Calculating Chi-Squared


\( 2 \times 2 \) Contingency Table

The following shows the results of a survey of a sample of \( 200 \) randomly chosen adults classified according to gender and sport. These are called observed values or observed frequencies.
$$ \begin{array}{|c|c|c|c|} \hline
& \text{loves sport} & \text{hates sport} & \text{sum} \\ \hline
\text{male} & 72 & 48 & 120 \\ \hline
\text{female} & 18 & 62 & 80 \\ \hline
\text{sum} & 90 & 110 & 200 \\ \hline
\end{array} $$

Expected Frequency Table

$$ \begin{array}{|c|c|c|c|} \hline
& \text{loves sport} & \text{hates sport} & \text{sum} \\ \hline
\text{male} & \dfrac{120 \times 90}{200} = 54 & \dfrac{120 \times 110}{200} = 66 & 120 \\ \hline
\text{female} & \dfrac{80 \times 90}{200} = 36 & \dfrac{80 \times 110}{200} = 44 & 80 \\ \hline
\text{sum} & 90 & 110 & 200 \\ \hline
\end{array} $$

Calculating \( \chi^{2} \)

$$ \chi^{2} = \sum \dfrac{(f_o – f_e)^2}{f_e} $$

\( f_o \) is an observed frequency
\( f_e \) is an expected frequency

$$ \begin{array}{|c|c|r|c|c|} \hline
f_o & f_e & f_o – f_e & (f_o – f_e)^2 & \dfrac{(f_o – f_e)^2}{f_e} \\ \hline
72 & 54 & 18 & 324 & 6.0 \\ \hline
48 & 66 & -18 & 324 & 4.9 \\ \hline
18 & 36 & -18 & 324 & 9.0 \\ \hline
62 & 44 & 18 & 324 & 7.4 \\ \hline
&&& \text{sum} & 27.3 \\ \hline
\end{array} \\
\therefore \chi^2 = 27.3
$$

\( 2 \times 3 \) Contingency Table

The following shows the results of a survey of a sample of \( 500 \) randomly chosen adults classified according to gender and political preferences. These are called observed values or observed frequencies.
$$ \begin{array}{|c|c|c|c|c|} \hline
& \text{Liberal} & \text{neutral} & \text{Democrates} & \text{sum} \\ \hline
\text{male} & 120 & 40 & 90 & 260 \\ \hline
\text{female} & 105 & 50 & 95 & 240 \\ \hline
\text{sum} & 225 & 90 & 185 & 500 \\ \hline
\end{array} $$

Expected Frequency Table

$$ \begin{array}{|r|r|r|r|r|} \hline
& \text{Liberal} & \text{neutral} & \text{Democrats} & \text{sum} \\ \hline
\text{male} & \dfrac{260 \times 225}{500} = 117 & \dfrac{260 \times 90}{500} = 46.8 & \dfrac{260 \times 185}{500} = 96.2 & 260 \\ \hline
\text{female} & \dfrac{240 \times 225}{500} = 108 & \dfrac{240 \times 90}{500} = 43.2 & \dfrac{240 \times 185}{500} = 88.8 & 240 \\ \hline
\text{sum} & 225 & 90 & 185 & 500 \\ \hline
\end{array} $$

Calculating \( \chi^{2} \)

$$ \chi^{2} = \sum \dfrac{(f_o – f_e)^2}{f_e} $$

\( f_o \) is an observed frequency
\( f_e \) is an expected frequency

$$ \begin{array}{|r|r|r|r|r|} \hline
f_o & f_e & f_o – f_e & (f_o – f_e)^2 & \dfrac{(f_o – f_e)^2}{f_e} \\ \hline
120 & 117.0 & 3.0 & 9.00 & 0.0769 \\ \hline
40 & 46.8 & -6.8 & 46.24 & 0.9880 \\ \hline
100 & 96.2 & 3.8 & 14.44 & 0.1501 \\ \hline
105 & 108.0 & -3.0 & 9.00 & 0.0833 \\ \hline
50 & 43.2 & 6.8 & 46.24 & 1.0704 \\ \hline
85 & 88.8 & -3.8 & 14.44 & 0.1626 \\ \hline
&&& \text{sum} & 2.5314 \\ \hline
\end{array} \\
\therefore \chi^2 = 2.5314
$$


Absolute Value Algebra Arithmetic Mean Arithmetic Sequence Binomial Expansion Binomial Theorem Chain Rule Circle Geometry Common Difference Common Ratio Compound Interest Cyclic Quadrilateral Differentiation Discriminant Double-Angle Formula Equation Exponent Exponential Function Factorials Functions Geometric Mean Geometric Sequence Geometric Series Inequality Integration Integration by Parts Kinematics Logarithm Logarithmic Functions Mathematical Induction Polynomial Probability Product Rule Proof Quadratic Quotient Rule Rational Functions Sequence Sketching Graphs Surds Transformation Trigonometric Functions Trigonometric Properties VCE Mathematics Volume




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