Summarizing Categorical Data: Two-Way Tables - Statistics
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Find the marginal relative frequency: row total $30$, grand total $120$.
Find the marginal relative frequency: row total $30$, grand total $120$.
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$\frac{30}{120}=0.25$. Divide the row total by the grand total.
$\frac{30}{120}=0.25$. Divide the row total by the grand total.
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Find $P(\text{Yes}\mid\text{Group A})$: Yes in Group A $18$, Group A total $60$.
Find $P(\text{Yes}\mid\text{Group A})$: Yes in Group A $18$, Group A total $60$.
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$\frac{18}{60}=0.30$. Divide Yes count by Group A total for conditional probability.
$\frac{18}{60}=0.30$. Divide Yes count by Group A total for conditional probability.
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What is a relative frequency?
What is a relative frequency?
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A proportion: $
frac{\text{count}}{\text{total}}$. Converts counts to proportions for comparison.
A proportion: $ frac{\text{count}}{\text{total}}$. Converts counts to proportions for comparison.
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Find the joint relative frequency: cell count $12$, grand total $80$.
Find the joint relative frequency: cell count $12$, grand total $80$.
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$\frac{12}{80}=0.15$. Apply the joint relative frequency formula.
$\frac{12}{80}=0.15$. Apply the joint relative frequency formula.
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Identify the association: $P(\text{Yes}\mid A)=0.70$ and $P(\text{Yes}\mid B)=0.20$.
Identify the association: $P(\text{Yes}\mid A)=0.70$ and $P(\text{Yes}\mid B)=0.20$.
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Association is suggested because the conditional proportions differ. Large difference (0.70 vs 0.20) indicates variables are related.
Association is suggested because the conditional proportions differ. Large difference (0.70 vs 0.20) indicates variables are related.
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What is a two-way frequency table used to summarize?
What is a two-way frequency table used to summarize?
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Counts for two categorical variables, organized by rows and columns. Two-way tables display relationships between two categorical variables.
Counts for two categorical variables, organized by rows and columns. Two-way tables display relationships between two categorical variables.
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What is the grand total in a two-way table?
What is the grand total in a two-way table?
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The sum of all cell frequencies in the table. Add all interior cells to get the overall total.
The sum of all cell frequencies in the table. Add all interior cells to get the overall total.
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What is a marginal frequency in a two-way table?
What is a marginal frequency in a two-way table?
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A row total or column total (a total along the margin). Found at the edges where row/column totals are displayed.
A row total or column total (a total along the margin). Found at the edges where row/column totals are displayed.
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What is the sum of all joint relative frequencies in a complete two-way table?
What is the sum of all joint relative frequencies in a complete two-way table?
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$1$. All proportions must sum to the whole.
$1$. All proportions must sum to the whole.
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What pattern in conditional relative frequencies suggests an association?
What pattern in conditional relative frequencies suggests an association?
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Conditional relative frequencies differ noticeably between groups. Different distributions indicate dependence between variables.
Conditional relative frequencies differ noticeably between groups. Different distributions indicate dependence between variables.
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What condition indicates no association between two categorical variables using conditional relative frequencies?
What condition indicates no association between two categorical variables using conditional relative frequencies?
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Conditional relative frequencies are (approximately) the same across groups. Independence means the condition doesn't affect the distribution.
Conditional relative frequencies are (approximately) the same across groups. Independence means the condition doesn't affect the distribution.
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In a two-way table, what is the sum of the relative frequencies within a fixed row when row-conditional frequencies are used?
In a two-way table, what is the sum of the relative frequencies within a fixed row when row-conditional frequencies are used?
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$1$. Row-conditional frequencies distribute the row total completely.
$1$. Row-conditional frequencies distribute the row total completely.
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Identify the meaning of $P(A)$ in a two-way table context.
Identify the meaning of $P(A)$ in a two-way table context.
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The marginal proportion for category $A$ (row or column total over grand total). Represents the overall probability of event $A$ occurring.
The marginal proportion for category $A$ (row or column total over grand total). Represents the overall probability of event $A$ occurring.
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Identify the meaning of $P(A\cap B)$ in a two-way table context.
Identify the meaning of $P(A\cap B)$ in a two-way table context.
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The joint proportion in the cell where $A$ and $B$ occur together. Represents the probability of both events occurring together.
The joint proportion in the cell where $A$ and $B$ occur together. Represents the probability of both events occurring together.
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Which denominator is used for $P(A\mid B)$ in a two-way table?
Which denominator is used for $P(A\mid B)$ in a two-way table?
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The total for category $B$ (the given condition). Conditional probability uses the condition's total as denominator.
The total for category $B$ (the given condition). Conditional probability uses the condition's total as denominator.
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State the formula for a conditional relative frequency of row $i$ given column $j$.
State the formula for a conditional relative frequency of row $i$ given column $j$.
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$\frac{\text{cell}(i,j)}{\text{column total } j}$. Shows distribution of rows within a specific column.
$\frac{\text{cell}(i,j)}{\text{column total } j}$. Shows distribution of rows within a specific column.
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State the formula for a conditional relative frequency of column $j$ given row $i$.
State the formula for a conditional relative frequency of column $j$ given row $i$.
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$\frac{\text{cell}(i,j)}{\text{row total } i}$. Shows distribution of columns within a specific row.
$\frac{\text{cell}(i,j)}{\text{row total } i}$. Shows distribution of columns within a specific row.
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State the formula for a marginal relative frequency for row $i$.
State the formula for a marginal relative frequency for row $i$.
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$\frac{\text{row total } i}{\text{grand total}}$. Shows what proportion of all data is in that row.
$\frac{\text{row total } i}{\text{grand total}}$. Shows what proportion of all data is in that row.
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State the formula for a joint relative frequency for cell $(i,j)$.
State the formula for a joint relative frequency for cell $(i,j)$.
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$\frac{\text{cell}(i,j)}{\text{grand total}}$. Divides the cell count by the grand total.
$\frac{\text{cell}(i,j)}{\text{grand total}}$. Divides the cell count by the grand total.
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What is a correct interpretation of a marginal relative frequency such as $\frac{n_{i\cdot}}{N}$?
What is a correct interpretation of a marginal relative frequency such as $\frac{n_{i\cdot}}{N}$?
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The proportion of all individuals in row category $R_i$. Marginal ignores the other variable and gives overall proportion.
The proportion of all individuals in row category $R_i$. Marginal ignores the other variable and gives overall proportion.
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What is a correct interpretation of a joint relative frequency such as $\frac{n_{ij}}{N}$?
What is a correct interpretation of a joint relative frequency such as $\frac{n_{ij}}{N}$?
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The proportion of all individuals in both categories $R_i$ and $C_j$. Joint means the proportion satisfying both conditions simultaneously.
The proportion of all individuals in both categories $R_i$ and $C_j$. Joint means the proportion satisfying both conditions simultaneously.
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What is the conditional relative frequency of row category $R_i$ given column category $C_j$?
What is the conditional relative frequency of row category $R_i$ given column category $C_j$?
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$\frac{n_{ij}}{n_{\cdot j}}$. Divide cell count by column total for proportion within that column.
$\frac{n_{ij}}{n_{\cdot j}}$. Divide cell count by column total for proportion within that column.
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What is the correct comparison to check association using conditional relative frequencies?
What is the correct comparison to check association using conditional relative frequencies?
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Compare $P(A\mid B_1)$ to $P(A\mid B_2)$ (and other $B$ categories). Different conditional probabilities indicate association between variables.
Compare $P(A\mid B_1)$ to $P(A\mid B_2)$ (and other $B$ categories). Different conditional probabilities indicate association between variables.
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What condition suggests no association between two categorical variables in a two-way table?
What condition suggests no association between two categorical variables in a two-way table?
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Conditional relative frequencies are (approximately) equal across groups. Independence means conditioning doesn't change the proportions.
Conditional relative frequencies are (approximately) equal across groups. Independence means conditioning doesn't change the proportions.
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Identify the denominator for $P(A\mid B)$ when computed from a two-way table.
Identify the denominator for $P(A\mid B)$ when computed from a two-way table.
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The total count (or total proportion) for category $B$. Condition on $B$ means we only consider those in category $B$.
The total count (or total proportion) for category $B$. Condition on $B$ means we only consider those in category $B$.
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Which phrase best matches $P(A\mid B)$ in a two-way table context?
Which phrase best matches $P(A\mid B)$ in a two-way table context?
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The proportion in category $A$ among those in category $B$. Conditional probability restricts to those already in category $B$.
The proportion in category $A$ among those in category $B$. Conditional probability restricts to those already in category $B$.
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Identify whether variables are associated if $P(A\mid B_1)=0.40$ and $P(A\mid B_2)=0.10$.
Identify whether variables are associated if $P(A\mid B_1)=0.40$ and $P(A\mid B_2)=0.10$.
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Associated (conditional relative frequencies differ substantially). Large difference (0.40 vs 0.10) indicates strong association.
Associated (conditional relative frequencies differ substantially). Large difference (0.40 vs 0.10) indicates strong association.
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Find the conditional relative frequency $P(C_1\mid R_2)$ if $n_{21}=9$ and $n_{2\cdot}=30$.
Find the conditional relative frequency $P(C_1\mid R_2)$ if $n_{21}=9$ and $n_{2\cdot}=30$.
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$\frac{9}{30}=0.30$. Conditional divides cell count by the conditioning category's total.
$\frac{9}{30}=0.30$. Conditional divides cell count by the conditioning category's total.
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Find the marginal relative frequency for a row total $18$ when the grand total is $72$.
Find the marginal relative frequency for a row total $18$ when the grand total is $72$.
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$\frac{18}{72}=0.25$. Marginal relative frequency equals margin total divided by grand total.
$\frac{18}{72}=0.25$. Marginal relative frequency equals margin total divided by grand total.
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What is the marginal relative frequency for a row total $n_{i\cdot}$ with grand total $N$?
What is the marginal relative frequency for a row total $n_{i\cdot}$ with grand total $N$?
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$\frac{n_{i\cdot}}{N}$. Divide row total by grand total for proportion in that row category.
$\frac{n_{i\cdot}}{N}$. Divide row total by grand total for proportion in that row category.
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