & \mathrm { C } & \mathrm { D } & \ & 1 & 3 & 4 \ ext { Total } & 16 & 8 & 24 \end{array} \begin{array}{l} ext { Nonminority Minority Total }\ \begin{array} { l c c c } ext { No arrest } & \mathrm { A } & \mathrm { B } & \ & 15 & 5 & 20 \ & ( 93.8 \% ) & ( 62.5 \% ) & ( 83.3 \% ) \ ext { Arrest } & \mathrm { C } & \mathrm { D } & \ & 1 & 3 & 4 \ & ( 63 \% ) & ( 37.5 \% ) & ( 16.7 \% ) \ ext { Total } & 16 & 8 & 24 \ & ( 100 \% ) & ( 100 \% ) & ( 100 \% ) \end{array} \end{array} E(A) = (16 × 20)/24 = 13.33 E(B) = (8 × 20)/24 = 6.67 E(C) = (16× 4)/24 = 2.67 E(D) = (8 × 4)/24 = 1.33 χ2(1) = 3.7688, p > 0.05, not significant. \begin{array} { | l | c | c | c | c | c | } \hline ext { Cell } & O & E & O - E & ( O - & ( O - \ &&&&) ^ { 2 } & E ) ^ { 2 } / E \ \hline \mathrm { A } & 15 & 13.33 & 1.67 & 2.7889 & 0.20922 \ \hline \mathrm { B } & 5 & 6.67 & - 1.67 & 2.7889 & 0.418126 \ \hline \mathrm { C } & 1 & 2.67 & - 1.67 & 2.7889 & 1.044532 \ \hline \mathrm { D } & 3 & 1.33 & 1.67 & 2.7889 & 2.096917 \ \hline & & & & \sum & 3.768795 \ \hline \end{array} Arrest× Race (severe violation) \begin{array} { l c c c } & ext { Nonminority } & ext { Minority } & ext { Total } \ ext { No arrest } & ext { A } & ext { B } & \ & 4 & 0 & 4 \ ext { Arrest } & \mathrm { C } & \mathrm { D } & \ & 3 & 4 & 7 \ ext { Total } & 7 & 4 &11 \end{array} \begin{array} { c c c c } & ext { Nonminority } & ext { Minority } & ext { Total } \ ext { No arrest } & \mathrm { A } & \mathrm { B } & \ & 4 & 0 & 4 \ & ( 57.1 \% ) & ( 0.0 \% ) & ( 36.4 \% ) \ ext { Arrest } & \mathrm { C } & \mathrm { D } & \ & 3 & 4 & 7 \ & ( 42.9 \% ) & ( 100 \% ) & ( 63.6 \% ) \ ext { Total } & 7 & 4 & 11 \ & ( 100 \% ) & ( 100 \% ) & ( 100 \% ) \end{array} E(A) = (7 × 4)/11 = 2.55 E(B) = (4 × 4)/11 = 1.45 E(C) = (7 × 7)/11 = 4.45 E(D) = (4 × 7)/11 = 2.55 χ2(1) = 3.5715, p > 0.05, not significant. \begin{array} { | l | c | c | c | c | c | } \hline ext { Cell } & O & E & O - E & ( O - & ( O - \ &&&&E ) ^ { 2 } & E ) ^ { 2 } / E \ \hline \mathrm { A } & 4 & 2.55 & 1.45 & 2.1025 & 0.82451 \ \hline \mathrm { B } & 0 & 1.45 & - 1.45 & 2.1025 & 1.45 \ \hline \mathrm { C } & 3 & 4.45 & - 1.45 & 2.1025 & 0.472472 \ \hline \mathrm { D } & 4 & 2.55 & 1.45 & 2.1025 & 0.82451 \ \hline & & & & \sum & 3.571492 \ \hline \end{array} When we take the seriousness of the traffic violation into account, the relationship between being a minority and arrest disappears, with χ2 values not reaching the critical value of 3.841. Thus, the hypothesis that minorities have a higher overall arrest rate after a traffic stop is not supported with these data. Recall that the scenarios are purely hypothetical.$

\begin{array}{|l|c|c|c|c|c|}\hline ext { Cell } & O & E & O-E & (O- & (O- \\hline \mathrm{A} & 20 & 16.46 & 3.54 & 12.5316 & 0.761337 \\hline \mathrm{B} & 4 & 7.54 & -3.54 & 12.5316 & 1.662016 \\hline \mathrm{C} & 4 & 7.54 & -3.54 & 12.5316 & 1.662016 \\hline \mathrm{D} & 7 & 3.46 & 3.54 & 12.5316 & 3.62185 \\hline & & & & \sum & 7.707218 \\hline\end{array}

\begin{aligned}\chi ^ { 2 } & = 7.7072 \\phi & = \sqrt { \frac { 7.7072 } { 35 } } = 0.4693\end{aligned}

f. You reject the null hypothesis (alpha 0.05) because χ (7.7072) exceeds 3.841, meaning that there is a statistically significant moderate (phi = 0.4693) relationship between the severity of a traffic violation and arrest (χ²(1) = 7.7072, p < 0.05). Whereas 63.6% of individuals stopped for a severe traffic violation were subsequently arrested, only 16.7% of individuals stopped for a minor traffic violation were. The value of chi square also indicates that severity of the traffic violation has a greater influence on an arrest decision than being a minority driver.
Arrest × Race (minor violation)

\begin{array} { l c c c } & ext { Nonminority } & ext { Minority } & ext { Total } \ ext { No arrest } & ext { A } & ext { B } & \& 15 & 5 & 20 \ ext { Arrest } & \mathrm { C } & \mathrm { D } & \& 1 & 3 & 4 \ ext { Total } & 16 & 8 & 24\end{array}

\begin{array}{l} ext { Nonminority Minority Total }\\begin{array} { l c c c } ext { No arrest } & \mathrm { A } & \mathrm { B } & \& 15 & 5 & 20 \& ( 93.8 \% ) & ( 62.5 \% ) & ( 83.3 \% ) \ ext { Arrest } & \mathrm { C } & \mathrm { D } & \& 1 & 3 & 4 \& ( 63 \% ) & ( 37.5 \% ) & ( 16.7 \% ) \ ext { Total } & 16 & 8 & 24 \& ( 100 \% ) & ( 100 \% ) & ( 100 \% )\end{array}\end{array}

E(A) = (16 × 20)/24 = 13.33
E(B) = (8 × 20)/24 = 6.67
E(C) = (16× 4)/24 = 2.67
E(D) = (8 × 4)/24 = 1.33
χ²(1) = 3.7688, p > 0.05, not significant.

\begin{array} { | l | c | c | c | c | c | } \hline ext { Cell } & O & E & O - E & ( O - & ( O - \&&&&) ^ { 2 } & E ) ^ { 2 } / E \\hline \mathrm { A } & 15 & 13.33 & 1.67 & 2.7889 & 0.20922 \\hline \mathrm { B } & 5 & 6.67 & - 1.67 & 2.7889 & 0.418126 \\hline \mathrm { C } & 1 & 2.67 & - 1.67 & 2.7889 & 1.044532 \\hline \mathrm { D } & 3 & 1.33 & 1.67 & 2.7889 & 2.096917 \\hline & & & & \sum & 3.768795 \\hline\end{array}

Arrest× Race (severe violation)

\begin{array} { l c c c } & ext { Nonminority } & ext { Minority } & ext { Total } \ ext { No arrest } & ext { A } & ext { B } & \& 4 & 0 & 4 \ ext { Arrest } & \mathrm { C } & \mathrm { D } & \& 3 & 4 & 7 \ ext { Total } & 7 & 4 &11\end{array}

\begin{array} { c c c c } & ext { Nonminority } & ext { Minority } & ext { Total } \ ext { No arrest } & \mathrm { A } & \mathrm { B } & \& 4 & 0 & 4 \& ( 57.1 \% ) & ( 0.0 \% ) & ( 36.4 \% ) \ ext { Arrest } & \mathrm { C } & \mathrm { D } & \& 3 & 4 & 7 \& ( 42.9 \% ) & ( 100 \% ) & ( 63.6 \% ) \ ext { Total } & 7 & 4 & 11 \& ( 100 \% ) & ( 100 \% ) & ( 100 \% )\end{array}

E(A) = (7 × 4)/11 = 2.55
E(B) = (4 × 4)/11 = 1.45
E(C) = (7 × 7)/11 = 4.45
E(D) = (4 × 7)/11 = 2.55
χ²(1) = 3.5715, p > 0.05, not significant.

\begin{array} { | l | c | c | c | c | c | } \hline ext { Cell } & O & E & O - E & ( O - & ( O - \&&&&E ) ^ { 2 } & E ) ^ { 2 } / E \\hline \mathrm { A } & 4 & 2.55 & 1.45 & 2.1025 & 0.82451 \\hline \mathrm { B } & 0 & 1.45 & - 1.45 & 2.1025 & 1.45 \\hline \mathrm { C } & 3 & 4.45 & - 1.45 & 2.1025 & 0.472472 \\hline \mathrm { D } & 4 & 2.55 & 1.45 & 2.1025 & 0.82451 \\hline & & & & \sum & 3.571492 \\hline\end{array}

When we take the seriousness of the traffic violation into account, the relationship between being a minority and arrest disappears, with χ² values not reaching the critical value of 3.841. Thus, the hypothesis that minorities have a higher overall arrest rate after a traffic stop is not supported with these data. Recall that the scenarios are purely hypothetical.

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