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Construct two-way tables

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5101438
A health survey of \(1000\) participants records whether each person exercises regularly, event \(S\), and eats a balanced diet, event \(G\). The survey finds that \(600\) participants exercise regularly, \(550\) eat a balanced diet, and \(400\) do both. a) Complete a two-way table for the data. b) Explain the meanings of \(S\cap G\) and \(S^c\cap G^c\) in context. c) How many participants exercise regularly or eat a balanced diet, or both? Give a formula. d) Find \(n(S^c\cup G)\).

Hints

- Enter the given intersection and marginal totals in the two-way table first. - Use row and column totals to determine the missing cells. - For a union, use inclusion-exclusion or the complement of the “neither” cell.

Solution

1. Start with \(n(S\cap G)=400\). Since \(n(S)=600\), \(n(S\cap G^c)=600-400=200\). 2. Since \(n(G)=550\), \(n(S^c\cap G)=550-400=150\). 3. The number who do neither is \(1000-(400+200+150)=250\). The remaining row and column totals are \(450\) and \(400\). 4. In context, \(S\cap G\) means the participant exercises regularly and eats a balanced diet. The event \(S^c\cap G^c\) means the participant does neither. 5. Use inclusion-exclusion: \(n(S\cup G)=n(S)+n(G)-n(S\cap G)=600+550-400=750\). 6. Use the completed table or inclusion-exclusion: \(n(S^c\cup G)=n(S^c)+n(G)-n(S^c\cap G)=400+550-150=800\).

Answer

a) <table><tr><th></th><th>\(S\)</th><th>\(S^c\)</th><th>Total</th></tr><tr><th>\(G\)</th><td>\(400\)</td><td>\(150\)</td><td>\(550\)</td></tr><tr><th>\(G^c\)</th><td>\(200\)</td><td>\(250\)</td><td>\(450\)</td></tr><tr><th>Total</th><td>\(600\)</td><td>\(400\)</td><td>\(1000\)</td></tr></table> b) \(S\cap G\) means the participant does both; \(S^c\cap G^c\) means the participant does neither. c) \(n(S\cup G)=n(S)+n(G)-n(S\cap G)=750\). d) \(n(S^c\cup G)=800\).
5101448
A veterinary clinic surveys \(200\) pet owners about whether they own a dog, event \(D\), or a cat, event \(C\). <table><tr><th></th><th>\(D\)</th><th>\(D^c\)</th><th>Total</th></tr><tr><th>\(C\)</th><td>\(30\)</td><td></td><td></td></tr><tr><th>\(C^c\)</th><td></td><td>\(55\)</td><td>\(110\)</td></tr><tr><th>Total</th><td>\(85\)</td><td></td><td>\(200\)</td></tr></table> a) How many respondents own neither a dog nor a cat? b) Complete the two-way table. c) Explain the meaning of \(D^c\cap C\) in context. d) Find \(n(D\cup C)\) and \(n(D\cup C^c)\).

Hints

- Identify the table cell that represents owning neither pet. - Use row and column totals to fill the missing entries. - For each union, either add the relevant cells or use inclusion-exclusion.

Solution

1. The “neither” cell is \(D^c\cap C^c\), so \(55\) respondents own neither pet. 2. The cat-owner row total is \(200-110=90\), and the no-dog column total is \(200-85=115\). 3. The missing interior cells are \(n(D^c\cap C)=115-55=60\) and \(n(D\cap C^c)=110-55=55\). 4. The event \(D^c\cap C\) means the respondent owns a cat but does not own a dog. 5. Use the table or inclusion-exclusion: \(n(D\cup C)=85+90-30=145\). 6. Similarly, \(n(D\cup C^c)=85+110-55=140\).

Answer

a) \(55\) respondents own neither a dog nor a cat. b) <table><tr><th></th><th>\(D\)</th><th>\(D^c\)</th><th>Total</th></tr><tr><th>\(C\)</th><td>\(30\)</td><td>\(60\)</td><td>\(90\)</td></tr><tr><th>\(C^c\)</th><td>\(55\)</td><td>\(55\)</td><td>\(110\)</td></tr><tr><th>Total</th><td>\(85\)</td><td>\(115\)</td><td>\(200\)</td></tr></table> c) The respondent owns a cat but does not own a dog. d) \(n(D\cup C)=145\), and \(n(D\cup C^c)=140\).
5101458
A school surveys \(500\) students about their interest in mathematics, event \(M\), and physics, event \(P\). Some values are shown in the two-way table. <table><tr><th></th><th>\(M\)</th><th>\(M^c\)</th><th>Total</th></tr><tr><th>\(P\)</th><td>\(120\)</td><td></td><td>\(250\)</td></tr><tr><th>\(P^c\)</th><td></td><td>\(70\)</td><td></td></tr><tr><th>Total</th><td>\(300\)</td><td></td><td>\(500\)</td></tr></table> a) Complete the table. b) Explain the meanings of \(120\) and \(70\) in context and name their corresponding intersections. c) Describe \(M\cap P^c\) in words. d) Find \(n(M\cup P)\) and \(n(M^c\cup P)\).

Hints

- Start with the given marginal totals to determine the missing row and column totals. - Match each interior cell with an intersection of two events. - When finding a union, subtract the intersection so it is not counted twice.

Solution

1. The no-math column total is \(500-300=200\), and the no-physics row total is \(500-250=250\). 2. The missing interior cells are \(n(M^c\cap P)=250-120=130\) and \(n(M\cap P^c)=300-120=180\). 3. The value \(120\) represents students interested in both subjects, \(M\cap P\). The value \(70\) represents students interested in neither subject, \(M^c\cap P^c\). 4. The event \(M\cap P^c\) means the student is interested in mathematics but not physics. 5. Use inclusion-exclusion: \(n(M\cup P)=300+250-120=430\). 6. Also, \(n(M^c\cup P)=200+250-130=320\).

Answer

a) <table><tr><th></th><th>\(M\)</th><th>\(M^c\)</th><th>Total</th></tr><tr><th>\(P\)</th><td>\(120\)</td><td>\(130\)</td><td>\(250\)</td></tr><tr><th>\(P^c\)</th><td>\(180\)</td><td>\(70\)</td><td>\(250\)</td></tr><tr><th>Total</th><td>\(300\)</td><td>\(200\)</td><td>\(500\)</td></tr></table> b) \(120\) is \(n(M\cap P)\), the number interested in both subjects. \(70\) is \(n(M^c\cap P^c)\), the number interested in neither subject. c) The student is interested in mathematics but not physics. d) \(n(M\cup P)=430\), and \(n(M^c\cup P)=320\).
5147288
A school surveys \(25\) students about two activities. Event \(M\) is “plays a musical instrument,” and event \(S\) is “plays a sport regularly.” The survey finds that \(12\) students play an instrument, \(15\) play a sport regularly, and \(4\) do neither. a) Create a complete two-way table of counts. b) How many students both play an instrument and play a sport regularly? c) What percentage of the students play a sport regularly but do not play an instrument?

Hints

- Identify the total number surveyed and the row and column totals. - Use a known total and one interior cell to find the other cell in that row or column. - For the percentage, divide the relevant cell by the total number surveyed.

Solution

1. The total is \(25\). The no-instrument total is \(25-12=13\), and the no-sport total is \(25-15=10\). 2. Since \(4\) students do neither, the number who play a sport but no instrument is \(13-4=9\). 3. The number who do both is \(15-9=6\). 4. The number who play an instrument but no sport is \(12-6=6\). 5. For part c, the required percentage is \(\frac{9}{25}=0.36=36\%\).

Answer

a) <table><tr><th></th><th>\(S\)</th><th>\(S^c\)</th><th>Total</th></tr><tr><th>\(M\)</th><td>\(6\)</td><td>\(6\)</td><td>\(12\)</td></tr><tr><th>\(M^c\)</th><td>\(9\)</td><td>\(4\)</td><td>\(13\)</td></tr><tr><th>Total</th><td>\(15\)</td><td>\(10\)</td><td>\(25\)</td></tr></table> b) \(6\) students do both activities. c) \(36\%\) play a sport regularly but do not play an instrument.
5147298
An electronics store reviews \(200\) smartphones that were sold. Event \(A\) is “the phone uses Android,” and event \(S\) is “the phone was sold with a screen protector.” Of the phones, \(120\) use Android, \(90\) were sold with a screen protector, and \(70\%\) of the Android phones were sold without a screen protector. a) Find the number of Android phones sold with a screen protector. b) Complete a two-way table for the data. c) How many non-Android phones were sold with a screen protector?

Hints

- Convert the percentage of the Android subgroup to a count first. - Check that every row and column total is consistent. - Decide which table cell is described by “\(70\%\) of the Android phones.”

Solution

1. The number of Android phones sold without a screen protector is \(0.70\cdot120=84\). 2. Therefore, the number of Android phones sold with a screen protector is \(120-84=36\). 3. The number of non-Android phones is \(200-120=80\), and the number sold without a screen protector is \(200-90=110\). 4. The number of non-Android phones sold with a screen protector is \(90-36=54\). 5. The remaining cell is \(80-54=26\).

Answer

a) \(36\) Android phones were sold with a screen protector. b) <table><tr><th></th><th>\(S\)</th><th>\(S^c\)</th><th>Total</th></tr><tr><th>\(A\)</th><td>\(36\)</td><td>\(84\)</td><td>\(120\)</td></tr><tr><th>\(A^c\)</th><td>\(54\)</td><td>\(26\)</td><td>\(80\)</td></tr><tr><th>Total</th><td>\(90\)</td><td>\(110\)</td><td>\(200\)</td></tr></table> c) \(54\) non-Android phones were sold with a screen protector.
5147308
A quality-control team checks \(500\) components for two possible defects, \(D_1\) and \(D_2\). The data show that \(465\) components have neither defect, \(20\) have defect \(D_1\), and \(25\) have defect \(D_2\). a) Create a two-way table of counts. b) How many components have both defects? c) A component is called “slightly damaged” if it has exactly one of the two defects. How many components are slightly damaged?

Hints

- Identify the cell that represents having neither defect. - Determine the two exclusive defect cells before finding the overlap. - “Exactly one defect” includes the two nonoverlapping defect cells.

Solution

1. The no-\(D_1\) total is \(500-20=480\), and the no-\(D_2\) total is \(500-25=475\). 2. The number with \(D_1\) but not \(D_2\) is \(475-465=10\). 3. The number with both defects is \(20-10=10\). 4. The number with \(D_2\) but not \(D_1\) is \(25-10=15\). 5. Exactly one defect means one of the two exclusive cells, so the total is \(10+15=25\).

Answer

a) <table><tr><th></th><th>\(D_2\)</th><th>\(D_2^c\)</th><th>Total</th></tr><tr><th>\(D_1\)</th><td>\(10\)</td><td>\(10\)</td><td>\(20\)</td></tr><tr><th>\(D_1^c\)</th><td>\(15\)</td><td>\(465\)</td><td>\(480\)</td></tr><tr><th>Total</th><td>\(25\)</td><td>\(475\)</td><td>\(500\)</td></tr></table> b) \(10\) components have both defects. c) \(25\) components are slightly damaged.
5147398
A survey asked \(100\) households about their pets. Of those households, \(40\) have at least one dog, \(30\) have at least one cat, and \(15\) have both at least one dog and at least one cat. a) Organize the data in a complete two-way table for the categories “has a dog” and “has a cat.” b) How many households have neither a dog nor a cat?

Hints

- Begin with the cell where the dog and cat categories overlap. - Subtract the overlap from each category total. - The four interior cells must add to \(100\).

Solution

1. The intersection of the “has a dog” and “has a cat” categories contains \(15\) households. 2. The number that have a dog but no cat is \(40-15=25\). 3. The number that have a cat but no dog is \(30-15=15\). 4. The number that have neither pet is \(100-(15+25+15)=45\). 5. The row and column totals are found by addition.

Answer

a) <table><tr><th></th><th>Has a cat</th><th>Does not have a cat</th><th>Total</th></tr><tr><th>Has a dog</th><td>\(15\)</td><td>\(25\)</td><td>\(40\)</td></tr><tr><th>Does not have a dog</th><td>\(15\)</td><td>\(45\)</td><td>\(60\)</td></tr><tr><th>Total</th><td>\(30\)</td><td>\(70\)</td><td>\(100\)</td></tr></table> b) \(45\) households have neither a dog nor a cat.
5147418
A quality-control team inspects \(1000\) components for two possible defects: a material defect, represented by \(M\), and a shape defect, represented by \(S\). The team finds that: - \(920\) components do not have a material defect. - \(35\) components have a shape defect but no material defect. - \(20\) components have both defects. a) Complete a two-way table of counts. b) How many components have a shape defect? c) Find \(n(M^c\cup S^c)\). Explain what this number means in context.

Hints

- Place the three given counts in the table before finding the missing cells. - The total with a shape defect is the sum of the two cells in that row. - For part c, identify the complement of “has both defects.”

Solution

1. Since \(920\) components do not have a material defect, \(1000-920=80\) have a material defect. 2. Of those \(80\), \(20\) also have a shape defect, so \(80-20=60\) have a material defect but no shape defect. 3. The number with a shape defect is \(20+35=55\). 4. The number with neither defect is \(920-35=885\). 5. By De Morgan's law, \(M^c\cup S^c=(M\cap S)^c\). Therefore, \(n(M^c\cup S^c)=1000-20=980\). 6. These \(980\) components do not have both defects; each has at most one of the two defects.

Answer

a) <table><tr><th></th><th>\(M\)</th><th>\(M^c\)</th><th>Total</th></tr><tr><th>\(S\)</th><td>\(20\)</td><td>\(35\)</td><td>\(55\)</td></tr><tr><th>\(S^c\)</th><td>\(60\)</td><td>\(885\)</td><td>\(945\)</td></tr><tr><th>Total</th><td>\(80\)</td><td>\(920\)</td><td>\(1000\)</td></tr></table> b) \(55\) components have a shape defect. c) \(n(M^c\cup S^c)=980\). This means that \(980\) components do not have both defects, so each has at most one of the two defects.
5386768
A survey of \(80\) students records whether each student plays an instrument, represented by \(I\), and belongs to a club, represented by \(C\). Complete the two-way table. <table><tr><th></th><th>\(C\)</th><th>\(C^c\)</th><th>Total</th></tr><tr><th>\(I\)</th><td>\(18\)</td><td>\(14\)</td><td></td></tr><tr><th>\(I^c\)</th><td>\(22\)</td><td></td><td></td></tr><tr><th>Total</th><td></td><td></td><td>\(80\)</td></tr></table>

Hints

- Begin with a row or column that has only one missing value. - Check that both sets of totals lead to the same grand total.

Solution

1. The total for row \(I\) is \(18+14=32\). 2. The total for column \(C\) is \(18+22=40\). 3. Therefore, the total for column \(C^c\) is \(80-40=40\). 4. The missing interior value is \(40-14=26\). 5. The total for row \(I^c\) is \(22+26=48\).

Answer

<table><tr><th></th><th>\(C\)</th><th>\(C^c\)</th><th>Total</th></tr><tr><th>\(I\)</th><td>\(18\)</td><td>\(14\)</td><td>\(32\)</td></tr><tr><th>\(I^c\)</th><td>\(22\)</td><td>\(26\)</td><td>\(48\)</td></tr><tr><th>Total</th><td>\(40\)</td><td>\(40\)</td><td>\(80\)</td></tr></table>

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