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Expected counts in two-way tables

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55039512
A survey records \(100\) students in a \(2 \times 2\) table. One row total is \(50\), and one column total is \(30\). If the row and column variables are independent, what is the expected count in the cell where that row and column meet?

Hints

- Identify the row total, column total, and grand total associated with the cell. - Expected counts under independence come from the table margins rather than the observed count in the cell. - Combine the two relevant margins as a proportion of the grand total.

Solution

1. Under independence, multiply the cell's row total by its column total and divide by the grand total. 2. The expected count is \(\frac{50\cdot 30}{100}=15\).

Answer

\(15\)
55039612
In a two-way table, a student says, “To find an expected count under independence, I need the observed count from that same cell.” Is the student correct? State the information that actually determines an expected cell count.

Hints

- Separate the observed data inside a cell from the totals around the edges of the table. - Think about what information describes the two marginal distributions. - Ask which quantities would remain available if the interior cell counts were hidden.

Solution

1. The statement is incorrect. 2. Under independence, an expected cell count is determined by the cell's row total, the cell's column total, and the grand total. 3. The observed count in that individual cell is not used to calculate its expected count.

Answer

No. The expected count is determined by the relevant row total, column total, and grand total.
55039712
A table contains \(120\) students. One row contains \(\frac{1}{4}\) of all students, and one column contains \(\frac{1}{3}\) of all students. If the row and column variables are independent, what is the expected count where that row and column meet?

Hints

- Convert the row and column information into proportions of the whole table. - Independence connects the two marginal proportions multiplicatively. - After finding the expected cell proportion, convert it back to a count.

Solution

1. Under independence, the expected proportion in the cell is the product of the row and column proportions. 2. The expected count is \(120\cdot\frac{1}{4}\cdot\frac{1}{3}=10\).

Answer

\(10\)
55039812
A two-way table has these margins: <table><tr><th></th><th>Option A</th><th>Option B</th><th>Total</th></tr><tr><th>Group 1</th><td>?</td><td>?</td><td>\(40\)</td></tr><tr><th>Group 2</th><td>?</td><td>?</td><td>\(60\)</td></tr><tr><th>Total</th><td>\(50\)</td><td>\(50\)</td><td>\(100\)</td></tr></table> Complete the four expected counts under independence.

Hints

- Work from the margins shown in the last row and last column. - Each expected count uses one row total and one column total. - Check that your expected counts add back to the given row and column totals.

Solution

1. For each cell, use \(E=\frac{(\text{row total})(\text{column total})}{\text{grand total}}\). 2. Group 1 has expected counts \(\frac{40\cdot 50}{100}=20\) and \(20\). 3. Group 2 has expected counts \(\frac{60\cdot 50}{100}=30\) and \(30\).

Answer

<table><tr><th></th><th>Option A</th><th>Option B</th></tr><tr><th>Group 1</th><td>\(20\)</td><td>\(20\)</td></tr><tr><th>Group 2</th><td>\(30\)</td><td>\(30\)</td></tr></table>
55039912
A \(2 \times 3\) table has row totals \(30\) and \(70\), column totals \(20\), \(50\), and \(30\), and grand total \(100\). Find all six expected counts under independence.

Hints

- Keep the same row total while moving across a row, and change only the column total. - The grand total is the denominator for every expected count. - Use the margins to check your completed expected table.

Solution

1. Use \(E=\frac{(\text{row total})(\text{column total})}{100}\) for each cell. 2. The first row expectations are \(6\), \(15\), and \(9\). 3. The second row expectations are \(14\), \(35\), and \(21\). 4. These expected counts reproduce the given row and column totals.

Answer

First row: \((6, 15, 9)\). Second row: \((14, 35, 21)\).
55040012
A \(2 \times 2\) table has row totals \(45\) and \(55\), column totals \(30\) and \(70\), and grand total \(100\). Find the expected count in every cell under independence.

Hints

- Expected counts do not have to be whole numbers. - Use one row total and one column total for each cell. - Check that each row of expected counts sums to its row total.

Solution

1. Apply \(E=\frac{(\text{row total})(\text{column total})}{100}\) to each cell. 2. The first row expectations are \(\frac{45\cdot 30}{100}=13.5\) and \(\frac{45\cdot 70}{100}=31.5\). 3. The second row expectations are \(\frac{55\cdot 30}{100}=16.5\) and \(\frac{55\cdot 70}{100}=38.5\).

Answer

First row: \((13.5, 31.5)\). Second row: \((16.5, 38.5)\).
55040112
A \(3 \times 2\) table has row totals \(20\), \(30\), and \(50\), column totals \(40\) and \(60\), and grand total \(100\). Complete the expected-count table under independence.

Hints

- The same two column proportions apply to every row under independence. - Compute one row at a time from its row total. - Verify the column sums after completing the table.

Solution

1. Multiply each row total by each column total and divide by \(100\). 2. The first row expectations are \((8, 12)\). 3. The second row expectations are \((12, 18)\). 4. The third row expectations are \((20, 30)\).

Answer

Expected rows: \((8, 12)\), \((12, 18)\), and \((20, 30)\).
55040212
In a survey of \(180\) people, one response category contains \(84\) people and one age category contains \(45\) people. If response and age category are independent, what is the expected count in their intersection?

Hints

- Treat the two category totals as margins of a two-way table. - Independence determines the intersection from the two marginal totals. - Simplify the fraction before multiplying if that makes the arithmetic easier.

Solution

1. Use the relevant row total \(84\), column total \(45\), and grand total \(180\). 2. The expected count is \(\frac{84\cdot 45}{180}=21\).

Answer

\(21\)
55040312
A \(2 \times 3\) table has row totals \(80\) and \(120\), column totals \(50\), \(70\), and \(80\), and grand total \(200\). Find the three expected counts in the first row under independence, then identify which column has the largest expected first-row count.

Hints

- You only need the first-row total, each column total, and the grand total. - The expected counts in one row are proportional to the column totals. - Compare the three completed expectations after calculating them.

Solution

1. Use the first-row total \(80\) with each column total. 2. The expected counts are \(\frac{80\cdot 50}{200}=20\), \(\frac{80\cdot 70}{200}=28\), and \(\frac{80\cdot 80}{200}=32\). 3. The largest expected count is \(32\), in the third column.

Answer

Expected first row: \((20, 28, 32)\). The third column has the largest expected count.
55040412
A \(2 \times 2\) table has row totals \(12\) and \(88\), column totals \(20\) and \(80\), and grand total \(100\). a) Find all four expected counts under independence. b) Which expected counts are below \(5\)? c) Does the usual high-school chi-square condition that every expected count be at least \(5\) hold?

Hints

- Compute the entire expected table before checking the condition. - Expected-count conditions concern the expected values, not the observed cell counts. - Compare each expectation with the stated threshold separately.

Solution

1. The expected counts are \(\frac{12\cdot 20}{100}=2.4\), \(\frac{12\cdot 80}{100}=9.6\), \(\frac{88\cdot 20}{100}=17.6\), and \(\frac{88\cdot 80}{100}=70.4\). 2. Only the cell with expected count \(2.4\) is below \(5\). 3. Therefore, the stated expected-count condition does not hold.

Answer

a) Expected rows: \((2.4, 9.6)\) and \((17.6, 70.4)\). b) Only \(2.4\) is below \(5\). c) No.
55040512
In a two-way table with grand total \(150\), one column total is \(45\). Under independence, the expected count in a cell in that column is \(18\). What is the total for the row containing that cell?

Hints

- Write the expected-count relationship with the row total as the unknown. - The known expected count, column total, and grand total determine the missing margin. - Rearrange the equation before substituting or simplifying.

Solution

1. Let the unknown row total be \(r\). 2. The expected-count equation is \(18=\frac{45r}{150}\). 3. Solving gives \(r=\frac{18\cdot 150}{45}=60\).

Answer

\(60\)
55040612
A two-way table has grand total \(100\). One row total is \(35\), and the expected count in one cell of that row is \(14\). Under independence, find the column total for that cell.

Hints

- Treat the expected-count formula as an equation rather than a one-way calculation. - The row total and grand total are already known. - Isolate the missing column total algebraically.

Solution

1. Let the unknown column total be \(c\). 2. Use \(14=\frac{35c}{100}\). 3. Solving gives \(c=\frac{14\cdot 100}{35}=40\).

Answer

\(40\)
54737612
A \(2 \times 3\) table has row totals \(40\) and \(60\), column totals \(30\), \(50\), and \(20\), and grand total \(100\). The observed first row is \((18, 14, 8)\). a) Find all expected cell counts under independence. b) Compute observed minus expected for every cell. c) Identify which column shows the largest absolute departure in the first row.

Hints

- Under independence, expected counts are determined entirely by the margins. - Recover the unlisted observed row from the column totals. - Residuals compare observed counts with independence expectations cell by cell.

Solution

1. Expected counts are row total times column total divided by the grand total. The first row expectations are \((12, 20, 8)\), and the second row expectations are \((18, 30, 12)\). 2. The observed second row is obtained from column totals: \((12, 36, 12)\). 3. Residuals are first row \((6, -6, 0)\) and second row \((-6, 6, 0)\). 4. The first two columns tie for the largest absolute first-row residual, each with magnitude \(6\).

Answer

a) Expected table: first row \((12, 20, 8)\), second row \((18, 30, 12)\). b) Residuals: first row \((6, -6, 0)\), second row \((-6, 6, 0)\). c) Columns 1 and 2 tie, with absolute residual \(6\).
55040712
Under independence, a cell has expected count \(24\). Its row total is \(60\), and its column total is \(80\). Find the grand total of the two-way table.

Hints

- This time the denominator of the expected-count formula is the unknown. - Write an equation that uses the two known margins and the expected count. - Rearrange carefully so the grand total is isolated.

Solution

1. Let the grand total be \(N\). 2. The expected-count equation is \(24=\frac{60\cdot 80}{N}\). 3. Solving gives \(N=\frac{60\cdot 80}{24}=200\).

Answer

\(200\)
55040812
A two-way table contains \(200\) observations. Row A contains \(80\) observations, and column X contains \(50\). Under independence, find the expected count in cell A-X in two ways: a) using the expected-count formula; b) using the idea that row A should make up the same proportion of column X as it does of the whole table. Explain why the two methods agree.

Hints

- First determine what fraction of the whole table belongs to row A. - Under independence, that row fraction should not change when you restrict attention to column X. - Compare the algebraic form of the proportion method with the standard expected-count formula.

Solution

1. Formula method: \(E=\frac{80\cdot 50}{200}=20\). 2. Proportion method: row A is \(\frac{80}{200}=0.4\) of the whole table, so under independence it should be \(0.4\) of column X. Thus \(0.4\cdot 50=20\). 3. The methods agree because the formula \(\frac{80\cdot 50}{200}\) is exactly the row proportion \(\frac{80}{200}\) multiplied by the column total \(50\).

Answer

a) \(20\) b) \(20\). Both methods express the same independence model: the row proportion is preserved within the column.
55040912
A cell belongs to a row with total \(30\) and a column with total \(40\) in a table with grand total \(100\). A student calculates its expected count as \(\frac{30\cdot 40}{30}=40\). Identify the error and give the correct expected count under independence.

Hints

- Check each part of the expected-count formula against the table margins. - Ask what quantity should normalize the product of a row total and a column total. - Test whether the student's result is plausible relative to the row and column totals.

Solution

1. The denominator in the expected-count formula must be the grand total, not the row total. 2. The correct calculation is \(\frac{30\cdot 40}{100}=12\). 3. The student's value \(40\) would incorrectly assign the entire column total to this one row.

Answer

The student divided by the row total instead of the grand total. The correct expected count is \(12\).
55041012
In one row of a two-way table, the expected counts under independence are \((10, 20, 30)\). The observed counts in the first two cells are \(14\) and \(17\), and the observed row total is \(60\). a) Find the third observed count. b) Find the three residuals \(O-E\). c) Explain why the residuals in this row add to \(0\).

Hints

- Use the observed row total before calculating residuals. - Compare each observed cell with its matching expected cell. - Think about what happens when you subtract two rows that have the same total.

Solution

1. The third observed count is \(60-14-17=29\). 2. The residuals are \(14-10=4\), \(17-20=-3\), and \(29-30=-1\). 3. Their sum is \(4-3-1=0\). 4. Expected counts preserve the row total, so the observed and expected row sums are both \(60\); therefore the row's observed-minus-expected residuals must sum to \(0\).

Answer

a) \(29\) b) \((4, -3, -1)\) c) They sum to \(0\) because the observed and expected counts have the same row total.
55041112
In a \(2 \times 3\) table with grand total \(200\), the first row total is \(60\). Under independence, the expected counts in the first row are \((12, 18, 30)\). a) Recover the three column totals. b) Find the expected counts in the second row.

Hints

- Work backward from each first-row expected count to its column margin. - All three reverse equations share the same row total and grand total. - After finding the column totals, determine the second row total from the grand total.

Solution

1. For each column total \(c\), the first-row expectation satisfies \(E=\frac{60c}{200}\), so \(c=\frac{200E}{60}\). 2. The column totals are \(40\), \(60\), and \(100\). 3. The second row total is \(200-60=140\). 4. Its expected counts are \(\frac{140\cdot 40}{200}=28\), \(\frac{140\cdot 60}{200}=42\), and \(\frac{140\cdot 100}{200}=70\).

Answer

a) Column totals: \((40, 60, 100)\). b) Second-row expected counts: \((28, 42, 70)\).
55041212
A \(2 \times 2\) observed table has row totals \(50\) and \(50\), column totals \(30\) and \(70\), and grand total \(100\). A student proposes this expected table under independence: <table><tr><th></th><th>Column 1</th><th>Column 2</th></tr><tr><th>Row 1</th><td>\(20\)</td><td>\(30\)</td></tr><tr><th>Row 2</th><td>\(10\)</td><td>\(40\)</td></tr></table> The proposed table has the correct row and column totals. Is it nevertheless a valid expected table under independence? Explain and give the correct expected counts.

Hints

- Preserving margins is one check, but independence imposes an additional proportional structure. - Compare each row's share of the grand total with its share of each column. - Recalculate at least one proposed cell from the expected-count formula.

Solution

1. Matching the margins is necessary but not sufficient for an independence expected table. 2. Because both row totals are \(50\), each row is half of the grand total, so each row should receive half of each column total. 3. The correct expectations are \(15\) and \(35\) in each row. 4. The proposed table is invalid because its interior counts do not follow the independence formula even though its margins match.

Answer

No. The correct expected table has rows \((15, 35)\) and \((15, 35)\).
55041312
The two panels show observed counts for Group A and Group B across categories X, Y, and Z. Panel a) is Group A and panel b) is Group B. a) Read the six observed counts and find the row totals, column totals, and grand total. b) Find all expected counts under independence. c) Which cell has the largest absolute residual \(|O-E|\)?
Figure for problem 550413

Hints

- Read each bar height from the count scale before doing any expected-count calculations. - Build the margins from the six observed counts you read from the panels. - After finding expectations, compare observed and expected counts cell by cell rather than comparing raw bar heights alone.

Solution

1. From the chart, Group A is \((28, 21, 21)\) and Group B is \((35, 42, 28)\). The row totals are \(70\) and \(105\); the column totals are \(63\), \(63\), and \(49\); the grand total is \(175\). 2. The expected counts for Group A are \((25.2, 25.2, 19.6)\), and those for Group B are \((37.8, 37.8, 29.4)\). 3. The residuals for Group A are \((2.8, -4.2, 1.4)\), and those for Group B are \((-2.8, 4.2, -1.4)\). 4. The largest absolute residual is \(4.2\), occurring in category Y for both groups.

Answer

a) Observed rows: \((28, 21, 21)\) and \((35, 42, 28)\); row totals \((70, 105)\); column totals \((63, 63, 49)\); grand total \(175\). b) Expected rows: \((25.2, 25.2, 19.6)\) and \((37.8, 37.8, 29.4)\). c) Category Y in both groups, with \(|O-E|=4.2\).

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