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Two-way tables for two categorical variables

Learning goals

  • Organizes paired categorical data into a two-way table.
  • Finds row, column, and overall totals in a two-way table.
  • Identifies joint and marginal counts from the table.
  • Uses the table to compare categorical distributions across groups.

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Two-way tables for two categorical variables 1
A survey records response channel and satisfaction. Eight respondents did not report satisfaction, but their response channels are known: \(5\) used chat and \(3\) used phone. Explain how to retain all respondents in a two-way table. What new column should be added, and what counts enter it?

Hints

- Missing values can be represented as an explicit category when retaining records is important. - Keep the known variable classification for each incomplete record. - Ensure the added category is distinct from every substantive response.

Solution

1. Add a satisfaction category labeled “not reported.” 2. Enter \(5\) in the chat/not-reported cell and \(3\) in the phone/not-reported cell. 3. This preserves the channel information and keeps each respondent in exactly one cell.

Answer

Add a “satisfaction not reported” column. Put \(5\) in the chat row and \(3\) in the phone row of that column.
Two-way tables for two categorical variables 2
A two-way table has column categories “low,” “medium,” and “high” and row categories “amber,” “blue,” and “green.” A report says, “The row variable increases from amber to green.” Is that interpretation justified by the row categories? Give a defensible way to order the rows in the display and explain why that ordering does not create an ordinal meaning.

Hints

- Ask whether the row labels themselves encode magnitude or rank. - Separate the order used to display categories from the measurement scale of the variable. - A reproducible display convention does not have to imply “less than” or “greater than.”

Solution

1. The row labels amber, blue, and green are names of categories; they do not encode numerical size or rank. 2. Therefore, describing the row variable as increasing from amber to green is not justified. 3. A neutral reproducible convention, such as alphabetical order, is defensible because it organizes the display without claiming that one category is larger than another.

Answer

No. The row categories are nominal, so “increases from amber to green” invents an unsupported ordering. A neutral order such as alphabetical order is defensible because it changes only the display, not the meaning of the categories.
Two-way tables for two categorical variables 3
A bakery counts orders by time and payment method. <table> <tr><th></th><th>Cash</th><th>Card</th></tr> <tr><td>Before noon</td><td>\(14\)</td><td>\(8\)</td></tr> <tr><td>Noon or later</td><td>\(5\)</td><td>\(12\)</td></tr> </table> How many orders were placed at noon or later and paid for with cash?

Hints

- Would the cash column alone tell you when an order was placed? - What information changes as you move from one row to another?

Solution

1. Find the row for noon or later and the column for cash. 2. The cell where they meet shows \(5\) orders.

Answer

\(5\) orders.
Two-way tables for two categorical variables 4
Each of these six art-club projects has a project type and a material: \((\text{solo},\text{paper})\), \((\text{team},\text{wood})\), \((\text{team},\text{paper})\), \((\text{solo},\text{wood})\), \((\text{team},\text{wood})\), \((\text{team},\text{paper})\). If project type labels the rows and material labels the columns, what number belongs in the team-and-wood cell?

Hints

- A pair contributes to a cell only if both of its labels match. - Read the project pairs one at a time and tally matches.

Solution

1. Check each project for both 'team' and 'wood'. 2. This pair appears twice, so that cell contains \(2\).

Answer

\(2\).
Two-way tables for two categorical variables 5
A phone provider records each customer's plan in April and May. Each customer had either the Basic or Plus plan in each month. <table> <tr><th>April / May</th><th>Basic</th><th>Plus</th></tr> <tr><td>Basic</td><td>\(12\)</td><td>\(5\)</td></tr> <tr><td>Plus</td><td>\(3\)</td><td>\(10\)</td></tr> </table> a) How many customers kept the same plan? b) How many switched from Basic to Plus? How many switched the other way?

Hints

- A customer who kept the plan has the same label in both months. - What tells you whether a change is from Basic to Plus or vice versa? - Which date labels the rows, and which labels the columns?

Solution

1. The Basic-to-Basic and Plus-to-Plus cells count customers who kept their plans: \(12+10=22\). 2. The Basic-to-Plus cell is \(5\); Plus-to-Basic is \(3\).

Answer

a) \(22\) customers. b) Basic to Plus: \(5\); Plus to Basic: \(3\).
Two-way tables for two categorical variables 6
A report at Northbridge University crosses campus location with course format, but each row of the source file represents one course section, not one student. Daria labels the grand total “students surveyed.” Explain the observational-unit error. What may the table legitimately count, and what additional data would be needed to create a student-level table?

Hints

- Identify what one row of the source file represents. - Ask whether one person can contribute more than one source row. - Match the table’s grand-total label to the observational unit.

Solution

1. Each source row represents a course section, so each interior frequency counts sections classified by campus location and format. 2. The grand total therefore counts course sections, not students. 3. A student-level table would require one record per student, or a rule for assigning students who take more than one section to a single observational unit without double-counting.

Answer

The table may legitimately count course sections. It cannot be labeled as a count of students without student-level records and a rule that prevents students enrolled in multiple sections from being counted more than once.
Two-way tables for two categorical variables 7
A report gives this two-way frequency table for school newspaper readers. <table> <tr><th></th><th>read the sports section</th><th>did not read the sports section</th><th>Total</th></tr> <tr><td>digital edition</td><td>\(39\)</td><td>\(26\)</td><td>\(65\)</td></tr> <tr><td>print edition</td><td>\(41\)</td><td>\(16\)</td><td>\(57\)</td></tr> <tr><td>Total</td><td>\(80\)</td><td>\(42\)</td><td>\(125\)</td></tr> </table> Is the table internally consistent? Identify the incorrect entry and give the value it must have.

Hints

- Add the row totals and compare that sum with the displayed grand total. - Add the column totals as an independent check. - When both margin sums agree, the inconsistent grand total is the entry that must change.

Solution

1. The row totals sum to \(65+57=122\). 2. The column totals also sum to \(80+42=122\). 3. Therefore the displayed grand total \(125\) is inconsistent and must be \(122\).

Answer

No. The grand total is incorrect; it must be \(122\), not \(125\).
Two-way tables for two categorical variables 8
A data clerk is building a two-way table for science fair projects. The four observed category pairs occurred \(22\), \(36\), \(22\), and \(38\) times for \((\text{team project, used a physical prototype})\), \((\text{team project, did not use a physical prototype})\), \((\text{individual project, used a physical prototype})\), and \((\text{individual project, did not use a physical prototype})\), respectively. Find every row and column total and verify the grand total in two different ways.

Hints

- Organize the four category combinations before computing totals. - A column total accounts for every cell in that column. - Check that the row totals and column totals lead to the same grand total.

Solution

1. Row totals are \(22+36=58\) and \(22+38=60\). 2. Column totals are \(22+22=44\) and \(36+38=74\). 3. Both \(58+60\) and \(44+74\) equal \(118\).

Answer

Row totals: \(58\), \(60\). Column totals: \(44\), \(74\). Grand total: \(118\).
Two-way tables for two categorical variables 9
A proposed \(2\times2\) table has row totals \(40\) and \(60\), column totals \(70\) and \(30\), and upper-left cell \(45\). Can such a table exist? Identify the violated constraint without completing all four cells.

Hints

- A joint count is contained within both its row and its column. - Compare the cell with the smaller containing margin. - An impossibility can be detected before reconstructing the table.

Solution

1. The upper-left cell lies in the first row, whose total is only \(40\). 2. A cell count cannot exceed either margin containing it. 3. Since \(45>40\), the proposed information is impossible.

Answer

No. The upper-left cell cannot be \(45\) because its row total is \(40\).
Two-way tables for two categorical variables 10
The recreation center recorded \(120\) separate visits. Each visit was classified by day and pool use. <table> <tr><th></th><th>used the pool</th><th>did not use the pool</th><th>Total</th></tr> <tr><td>weekday visit</td><td>\(36\)</td><td>\(17\)</td><td>\(53\)</td></tr> <tr><td>weekend visit</td><td>\(40\)</td><td>\(27\)</td><td>\(67\)</td></tr> <tr><td>Total</td><td>\(76\)</td><td>\(44\)</td><td>\(120\)</td></tr> </table> Amina says the \(40\) belongs in the weekday row because it is in the “used the pool” column. Explain what the cell counts and why Amina is wrong.

Hints

- Read both the row label and the column label for the cell. - Would moving the count to another row keep its day category correct? - Check the row totals as a second way to spot the mistake.

Solution

1. Each cell describes both a day category and a pool-use category. 2. The \(40\) counts weekend visits when the pool was used. 3. Moving the count to the weekday row would mislabel those visits and make the row totals incorrect.

Answer

The \(40\) counts weekend visits when the pool was used. The column tells us about pool use, but the row tells us the day category.
Two-way tables for two categorical variables 11
A campus café recorded these 12 orders in the order \((\text{service type},\text{cup type})\): \( \begin{aligned} &(\text{counter},\text{reusable}),(\text{mobile},\text{disposable}),(\text{counter},\text{disposable}),\\ &(\text{mobile},\text{reusable}),(\text{mobile},\text{disposable}),(\text{counter},\text{reusable}),\\ &(\text{counter},\text{disposable}),(\text{mobile},\text{disposable}),(\text{counter},\text{reusable}),\\ &(\text{mobile},\text{reusable}),(\text{counter},\text{disposable}),(\text{mobile},\text{disposable}). \end{aligned} \) Create a two-way frequency table with service type as the row variable and cup type as the column variable. Include all margins.

Hints

- Treat each ordered pair as one observation in exactly one cell. - Tally the four possible category combinations before finding any totals. - Check that both sets of margins give the same grand total.

Solution

1. Tally each ordered pair: counter/reusable \(=3\), counter/disposable \(=3\), mobile/reusable \(=2\), and mobile/disposable \(=4\). 2. The row totals are \(6\) counter orders and \(6\) mobile orders. 3. The column totals are \(5\) reusable cups and \(7\) disposable cups, for a grand total of \(12\).

Answer

<table> <tr><th></th><th>reusable</th><th>disposable</th><th>Total</th></tr> <tr><td>counter</td><td>\(3\)</td><td>\(3\)</td><td>\(6\)</td></tr> <tr><td>mobile</td><td>\(2\)</td><td>\(4\)</td><td>\(6\)</td></tr> <tr><td>Total</td><td>\(5\)</td><td>\(7\)</td><td>\(12\)</td></tr> </table>
Two-way tables for two categorical variables 12
A survey at Cedar Grove High School originally used rows “walk,” “bike,” and “motorized” and columns “arrived before 8 a.m.” and “arrived at or after 8 a.m.” Tomás proposes combining “walk” and “bike” into a row called “active travel.” The original interior counts are: <table> <tr><th></th><th>before 8 a.m.</th><th>at or after 8 a.m.</th></tr> <tr><td>walk</td><td>\(18\)</td><td>\(7\)</td></tr> <tr><td>bike</td><td>\(12\)</td><td>\(9\)</td></tr> <tr><td>motorized</td><td>\(31\)</td><td>\(23\)</td></tr> </table> Create the recoded table and explain which dimension changes and which dimension stays unchanged.

Hints

- Combine categories only within the same column. - Ask which variable is being recoded. - Verify that recoding does not change the grand total.

Solution

1. Add the walk and bike counts within each arrival-time column: \(18+12=30\) and \(7+9=16\). 2. The motorized row remains \((31,23)\). 3. The row variable changes from three categories to two; the two arrival-time columns remain unchanged.

Answer

<table> <tr><th></th><th>before 8 a.m.</th><th>at or after 8 a.m.</th><th>Total</th></tr> <tr><td>active travel</td><td>\(30\)</td><td>\(16\)</td><td>\(46\)</td></tr> <tr><td>motorized</td><td>\(31\)</td><td>\(23\)</td><td>\(54\)</td></tr> <tr><td>Total</td><td>\(61\)</td><td>\(39\)</td><td>\(100\)</td></tr> </table> The row dimension is reduced from three categories to two; the column dimension is unchanged.
Two-way tables for two categorical variables 13
A draft table uses rows “all subscriptions,” “monthly subscriptions,” and “annual subscriptions,” crossed with renewal status. Monthly and annual subscriptions are the only subscription types. Explain why “all subscriptions” must not appear as a third row category. How should the information be displayed instead?

Hints

- Check whether one proposed row contains observations from other rows. - Interior categories should classify each observation exactly once. - Distinguish a category from a subtotal.

Solution

1. “All subscriptions” contains both monthly and annual subscriptions, so it overlaps both proposed row categories. 2. Treating it as a third interior row would count every subscription once in its type row and again in the all-subscriptions row. 3. Monthly and annual should be the two row categories; “all subscriptions” should appear only as the row-total margin or as a separate summary outside the interior table.

Answer

Use monthly and annual as the mutually exclusive interior rows. Display “all subscriptions” only as a total, not as another category row, because it is a subtotal that contains the other rows.
Two-way tables for two categorical variables 14
At a concert, \(40\) tickets were bought online and \(35\) at the door. There were \(32\) tickets for the early show and \(43\) for the late show. The ticket clerk knows that \(18\) online tickets were for the early show. Complete a two-way table with purchase method as the rows and show time as the columns. Explain why the row and column totals alone would not tell you how many online tickets were for the early show.

Hints

- Put the two purchase methods in separate rows and the two show times in columns. - Which cells share a row or column with the known ticket count? - Could moving one ticket between matching category pairs keep the totals unchanged?

Solution

1. Online late-show tickets: \(40-18=22\). 2. At-the-door early-show tickets: \(32-18=14\). 3. At-the-door late-show tickets: \(35-14=21\). Both column totals check: \(18+14=32\), \(22+21=43\). 4. Without the extra cell count, different assignments of tickets can have the same row and column totals.

Answer

<table> <tr><th></th><th>Early show</th><th>Late show</th><th>Total</th></tr> <tr><td>Online</td><td>\(18\)</td><td>\(22\)</td><td>\(40\)</td></tr> <tr><td>At the door</td><td>\(14\)</td><td>\(21\)</td><td>\(35\)</td></tr> <tr><td>Total</td><td>\(32\)</td><td>\(43\)</td><td>\(75\)</td></tr> </table> The totals alone do not fix an interior cell; the given \(18\) makes the table unique.
Two-way tables for two categorical variables 15
A completed table originally showed \(26\) evening customers who paid cash and \(34\) evening customers who paid by card. A later audit found that \(5\) of the evening cash payments had been miscoded; they were actually evening card payments. The original table was: <table> <tr><th></th><th>cash</th><th>card</th><th>Total</th></tr> <tr><td>morning</td><td>\(31\)</td><td>\(29\)</td><td>\(60\)</td></tr> <tr><td>evening</td><td>\(26\)</td><td>\(34\)</td><td>\(60\)</td></tr> <tr><td>Total</td><td>\(57\)</td><td>\(63\)</td><td>\(120\)</td></tr> </table> Correct the table and identify which margins change.

Hints

- A recoding within one row transfers observations between columns. - Track the decrease and increase as equal amounts. - Decide which totals are unaffected by moving rather than adding observations.

Solution

1. Move \(5\) observations within the evening row: evening/cash becomes \(26-5=21\), and evening/card becomes \(34+5=39\). 2. The evening row total and grand total do not change. 3. The cash column total becomes \(52\), and the card column total becomes \(68\).

Answer

<table> <tr><th></th><th>cash</th><th>card</th><th>Total</th></tr> <tr><td>morning</td><td>\(31\)</td><td>\(29\)</td><td>\(60\)</td></tr> <tr><td>evening</td><td>\(21\)</td><td>\(39\)</td><td>\(60\)</td></tr> <tr><td>Total</td><td>\(52\)</td><td>\(68\)</td><td>\(120\)</td></tr> </table> Only the two column totals change.
Two-way tables for two categorical variables 16
A poll asks each respondent to select every streaming service they use. The report proposes a two-way table with rows “uses Service A” and “uses Service B” and columns “uses Service C” and “does not use Service C.” Explain why the proposed row categories do not define a standard two-way frequency table. Give one valid way to revise the row variable while preserving all respondents.

Hints

- Check whether one respondent can satisfy both proposed row labels. - A table cell should represent one unique category pair for each observation. - Consider creating exhaustive, nonoverlapping combinations of A and B use.

Solution

1. “Uses Service A” and “uses Service B” are not mutually exclusive; one respondent can belong to both rows. 2. Standard two-way table categories for one variable must place each observation in exactly one row and one column. 3. One valid revision is a four-category row variable: “A only,” “B only,” “both A and B,” and “neither A nor B.”

Answer

The proposed rows overlap, so respondents could be counted twice. A valid row variable is “A only,” “B only,” “both,” and “neither,” crossed with Service C use.
Two-way tables for two categorical variables 17
Two analysts display the same counts in different orientations. <table> <tr><th>Table A</th><th>approved</th><th>not approved</th></tr> <tr><td>new applicant</td><td>\(32\)</td><td>\(18\)</td></tr> <tr><td>returning applicant</td><td>\(41\)</td><td>\(9\)</td></tr> </table> <table> <tr><th>Table B</th><th>new applicant</th><th>returning applicant</th></tr> <tr><td>not approved</td><td>\(18\)</td><td>\(9\)</td></tr> <tr><td>approved</td><td>\(32\)</td><td>\(41\)</td></tr> </table> Determine whether the tables represent the same joint frequency information. Describe the row and column operations that connect them.

Hints

- Compare category pairs rather than physical cell positions. - Ask what transposing a table does to its variables. - Reordering categories changes presentation, not the underlying pairs.

Solution

1. Match counts by category pair: every new/approved, new/not-approved, returning/approved, and returning/not-approved count agrees. 2. Transposing Table A makes approval status the row variable and applicant type the column variable. 3. Reordering the transposed rows to not approved, approved produces Table B. Therefore the tables contain the same joint information.

Answer

Yes. Transpose Table A, then place the “not approved” row before the “approved” row. The category-pair counts are unchanged.
Two-way tables for two categorical variables 18
A company records work location and desk type for \(100\) employees. <table> <tr><th></th><th>Standing desk</th><th>Standard desk</th><th>Total</th></tr> <tr><td>Remote</td><td>\(26\)</td><td>\(34\)</td><td>\(60\)</td></tr> <tr><td>On-site</td><td>\(18\)</td><td>\(22\)</td><td>\(40\)</td></tr> <tr><td>Total</td><td>\(44\)</td><td>\(56\)</td><td>\(100\)</td></tr> </table> One employee was entered as “remote with a standing desk” but should have been “on-site with a standard desk.” Correct the entire table. State which margins change and explain why the grand total does not change.

Hints

- A correction can move one observation from one joint category to another rather than remove it. - Track the row and column lost by the original cell and gained by the corrected cell. - Check whether the number of employees in the data set changes.

Solution

1. Move one observation out of the remote/standing cell: \(26\to25\). 2. Move it into the on-site/standard cell: \(22\to23\). The other two interior cells do not change. 3. The remote total becomes \(59\), and the on-site total becomes \(41\). 4. The standing-desk total becomes \(43\), and the standard-desk total becomes \(57\). 5. The same employee remains in the data set, so the grand total stays \(100\).

Answer

<table> <tr><th></th><th>Standing desk</th><th>Standard desk</th><th>Total</th></tr> <tr><td>Remote</td><td>\(25\)</td><td>\(34\)</td><td>\(59\)</td></tr> <tr><td>On-site</td><td>\(18\)</td><td>\(23\)</td><td>\(41\)</td></tr> <tr><td>Total</td><td>\(43\)</td><td>\(57\)</td><td>\(100\)</td></tr> </table> Both row margins and both column margins change; the grand total does not.
Two-way tables for two categorical variables 19
A study has variables “housing” with categories apartment and house, and “internet service” with categories fiber and nonfiber. The four joint counts are \(46\), \(34\), \(29\), and \(51\), listed in this order: \( (\text{house}, \text{fiber}),\;(\text{apartment}, \text{nonfiber}),\;(\text{apartment}, \text{fiber}),\;(\text{house}, \text{nonfiber}). \) Create a table with internet service as the row variable and housing as the column variable. Include margins.

Hints

- Read the requested row and column variables before placing counts. - Attach each value to its labeled category pair. - Use margins to check that every observation was placed once.

Solution

1. Place each count by its labels, not by the order in which row and column categories might normally be listed. 2. The fiber row is apartment \(29\), house \(46\); the nonfiber row is apartment \(34\), house \(51\). 3. Row totals are \(75\) and \(85\); column totals are \(63\) and \(97\); the grand total is \(160\).

Answer

<table> <tr><th></th><th>apartment</th><th>house</th><th>Total</th></tr> <tr><td>fiber</td><td>\(29\)</td><td>\(46\)</td><td>\(75\)</td></tr> <tr><td>nonfiber</td><td>\(34\)</td><td>\(51\)</td><td>\(85\)</td></tr> <tr><td>Total</td><td>\(63\)</td><td>\(97\)</td><td>\(160\)</td></tr> </table>
Two-way tables for two categorical variables 20
A survey’s response variable has three defined categories: yes, no, and not reported. In one month, no respondent falls in “not reported.” An analyst deletes that zero-count column before publishing the table. Explain one advantage and one disadvantage of keeping the zero-count column. Does deleting it change the observed grand total or the defined variable?

Hints

- Separate a category’s definition from its observed frequency. - Ask what a zero count contributes to the grand total. - Consider comparability across different reporting periods.

Solution

1. Keeping the column documents the complete category system and makes month-to-month tables structurally comparable. 2. Deleting it creates a more compact display but can make readers think the variable was defined with only two categories. 3. Because the deleted column has count \(0\), the observed grand total is unchanged. 4. The variable’s defined domain still includes “not reported” even when that category is absent from this sample.

Answer

Keeping the zero column preserves the declared category structure; deleting it simplifies the display. The grand total is unchanged, but the variable still has three defined categories.
Two-way tables for two categorical variables 21
Two regional offices use the same row and column definitions. Their tables are: <table> <tr><th>Office A</th><th>approved</th><th>not approved</th></tr> <tr><td>new client</td><td>\(24\)</td><td>\(16\)</td></tr> <tr><td>returning client</td><td>\(31\)</td><td>\(9\)</td></tr> </table> <table> <tr><th>Office B</th><th>approved</th><th>not approved</th></tr> <tr><td>new client</td><td>\(18\)</td><td>\(22\)</td></tr> <tr><td>returning client</td><td>\(27\)</td><td>\(13\)</td></tr> </table> Combine the offices into one two-way table. State the condition that makes cell-by-cell addition legitimate.

Hints

- Add only cells representing the same category pair. - Confirm that no observation appears in both office datasets. - Recompute margins after combining the interior counts.

Solution

1. Add corresponding category-pair counts: \(24+18=42\), \(16+22=38\), \(31+27=58\), and \(9+13=22\). 2. Row totals are \(80\) and \(80\); column totals are \(100\) and \(60\); the grand total is \(160\). 3. Cell-by-cell addition is legitimate because both tables use identical category definitions and represent disjoint sets of observations.

Answer

<table> <tr><th></th><th>approved</th><th>not approved</th><th>Total</th></tr> <tr><td>new client</td><td>\(42\)</td><td>\(38\)</td><td>\(80\)</td></tr> <tr><td>returning client</td><td>\(58\)</td><td>\(22\)</td><td>\(80\)</td></tr> <tr><td>Total</td><td>\(100\)</td><td>\(60\)</td><td>\(160\)</td></tr> </table> The definitions must match, and the observation sets must not overlap.
Two-way tables for two categorical variables 22
A report lists the four interior counts of a \(2\times2\) table as the unlabeled multiset \(\{12,18,27,43\}\). The row and column category labels are known, but no count is attached to a category pair. Can the labeled table be reconstructed uniquely? Explain and give two different labeled tables consistent with the information.

Hints

- Counts without positions do not identify category pairs. - Try swapping two counts that are not in the same labeled cell. - One counterexample pair is enough to disprove uniqueness.

Solution

1. The multiset gives values but not their positions. 2. For example, \(\begin{pmatrix}12&18\\27&43\end{pmatrix}\) and \(\begin{pmatrix}12&27\\18&43\end{pmatrix}\) use the same four counts but assign them to different category pairs. 3. Because these labeled tables differ, reconstruction is not unique.

Answer

No. For example, both \(\begin{pmatrix}12&18\\27&43\end{pmatrix}\) and \(\begin{pmatrix}12&27\\18&43\end{pmatrix}\) use the same multiset but represent different labeled tables.
Two-way tables for two categorical variables 23
A hospital table separates discharges into “home,” “rehabilitation,” and “long-term care,” crossed with “readmitted within 30 days” and “not readmitted.” An administrator wants to merge rehabilitation and long-term care into “facility discharge.” Give one decision question for which the merged table is adequate and one for which it is inadequate.

Hints

- Identify exactly which distinction aggregation removes. - A question using only the combined category remains answerable. - A question comparing the merged components does not.

Solution

1. The merged table is adequate for comparing home discharges with all facility discharges. 2. It is inadequate for comparing rehabilitation with long-term care, because those categories are no longer separated. 3. Aggregation is acceptable only when the decision does not require the distinction being removed.

Answer

Adequate: “Is readmission more common for home or facility discharges?” Inadequate: “Is readmission more common after rehabilitation or long-term care?”
Two-way tables for two categorical variables 24
A district combines annual tables for the same students in 2025 and 2026. Each table classifies students by lunch type and bus use. The analyst adds the tables cell by cell and labels the grand total “number of students.” Explain why that label may be wrong even when the arithmetic is correct. Give a more accurate label.

Hints

- Identify the observational unit in each annual table. - Combining periods can create repeated records for the same person. - Choose a label that reflects what is actually counted.

Solution

1. The same student can appear once in each year, so cell-by-cell addition counts student-year records rather than necessarily distinct students. 2. Arithmetic addition is valid for totals of records if category definitions match. 3. A more accurate grand-total label is “student-year observations.”

Answer

The sum may count the same person twice, once per year. Label the total “student-year observations,” not necessarily “students.”
Two-way tables for two categorical variables 25
A school records each student's **one trip to school** by travel method (bus or walk) and whether the student used a school-bus pass on that trip. A school-bus pass can only be used on the bus. The table has a cell for each possible pair of answers. Which cell must be \(0\): bus and pass used, bus and pass not used, walk and pass used, or walk and pass not used? Why is a count of \(0\) different from a missing count?

Hints

- Does every combination of the two labels describe a possible trip? - Could a ticket be used on a trip that never boards the bus? - Would a cell remain missing if the school knew it had zero observations?

Solution

1. A student who walks cannot use a bus pass on that trip, so the walk-and-pass-used cell must be \(0\). 2. Riding the bus without using a pass is possible, as are riding with a pass and walking without one. 3. The required zero represents an impossible combination, while a missing count is a number the school did not record.

Answer

The “walk and pass used” cell must be \(0\). That combination is impossible; a missing count is unknown, not necessarily zero.
Two-way tables for two categorical variables 26
A school report counts lunch orders by meal and drink choice. One count is left blank. <table> <tr><th></th><th>Drink</th><th>No drink</th></tr> <tr><td>Salad</td><td>\(18\)</td><td>\(\square\)</td></tr> <tr><td>Sandwich</td><td>\(22\)</td><td>\(10\)</td></tr> </table> a) What would the total be if the blank meant a recorded \(0\)? What if it meant \(5\) unreported orders? b) Can the blank alone tell you which total is correct? What would you need to check?

Hints

- Does an empty spot tell you that nobody was counted? - Which three cells have visible counts? - Would a missing count and a recorded zero lead to the same total? - What part of the report could explain the blank?

Solution

1. If the blank stands for \(0\), the total is \(18+22+10=50\). 2. If it stands for \(5\), the total is \(18+5+22+10=55\). 3. Neither total is forced without a note explaining the blank or a check of the original lunch records.

Answer

a) \(50\) if the blank is \(0\); \(55\) if it is \(5\). b) No. Check the table's explanation of blank cells or the original lunch-order records.
Two-way tables for two categorical variables 27
Make a two-way frequency table with travel methods as rows and arrival status as columns. Include all totals.
Figure for problem 556189

Hints

- Locate the two bars for a single way of traveling. - How will you keep late arrivals separate from on-time arrivals? - Can you check the grand total using both sets of table margins?

Solution

1. Read bus/on time \(25\), bus/late \(10\), bike/on time \(15\), and bike/late \(5\). 2. The bus row totals \(25+10=35\), and the bike row totals \(15+5=20\). 3. The on-time column totals \(25+15=40\), and the late column totals \(10+5=15\). 4. The grand total is \(35+20=55\).

Answer

<table> <tr><th>Commute</th><th>On time</th><th>Late</th><th>Total</th></tr> <tr><td>Bus</td><td>\(25\)</td><td>\(10\)</td><td>\(35\)</td></tr> <tr><td>Bike</td><td>\(15\)</td><td>\(5\)</td><td>\(20\)</td></tr> <tr><td>Total</td><td>\(40\)</td><td>\(15\)</td><td>\(55\)</td></tr> </table>
Two-way tables for two categorical variables 28
A food truck recorded cash and card payments during morning (AM) and afternoon (PM) shifts. There were \(40\) payments in all, but one bar is missing. Find the missing PM card count, then make a two-way table with shifts as rows and payment methods as columns. Include totals.
Figure for problem 563281

Hints

- Which combination of shift and payment method is not accounted for yet? - How do the visible bar counts and the overall total constrain that missing cell? - Can you check the completed table's totals in both directions?

Solution

1. Read three paired-category bars: AM cash \(12\), AM card \(8\), and PM cash \(14\). 2. PM card is \(40-(12+8+14)=6\). Then the row totals are \(20\) and \(20\), and the column totals are \(26\) cash and \(14\) card.

Answer

<table> <tr><th>Shift</th><th>Cash</th><th>Card</th><th>Total</th></tr> <tr><td>AM</td><td>\(12\)</td><td>\(8\)</td><td>\(20\)</td></tr> <tr><td>PM</td><td>\(14\)</td><td>\(6\)</td><td>\(20\)</td></tr> <tr><td>Total</td><td>\(26\)</td><td>\(14\)</td><td>\(40\)</td></tr> </table>
Two-way tables for two categorical variables 29
A library table groups checkouts by time and type of book. <table> <tr><th></th><th>Fiction</th><th>Nonfiction</th><th>Total</th></tr> <tr><td>Morning</td><td>\(18\)</td><td>\(\square\)</td><td>\(25\)</td></tr> <tr><td>Afternoon</td><td>\(9\)</td><td>\(11\)</td><td>\(20\)</td></tr> <tr><td>Total</td><td>\(27\)</td><td>\(18\)</td><td>\(45\)</td></tr> </table> Find the missing count and check it using both a row and a column total.

Hints

- Could the totals be correct if the missing entry were larger than its row total? - What count remains after the known cells in its row are taken away? - Can the column information independently confirm your choice?

Solution

1. The missing morning-nonfiction count is \(25-18=7\). 2. The nonfiction column also gives \(18-11=7\), so both totals agree.

Answer

\(7\) morning nonfiction checkouts. Row check: \(18+7=25\). Column check: \(7+11=18\).
Two-way tables for two categorical variables 30
A school sorts students by how they arrived and whether they were late. <table> <tr><th></th><th>On time</th><th>Late</th><th>Total</th></tr> <tr><td>Walk</td><td>\(10\)</td><td>\(2\)</td><td>\(12\)</td></tr> <tr><td>Bus</td><td>\(13\)</td><td>\(5\)</td><td>\(18\)</td></tr> <tr><td>Car</td><td>\(20\)</td><td>\(5\)</td><td>\(25\)</td></tr> </table> Someone adds a fourth row, “walk or bus,” with \(23\) on time and \(7\) late. Why is that row not a new group? What is the correct grand total?

Hints

- Where did the counts in the proposed new row come from? - Would its addition make students appear in more than one row? - Which original rows classify each student once?

Solution

1. The new row combines the walk and bus cells: \(10+13=23\) on time and \(2+5=7\) late. 2. Those \(30\) students are already counted. The correct total is \(12+18+25=55\), not \(85\).

Answer

The new row combines counts already in the first two rows. The correct total is \(55\) students.
Two-way tables for two categorical variables 31
Find the total number of students using each travel method. Which method has the most students?
Figure for problem 563284

Hints

- Would looking at only the on-time bars miss any students? - Look for labels that share a travel method but differ in arrival result. - Check that the three method totals account for all six bars.

Solution

1. Walking totals \(13+2=15\) students. 2. Bus totals \(10+7=17\) students, and car totals \(9+3=12\) students. 3. The bus group has the largest total, \(17\).

Answer

Walk: \(15\) students; bus: \(17\); car: \(12\). Bus has the most students.
Two-way tables for two categorical variables 32
A wildlife team followed \(42\) tagged ducks. Each duck was at either North Pond or South Pond when checked last week and again this week. Sixteen ducks stayed at North Pond, \(12\) stayed at South Pond, and \(7\) moved from North to South. Exactly as many moved from South to North. Make a two-way table with last week's pond in the rows and this week's pond in the columns. Include all totals. What do you notice about each pond's row and column totals?

Hints

- Where would you place ducks that stayed at the same pond? - Could the North Pond count change if the same number of ducks arrived as left? - Check the total number of tagged ducks against both sets of table margins.

Solution

1. Seven ducks moved in each direction, so both off-diagonal cells are \(7\). 2. The North Pond row and column each total \(16+7=23\). 3. The South Pond row and column each total \(12+7=19\), giving \(42\) ducks. Equal movement in both directions leaves each pond's count unchanged.

Answer

<table> <tr><th>Last week / this week</th><th>North Pond</th><th>South Pond</th><th>Total</th></tr> <tr><td>North Pond</td><td>\(16\)</td><td>\(7\)</td><td>\(23\)</td></tr> <tr><td>South Pond</td><td>\(7\)</td><td>\(12\)</td><td>\(19\)</td></tr> <tr><td>Total</td><td>\(23\)</td><td>\(19\)</td><td>\(42\)</td></tr> </table> Each pond's row total equals its column total because equal numbers of ducks moved in both directions.
Two-way tables for two categorical variables 33
An image classifier and a human reviewer each label the same \(200\) photos as “damaged” or “not damaged.” Their classifications are summarized below. <table> <tr><th></th><th>Human: damaged</th><th>Human: not damaged</th><th>Total</th></tr> <tr><td>Classifier: damaged</td><td>\(68\)</td><td>\(24\)</td><td>\(92\)</td></tr> <tr><td>Classifier: not damaged</td><td>\(12\)</td><td>\(96\)</td><td>\(108\)</td></tr> <tr><td>Total</td><td>\(80\)</td><td>\(120\)</td><td>\(200\)</td></tr> </table> Using the human review as the reference classification: a) Identify the false-positive and false-negative cells. b) Find the total number of disagreements. c) Explain why the classifier labels \(12\) more photos as damaged than the human reviewer does.

Hints

- Decide what agreement and disagreement look like in a classification table. - The direction of an error depends on which classification is treated as the reference. - Compare the two damaged margins with the difference between the off-diagonal cells.

Solution

1. False positives are photos labeled damaged only by the classifier, so there are \(24\). False negatives are photos labeled damaged only by the human reviewer, so there are \(12\). 2. Disagreements occupy the two off-diagonal cells: \(24+12=36\). 3. The classifier’s damaged total exceeds the human damaged total by \(92-80=12\). Equivalently, false positives exceed false negatives by \(24-12=12\).

Answer

a) False positives: \(24\); false negatives: \(12\). b) There are \(36\) disagreements. c) The classifier has \(12\) more damaged labels because it has \(12\) more false positives than false negatives.
Two-way tables for two categorical variables 34
These tables show students' housing and meal plans in two class-year groups. <table> <tr><th>First-year students</th><th>meal plan</th><th>no meal plan</th></tr> <tr><td>on campus</td><td>\(48\)</td><td>\(12\)</td></tr> <tr><td>off campus</td><td>\(9\)</td><td>\(21\)</td></tr> </table> <table> <tr><th>Upper-year students</th><th>meal plan</th><th>no meal plan</th></tr> <tr><td>on campus</td><td>\(26\)</td><td>\(34\)</td></tr> <tr><td>off campus</td><td>\(18\)</td><td>\(52\)</td></tr> </table> Combine the two groups into one two-way table showing housing and meal plan. Include all row and column totals.

Hints

- Which cells describe the same housing and meal-plan combination? - What should happen to the number of columns when the groups are combined? - Check that the row and column totals give the same number of students.

Solution

1. Add matching cells from the two tables: \(48+26=74\), \(12+34=46\), \(9+18=27\), and \(21+52=73\). 2. The row totals are \(120\) and \(100\); the column totals are \(101\) and \(119\). 3. Both sets of totals give \(220\) students.

Answer

<table> <tr><th></th><th>meal plan</th><th>no meal plan</th><th>Total</th></tr> <tr><td>on campus</td><td>\(74\)</td><td>\(46\)</td><td>\(120\)</td></tr> <tr><td>off campus</td><td>\(27\)</td><td>\(73\)</td><td>\(100\)</td></tr> <tr><td>Total</td><td>\(101\)</td><td>\(119\)</td><td>\(220\)</td></tr> </table>
Two-way tables for two categorical variables 35
The rows of a two-way table show whether students chose choir or band last semester. Its columns show those same two choices this semester. Exactly as many students moved from choir to band as moved from band to choir. Last semester there were \(70\) in choir and \(50\) in band. This semester there are \(65\) in choir and \(55\) in band. Can all these claims be true? Explain using the row and column totals, without guessing every cell.

Hints

- What happens to a club's size if the same number joins it as leaves it? - Compare the earlier and later totals for choir. - Can the same number switch both ways if a club gains or loses members?

Solution

1. Equal numbers switching both directions would give no net gain or loss for either group. 2. Choir's total changes from \(70\) to \(65\), a loss of \(5\); band's total rises by \(5\). 3. Thus the two switching counts cannot be equal; the claims conflict.

Answer

No. Equal switching in both directions would leave the choir and band totals unchanged, but each changes by \(5\).
Two-way tables for two categorical variables 36
A registration report combined “mobile” and “desktop” into one row labeled “online.” <table> <tr><th></th><th>Attended</th><th>Did not attend</th><th>Total</th></tr> <tr><td>Online</td><td>\(54\)</td><td>\(36\)</td><td>\(90\)</td></tr> <tr><td>In person</td><td>\(28\)</td><td>\(22\)</td><td>\(50\)</td></tr> <tr><td>Total</td><td>\(82\)</td><td>\(58\)</td><td>\(140\)</td></tr> </table> An archived note says that \(40\) online registrants used mobile devices and that \(30\) of those mobile registrants attended. a) Reconstruct a table with separate mobile, desktop, and in-person rows. b) Explain why the collapsed table alone could not determine the mobile and desktop rows.

Hints

- Treat the archived mobile information as one complete row of the expanded table. - The desktop counts are the portions of the online counts not assigned to mobile. - Consider how many different pairs of rows could have the same combined row.

Solution

1. The mobile row is \(30\) attended and \(40-30=10\) did not attend. 2. Subtract the mobile row from the online row. The desktop row is \(54-30=24\) attended and \(36-10=26\) did not attend, for a total of \(50\). 3. The in-person row remains \((28,22)\). 4. The collapsed online row stores only column sums across mobile and desktop. Many different mobile-desktop splits could produce those same sums, so the archived information is necessary.

Answer

a) <table> <tr><th></th><th>Attended</th><th>Did not attend</th><th>Total</th></tr> <tr><td>Mobile</td><td>\(30\)</td><td>\(10\)</td><td>\(40\)</td></tr> <tr><td>Desktop</td><td>\(24\)</td><td>\(26\)</td><td>\(50\)</td></tr> <tr><td>In person</td><td>\(28\)</td><td>\(22\)</td><td>\(50\)</td></tr> <tr><td>Total</td><td>\(82\)</td><td>\(58\)</td><td>\(140\)</td></tr> </table> b) Collapsing mobile and desktop discards their separate cell counts, so the split is not recoverable without additional information.
Two-way tables for two categorical variables 37
Four sampled records have category pairs and survey weights: \( (A, \text{yes}, 2.0),\quad(A, \text{no}, 1.0),\quad (B, \text{yes}, 0.5),\quad(B, \text{no}, 1.5). \) Create both the unweighted \(2 \times 2\) frequency table and the weighted table. Include all margins, and explain why the weighted entries should not be described as numbers of sampled records.

Hints

- Tally records once for the unweighted table. - Sum record weights, rather than record indicators, for the weighted table. - Distinguish sample size from weighted population contribution.

Solution

1. Each category pair appears once, so every unweighted interior count is \(1\); row and column totals are \(2\), with grand total \(4\). 2. Summing weights by cell gives weighted entries \(2.0,1.0,0.5,1.5\). 3. Weighted row totals are \(3.0,2.0\); weighted column totals are \(2.5,2.5\); the weighted grand total is \(5.0\). 4. Weights represent estimated population contribution, so weighted entries are not literal counts of the four sampled records.

Answer

Unweighted table: <table> <tr><th></th><th>yes</th><th>no</th><th>Total</th></tr> <tr><td>A</td><td>\(1\)</td><td>\(1\)</td><td>\(2\)</td></tr> <tr><td>B</td><td>\(1\)</td><td>\(1\)</td><td>\(2\)</td></tr> <tr><td>Total</td><td>\(2\)</td><td>\(2\)</td><td>\(4\)</td></tr> </table> Weighted table: <table> <tr><th></th><th>yes</th><th>no</th><th>Total</th></tr> <tr><td>A</td><td>\(2.0\)</td><td>\(1.0\)</td><td>\(3.0\)</td></tr> <tr><td>B</td><td>\(0.5\)</td><td>\(1.5\)</td><td>\(2.0\)</td></tr> <tr><td>Total</td><td>\(2.5\)</td><td>\(2.5\)</td><td>\(5.0\)</td></tr> </table> The weighted values are estimated population contributions, not counts of sampled records.
Two-way tables for two categorical variables 38
A \(2\times3\) table records \(150\) deliveries by vehicle type and time window. The van row total is \(84\). The morning column total is \(51\), the afternoon column total is \(63\), and the evening column total is \(36\). Van deliveries include \(32\) morning and \(34\) afternoon deliveries. Complete all six interior cells and all margins.

Hints

- Complete the known row before using the column totals. - Each lower cell is the corresponding column remainder. - Check that the two row totals reproduce the grand total.

Solution

1. Van evening deliveries are \(84-32-34=18\). 2. Bicycle deliveries are \(51-32=19\) in the morning, \(63-34=29\) in the afternoon, and \(36-18=18\) in the evening. 3. The bicycle row total is \(19+29+18=66\), and \(84+66=150\).

Answer

<table> <tr><th></th><th>morning</th><th>afternoon</th><th>evening</th><th>Total</th></tr> <tr><td>van</td><td>\(32\)</td><td>\(34\)</td><td>\(18\)</td><td>\(84\)</td></tr> <tr><td>bicycle</td><td>\(19\)</td><td>\(29\)</td><td>\(18\)</td><td>\(66\)</td></tr> <tr><td>Total</td><td>\(51\)</td><td>\(63\)</td><td>\(36\)</td><td>\(150\)</td></tr> </table>
Two-way tables for two categorical variables 39
A school records whether students chose art or music in the fall and again in the spring. In the fall, \(60\) chose art and \(40\) chose music. In the spring, \(55\) chose art and \(45\) chose music. Exactly \(65\) kept the same elective. Make a two-way table with fall choices as rows and spring choices as columns. How can you check that the number who stayed in the same elective is correct?

Hints

- The two cells showing no change lie on the table's diagonal. - Express the other cells from one unknown and the row and column totals. - Check both the margins and the sum of the unchanged choices.

Solution

1. Let \(x\) be the number who took art both times. Then fall-art to spring-music is \(60-x\), and fall-music to spring-art is \(55-x\). 2. Fall-music to spring-music is \(40-(55-x)=x-15\). The two unchanged cells sum to \(65\), so \(x+(x-15)=65\) and \(x=40\). 3. The counts are \(40,20,15,25\); the diagonal sums to \(40+25=65\) and all four margins agree.

Answer

<table> <tr><th>Fall / spring</th><th>Art</th><th>Music</th><th>Total</th></tr> <tr><td>Art</td><td>\(40\)</td><td>\(20\)</td><td>\(60\)</td></tr> <tr><td>Music</td><td>\(15\)</td><td>\(25\)</td><td>\(40\)</td></tr> <tr><td>Total</td><td>\(55\)</td><td>\(45\)</td><td>\(100\)</td></tr> </table> The same-elective count is \(40+25=65\).
Two-way tables for two categorical variables 40
A school compared how students traveled to school last year and this year: by car, bus, or bike. The same number switched between any two methods in each direction. Of the students who did not change methods, \(40\) went by car, \(31\) by bus, and \(22\) by bike. The changes in one direction were \(9\) from car to bus, \(6\) from car to bike, and \(8\) from bus to bike. Make a two-way table showing last year's methods in rows and this year's in columns. Check its total.

Hints

- Which off-diagonal cell describes the reverse of a stated switch? - Fill the reverse switches before calculating row totals. - How could the row and column totals check each other?

Solution

1. The reverse switching counts must match: bus to car \(9\), bike to car \(6\), and bike to bus \(8\). 2. Row totals are \(55\), \(48\), and \(36\). 3. The total is \(55+48+36=139\). The column totals also add to \(139\).

Answer

<table> <tr><th>Last year / this year</th><th>Car ride</th><th>Bus</th><th>Bike</th><th>Total</th></tr> <tr><td>Car ride</td><td>\(40\)</td><td>\(9\)</td><td>\(6\)</td><td>\(55\)</td></tr> <tr><td>Bus</td><td>\(9\)</td><td>\(31\)</td><td>\(8\)</td><td>\(48\)</td></tr> <tr><td>Bike</td><td>\(6\)</td><td>\(8\)</td><td>\(22\)</td><td>\(36\)</td></tr> <tr><td>Total</td><td>\(55\)</td><td>\(48\)</td><td>\(36\)</td><td>\(139\)</td></tr> </table>
Two-way tables for two categorical variables 41
At a school festival, \(50\) tickets were bought online and \(70\) at the entrance. There were \(30\) concert tickets, \(40\) play tickets, and \(50\) tour tickets. Each ticket has one purchase method and one event type. Can you tell exactly how many online tickets were for the concert? Make **two different** valid two-way tables with purchase method as rows and event type as columns. Use the given totals to check both tables.

Hints

- A row total counts every event type bought through that method. - Choose a possible set of online counts, then use the column totals to find the entrance counts. - Can you change some paired counts without changing the row and column totals?

Solution

1. The row and column totals do not specify any of the six paired-category counts. 2. One possible online row is \((10,15,25)\), leaving entrance counts \((20,25,25)\). 3. Another possible online row is \((20,10,20)\), leaving entrance counts \((10,30,30)\). 4. Both tables have row totals \(50,70\) and column totals \(30,40,50\), yet their online concert counts differ.

Answer

No. For example, both tables fit the totals: <table> <tr><th></th><th>Concert</th><th>Play</th><th>Tour</th><th>Total</th></tr> <tr><td>Online</td><td>\(10\)</td><td>\(15\)</td><td>\(25\)</td><td>\(50\)</td></tr> <tr><td>Entrance</td><td>\(20\)</td><td>\(25\)</td><td>\(25\)</td><td>\(70\)</td></tr> <tr><td>Total</td><td>\(30\)</td><td>\(40\)</td><td>\(50\)</td><td>\(120\)</td></tr> </table> <table> <tr><th></th><th>Concert</th><th>Play</th><th>Tour</th><th>Total</th></tr> <tr><td>Online</td><td>\(20\)</td><td>\(10\)</td><td>\(20\)</td><td>\(50\)</td></tr> <tr><td>Entrance</td><td>\(10\)</td><td>\(30\)</td><td>\(30\)</td><td>\(70\)</td></tr> <tr><td>Total</td><td>\(30\)</td><td>\(40\)</td><td>\(50\)</td><td>\(120\)</td></tr> </table>
Two-way tables for two categorical variables 42
A privacy report replaces one small count with “suppressed” but prints every other interior count and all margins. <table> <tr><th></th><th>requested aid</th><th>did not request aid</th><th>Total</th></tr> <tr><td>Program A</td><td>suppressed</td><td>\(28\)</td><td>\(31\)</td></tr> <tr><td>Program B</td><td>\(19\)</td><td>\(42\)</td><td>\(61\)</td></tr> <tr><td>Total</td><td>\(22\)</td><td>\(70\)</td><td>\(92\)</td></tr> </table> Show why the suppression does not protect the count. State one kind of additional suppression that would be needed to prevent direct reconstruction.

Hints

- Try reconstructing the hidden value from its row. - Check whether its column provides the same reconstruction. - Privacy suppression must remove every direct arithmetic path to the value.

Solution

1. The Program A row reveals the hidden count as \(31-28=3\). 2. The requested-aid column independently reveals it as \(22-19=3\). 3. Suppressing only the small cell is ineffective while a containing row or column total and the companion cell remain visible. 4. At least one additional linked value, such as the Program A row total or another cell needed to derive that total, must also be suppressed under a valid complementary-suppression plan.

Answer

The hidden count is \(3\). The printed margins disclose it, so linked totals or companion cells must also be suppressed according to a complementary-suppression rule.
Two-way tables for two categorical variables 43
A fitness center tracked which class \(100\) members attended in September and again in October. In September, \(60\) chose yoga and \(40\) chose cycling. In October, \(50\) chose yoga and \(50\) chose cycling. Each member chose exactly one class each month. a) Must more members have switched from yoga to cycling or from cycling to yoga? By how many? b) Give two different two-way tables that fit these totals. What can you *not* find from the totals alone?

Hints

- Compare the number choosing yoga at the two dates. - Which two cells record the changes in opposite directions? - Can two different patterns of individual switches leave the same class totals?

Solution

1. Yoga lost a net \(10\) members, so yoga-to-cycling switches must outnumber cycling-to-yoga switches by \(10\). 2. One possible table (September rows, October columns) has cells \((50,10)\) in the yoga row and \((0,40)\) in the cycling row. 3. Another has \((45,15)\) and \((5,35)\). Both have the required margins, but different numbers switched in each direction.

Answer

a) Yoga to cycling, by \(10\) members. b) Two valid tables (September rows; October columns) are <table> <tr><th></th><th>Yoga</th><th>Cycling</th></tr> <tr><td>Yoga</td><td>\(50\)</td><td>\(10\)</td></tr> <tr><td>Cycling</td><td>\(0\)</td><td>\(40\)</td></tr> </table> and <table> <tr><th></th><th>Yoga</th><th>Cycling</th></tr> <tr><td>Yoga</td><td>\(45\)</td><td>\(15\)</td></tr> <tr><td>Cycling</td><td>\(5\)</td><td>\(35\)</td></tr> </table> The separate numbers switching in each direction cannot be determined from the margins alone.
Two-way tables for two categorical variables 44
Two completed \(2\times2\) tables have the same row totals and column totals: \( T_1=\begin{pmatrix}40&10\\10&40\end{pmatrix}, \qquad T_2=\begin{pmatrix}25&25\\25&25\end{pmatrix}. \) Explain what the shared margins do and do not determine. Which table shows a stronger association between the variables?

Hints

- Compare interior patterns after confirming the margins match. - Margins summarize each variable separately. - Association is visible in how cell proportions vary across rows.

Solution

1. Both tables have row totals \(50,50\) and column totals \(50,50\). 2. Margins determine one-variable distributions but not how categories pair within the table. 3. \(T_1\) concentrates observations on the diagonal, while \(T_2\) has the same distribution in both rows; \(T_1\) shows stronger association.

Answer

The margins are identical but the joint distributions differ. \(T_1\) shows stronger association; \(T_2\) shows no row-to-row change in the column distribution.
Two-way tables for two categorical variables 45
A school recorded whether juniors and seniors returned a library book by Friday. Later, the librarian finds that **five** “Not returned” answers should have been “Returned,” but does not know how many of those five came from each grade. <table> <tr><th></th><th>Returned</th><th>Not returned</th><th>Total</th></tr> <tr><td>Juniors</td><td>\(21\)</td><td>\(29\)</td><td>\(50\)</td></tr> <tr><td>Seniors</td><td>\(26\)</td><td>\(24\)</td><td>\(50\)</td></tr> <tr><td>Total</td><td>\(47\)</td><td>\(53\)</td><td>\(100\)</td></tr> </table> a) What must the two new column totals be? b) Give **two different corrected tables** that fit the information. Can the final number of juniors who returned books be determined exactly?

Hints

- What happens to the two column totals when five responses change category? - Could all five corrected responses come from one grade? The other grade? - Check which row totals stay unchanged in both possible tables.

Solution

1. The overall returned total rises from \(47\) to \(52\) and not-returned falls from \(53\) to \(48\). 2. If all five corrected records are juniors, the two rows become juniors \((26,24)\) and seniors \((26,24)\). 3. If all five are seniors, the rows instead become juniors \((21,29)\) and seniors \((31,19)\). 4. Both tables have the same corrected margins, so the final juniors' returned count cannot be fixed from the given information; it can be any integer from \(21\) to \(26\).

Answer

a) Returned: \(52\); not returned: \(48\). b) For example, either corrected table is possible: <table> <tr><th></th><th>Returned</th><th>Not returned</th><th>Total</th></tr> <tr><td>Juniors</td><td>\(26\)</td><td>\(24\)</td><td>\(50\)</td></tr> <tr><td>Seniors</td><td>\(26\)</td><td>\(24\)</td><td>\(50\)</td></tr> <tr><td>Total</td><td>\(52\)</td><td>\(48\)</td><td>\(100\)</td></tr> </table> <table> <tr><th></th><th>Returned</th><th>Not returned</th><th>Total</th></tr> <tr><td>Juniors</td><td>\(21\)</td><td>\(29\)</td><td>\(50\)</td></tr> <tr><td>Seniors</td><td>\(31\)</td><td>\(19\)</td><td>\(50\)</td></tr> <tr><td>Total</td><td>\(52\)</td><td>\(48\)</td><td>\(100\)</td></tr> </table> No. The juniors' corrected returned count can be from \(21\) through \(26\).
Two-way tables for two categorical variables 46
A table counts registrations by workshop and payment status. One record was duplicated in “ceramics and paid,” while another record was omitted from “painting and unpaid.” Describe the net changes to the grand total, the four affected margins, and the two affected interior cells when both errors are corrected.

Hints

- Treat the duplicate correction and omission correction separately. - Combine their effects only after tracking each cell. - Equal removal and addition leave the total number of observations unchanged.

Solution

1. Remove one from ceramics/paid and add one to painting/unpaid. 2. The grand total is unchanged because one record is removed and one is added. 3. Ceramics row and paid column each decrease by \(1\); painting row and unpaid column each increase by \(1\).

Answer

Ceramics/paid decreases by \(1\), and painting/unpaid increases by \(1\). The grand total is unchanged. The ceramics row and paid column decrease by \(1\); the painting row and unpaid column increase by \(1\).
Two-way tables for two categorical variables 47
A two-way table of \(200\) records has row margins \(120,80\) and column margins \(90,110\). The raw file is lost. Can the original order of the \(200\) records be reconstructed from the table? Can the number of records in each category pair be reconstructed from the margins alone? Explain the two different information losses.

Hints

- Ask what information a frequency table keeps about individual records. - Then ask what margins keep about row-column pairings. - Identify whether one or several tables can share the same margins.

Solution

1. A frequency table discards record order, so the original sequence cannot be reconstructed even if every interior count were known. 2. The displayed margins also omit how row and column categories pair. 3. Many interior tables can share these margins, so the four joint counts are not determined either. 4. Record-order loss and joint-association loss are separate: the first occurs when microdata are aggregated, and the second occurs when only margins are retained.

Answer

Neither the original record order nor the interior joint counts can be reconstructed. Aggregation removes order, and retaining only margins also removes the pairing information.
Two-way tables for two categorical variables 48
Two school help desks record how students contacted them and whether a question was answered. One desk tracks phone, email, and web form separately; the other combines email and web form as “online.” <table> <tr><th>Help desk 1</th><th>Answered</th><th>Not answered</th></tr> <tr><td>Phone</td><td>\(12\)</td><td>\(8\)</td></tr> <tr><td>Email</td><td>\(10\)</td><td>\(2\)</td></tr> <tr><td>Web form</td><td>\(6\)</td><td>\(4\)</td></tr> </table> <table> <tr><th>Help desk 2</th><th>Answered</th><th>Not answered</th></tr> <tr><td>Phone</td><td>\(8\)</td><td>\(7\)</td></tr> <tr><td>Online</td><td>\(15\)</td><td>\(5\)</td></tr> </table> Make one table for both desks together, using rows “phone” and “online.” Explain why the combined report cannot show separate email and web-form counts.

Hints

- Which two contact methods at the first desk match “online” at the second? - Combine categories only when their meanings match. - Can you split desk 2's online count without more information?

Solution

1. At desk 1, online combines email and web form: \(10+6=16\) answered and \(2+4=6\) not answered. 2. Add corresponding categories across desks: phone answered \(12+8=20\), phone not answered \(8+7=15\); online answered \(16+15=31\), online not answered \(6+5=11\). 3. Help desk 2 did not keep the email/web-form split, so the combined data cannot recover it.

Answer

<table> <tr><th></th><th>Answered</th><th>Not answered</th><th>Total</th></tr> <tr><td>Phone</td><td>\(20\)</td><td>\(15\)</td><td>\(35\)</td></tr> <tr><td>Online</td><td>\(31\)</td><td>\(11\)</td><td>\(42\)</td></tr> <tr><td>Total</td><td>\(51\)</td><td>\(26\)</td><td>\(77\)</td></tr> </table> The separate email and web-form counts at desk 2 are not known.
Two-way tables for two categorical variables 49
A report claims that exactly two interior entries in the table below were swapped and that every displayed row and column total is correct. <table> <tr><th></th><th>approved</th><th>not approved</th><th>Total</th></tr> <tr><td>online application</td><td>\(44\)</td><td>\(18\)</td><td>\(68\)</td></tr> <tr><td>paper application</td><td>\(24\)</td><td>\(14\)</td><td>\(32\)</td></tr> <tr><td>Total</td><td>\(58\)</td><td>\(42\)</td><td>\(100\)</td></tr> </table> Determine whether the claim is possible. Support your conclusion by testing the swap that fixes the row totals.

Hints

- Test candidate swaps against both row totals and column totals. - Fixing one set of margins is not enough. - A contradiction may mean the stated error description is incomplete.

Solution

1. The online row needs counts summing to \(68\). Swapping \(18\) and \(24\) gives \(44+24=68\). 2. The paper row then gives \(18+14=32\). 3. The corrected column totals are \(44+18=62\) and \(24+14=38\), not \(58\) and \(42\). Therefore at least one additional entry or margin is wrong, so the claim that all displayed margins are correct is impossible.

Answer

No swap of only two interior entries can make the displayed row and column totals all correct. Swapping \(18\) and \(24\) fixes the row totals but produces column totals \(62\) and \(38\), not \(58\) and \(42\). The report is internally contradictory.

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