Worked solution P27-Easy-1. The row-A conditional proportion is 22/(22+12)=0.647. Compare it with row B's 0.447; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{22}{34}=0.647.$$ The grand total 81 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Easy-2. The row-A conditional proportion is 19/(19+15)=0.559. Compare it with row B's 0.396; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{19}{34}=0.559.$$ The grand total 82 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Easy-3. The row-A conditional proportion is 21/(21+19)=0.525. Compare it with row B's 0.432; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{21}{40}=0.525.$$ The grand total 77 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Easy-4. The row-A conditional proportion is 23/(23+18)=0.561. Compare it with row B's 0.412; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{23}{41}=0.561.$$ The grand total 75 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Easy-5. The row-A conditional proportion is 27/(27+15)=0.643. Compare it with row B's 0.435; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{27}{42}=0.643.$$ The grand total 88 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Easy-6. The row-A conditional proportion is 28/(28+15)=0.651. Compare it with row B's 0.458; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{28}{43}=0.651.$$ The grand total 91 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Easy-7. The row-A conditional proportion is 21/(21+16)=0.568. Compare it with row B's 0.400; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{21}{37}=0.568.$$ The grand total 87 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Easy-8. The row-A conditional proportion is 24/(24+19)=0.558. Compare it with row B's 0.523; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{24}{43}=0.558.$$ The grand total 87 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Easy-9. The row-A conditional proportion is 29/(29+13)=0.690. Compare it with row B's 0.460; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{29}{42}=0.690.$$ The grand total 92 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Easy-10. The row-A conditional proportion is 22/(22+17)=0.564. Compare it with row B's 0.375; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{22}{39}=0.564.$$ The grand total 79 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Easy-11. The row-A conditional proportion is 20/(20+18)=0.526. Compare it with row B's 0.489; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{20}{38}=0.526.$$ The grand total 83 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Easy-12. The row-A conditional proportion is 18/(18+19)=0.486. Compare it with row B's 0.460; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{18}{37}=0.486.$$ The grand total 87 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Easy-13. The row-A conditional proportion is 24/(24+16)=0.600. Compare it with row B's 0.412; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{24}{40}=0.600.$$ The grand total 91 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Easy-14. The row-A conditional proportion is 24/(24+12)=0.667. Compare it with row B's 0.341; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{24}{36}=0.667.$$ The grand total 80 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Easy-15. The row-A conditional proportion is 27/(27+14)=0.659. Compare it with row B's 0.440; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{27}{41}=0.659.$$ The grand total 91 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Easy-16. The row-A conditional proportion is 24/(24+19)=0.558. Compare it with row B's 0.442; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{24}{43}=0.558.$$ The grand total 86 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Tough-1. The row-A conditional proportion is 19/(19+18)=0.514. Compare it with row B's 0.439; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{19}{37}=0.514.$$ The grand total 78 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Tough-2. The row-A conditional proportion is 22/(22+16)=0.579. Compare it with row B's 0.333; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{22}{38}=0.579.$$ The grand total 83 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Tough-3. The row-A conditional proportion is 21/(21+14)=0.600. Compare it with row B's 0.479; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{21}{35}=0.600.$$ The grand total 83 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Tough-4. The row-A conditional proportion is 22/(22+20)=0.524. Compare it with row B's 0.378; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{22}{42}=0.524.$$ The grand total 87 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Tough-5. The row-A conditional proportion is 28/(28+19)=0.596. Compare it with row B's 0.455; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{28}{47}=0.596.$$ The grand total 91 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Tough-6. The row-A conditional proportion is 22/(22+14)=0.611. Compare it with row B's 0.404; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{22}{36}=0.611.$$ The grand total 88 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Tough-7. The row-A conditional proportion is 27/(27+12)=0.692. Compare it with row B's 0.474; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{27}{39}=0.692.$$ The grand total 77 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Tough-8. The row-A conditional proportion is 22/(22+14)=0.611. Compare it with row B's 0.292; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{22}{36}=0.611.$$ The grand total 84 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Tough-9. The row-A conditional proportion is 19/(19+14)=0.576. Compare it with row B's 0.429; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{19}{33}=0.576.$$ The grand total 82 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Tough-10. The row-A conditional proportion is 24/(24+12)=0.667. Compare it with row B's 0.350; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{24}{36}=0.667.$$ The grand total 76 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Tough-11. The row-A conditional proportion is 20/(20+12)=0.625. Compare it with row B's 0.409; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{20}{32}=0.625.$$ The grand total 76 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Tough-12. The row-A conditional proportion is 30/(30+18)=0.625. Compare it with row B's 0.439; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{30}{48}=0.625.$$ The grand total 89 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Tough-13. The row-A conditional proportion is 25/(25+17)=0.595. Compare it with row B's 0.358; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{25}{42}=0.595.$$ The grand total 95 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Tough-14. The row-A conditional proportion is 30/(30+17)=0.638. Compare it with row B's 0.359; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{30}{47}=0.638.$$ The grand total 86 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Tough-15. The row-A conditional proportion is 21/(21+18)=0.538. Compare it with row B's 0.407; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{21}{39}=0.538.$$ The grand total 93 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Tough-16. The row-A conditional proportion is 23/(23+19)=0.548. Compare it with row B's 0.375; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{23}{42}=0.548.$$ The grand total 90 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Toughest-1. The row-A conditional proportion is 20/(20+16)=0.556. Compare it with row B's 0.386; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{20}{36}=0.556.$$ The grand total 80 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Toughest-2. The row-A conditional proportion is 30/(30+20)=0.600. Compare it with row B's 0.370; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{30}{50}=0.600.$$ The grand total 104 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Toughest-3. The row-A conditional proportion is 27/(27+20)=0.574. Compare it with row B's 0.380; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{27}{47}=0.574.$$ The grand total 97 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Toughest-4. The row-A conditional proportion is 21/(21+12)=0.636. Compare it with row B's 0.425; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{21}{33}=0.636.$$ The grand total 73 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Toughest-5. The row-A conditional proportion is 27/(27+20)=0.574. Compare it with row B's 0.389; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{27}{47}=0.574.$$ The grand total 83 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Toughest-6. The row-A conditional proportion is 20/(20+19)=0.513. Compare it with row B's 0.411; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{20}{39}=0.513.$$ The grand total 95 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Toughest-7. The row-A conditional proportion is 27/(27+17)=0.614. Compare it with row B's 0.457; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{27}{44}=0.614.$$ The grand total 90 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Toughest-8. The row-A conditional proportion is 19/(19+16)=0.543. Compare it with row B's 0.442; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{19}{35}=0.543.$$ The grand total 78 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Toughest-9. The row-A conditional proportion is 28/(28+12)=0.700. Compare it with row B's 0.319; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{28}{40}=0.700.$$ The grand total 87 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Toughest-10. The row-A conditional proportion is 26/(26+12)=0.684. Compare it with row B's 0.396; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{26}{38}=0.684.$$ The grand total 86 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Toughest-11. The row-A conditional proportion is 18/(18+19)=0.486. Compare it with row B's 0.434; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{18}{37}=0.486.$$ The grand total 90 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Toughest-12. The row-A conditional proportion is 23/(23+14)=0.622. Compare it with row B's 0.326; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{23}{37}=0.622.$$ The grand total 83 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Toughest-13. The row-A conditional proportion is 21/(21+19)=0.525. Compare it with row B's 0.462; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{21}{40}=0.525.$$ The grand total 79 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Toughest-14. The row-A conditional proportion is 30/(30+15)=0.667. Compare it with row B's 0.435; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{30}{45}=0.667.$$ The grand total 91 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Toughest-15. The row-A conditional proportion is 25/(25+14)=0.641. Compare it with row B's 0.450; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{25}{39}=0.641.$$ The grand total 79 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.

Worked solution P27-Toughest-16. The row-A conditional proportion is 19/(19+13)=0.594. Compare it with row B's 0.488; unequal conditional proportions indicate association in the table. $$P(\text{Yes}\mid A)=\frac{19}{32}=0.594.$$ The grand total 73 is not the conditional denominator. Interpretation: The comparison concerns distributions within row groups and does not by itself establish causation. Validity: Use the group named after the conditioning bar as the denominator and report which variable defines each row and column. Error to reject: Dividing by the grand total produces a joint relative frequency, not a conditional relative frequency.