Two-Way Tables and Relative Frequency: Marginal and Conditional
Two-Way Tables and Relative Frequency: Marginal and Conditional practice bank: Solve the visible multiple-choice and free-response questions, then compare every step with the worked answers.
Two-Way Tables and Relative Frequency: Marginal and Conditional: formulas and targets
| Conditional relative frequency | cell count ÷ conditioning row or column total |
|---|---|
| Joint relative frequency | cell count ÷ grand total |
Two-Way Tables and Conditional Relative Frequency multiple-choice practice
Question 1. Two-Way Tables and Conditional Relative Frequency
A two-way table for a regional hospital in Metro East during a school-year data collection has Group A counts 60 meeting and 19 not meeting the appointment completion criterion; Group B counts are 48 and 61. Compare the conditional proportions that meet the criterion and comment on association.
Answer: C
Within Group A, p̂A=60(60+19)=0.759. Within Group B, p̂B=48(48+61)=0.44. The difference is 0.319 (31.9% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice A: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.759 for A and 0.44 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice B: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.759 for A and 0.44 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.759 for A and 0.44 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 2. Two-Way Tables and Conditional Relative Frequency
A two-way table for a regional hospital in Metro East during a monthly quality review has Group A counts 38 meeting and 22 not meeting the appointment completion criterion; Group B counts are 55 and 44. Compare the conditional proportions that meet the criterion and comment on association.
Answer: D
Within Group A, p̂A=38(38+22)=0.633. Within Group B, p̂B=55(55+44)=0.556. The difference is 0.078 (7.8% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice A: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.633 for A and 0.556 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice B: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.633 for A and 0.556 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice C: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.633 for A and 0.556 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 3. Two-Way Tables and Conditional Relative Frequency
A two-way table for a housing authority in Pacific Northwest during a community outreach cycle has Group A counts 30 meeting and 32 not meeting the application processing time criterion; Group B counts are 55 and 55. Compare the conditional proportions that meet the criterion and comment on association.
Answer: A
Within Group A, p̂A=30(30+32)=0.484. Within Group B, p̂B=55(55+55)=0.5. The difference is -0.016 (1.6% percentage points in magnitude). The conditional proportions are very similar, so the table gives little evidence of association.
Why the other choices fail
- Choice B: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.484 for A and 0.5 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice C: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.484 for A and 0.5 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.484 for A and 0.5 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 4. Two-Way Tables and Conditional Relative Frequency
A two-way table for a solar installer in Atlantic Corridor during a multiweek validation study has Group A counts 56 meeting and 23 not meeting the daily energy output criterion; Group B counts are 58 and 25. Compare the conditional proportions that meet the criterion and comment on association.
Answer: C
Within Group A, p̂A=56(56+23)=0.709. Within Group B, p̂B=58(58+25)=0.699. The difference is 0.01 (1.0% percentage points in magnitude). The conditional proportions are very similar, so the table gives little evidence of association.
Why the other choices fail
- Choice A: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.709 for A and 0.699 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice B: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.709 for A and 0.699 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.709 for A and 0.699 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 5. Two-Way Tables and Conditional Relative Frequency
A two-way table for a university advising center in Metro East during a community outreach cycle has Group A counts 31 meeting and 18 not meeting the appointment wait time criterion; Group B counts are 54 and 22. Compare the conditional proportions that meet the criterion and comment on association.
Answer: C
Within Group A, p̂A=31(31+18)=0.633. Within Group B, p̂B=54(54+22)=0.711. The difference is -0.078 (7.8% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice A: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.633 for A and 0.711 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice B: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.633 for A and 0.711 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.633 for A and 0.711 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 6. Two-Way Tables and Conditional Relative Frequency
A two-way table for a community college in Mountain Region during a community outreach cycle has Group A counts 37 meeting and 24 not meeting the course completion criterion; Group B counts are 31 and 53. Compare the conditional proportions that meet the criterion and comment on association.
Answer: A
Within Group A, p̂A=37(37+24)=0.607. Within Group B, p̂B=31(31+53)=0.369. The difference is 0.238 (23.8% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice B: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.607 for A and 0.369 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice C: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.607 for A and 0.369 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.607 for A and 0.369 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 7. Two-Way Tables and Conditional Relative Frequency
A two-way table for a housing authority in Pacific Northwest during a multiweek validation study has Group A counts 50 meeting and 58 not meeting the application processing time criterion; Group B counts are 47 and 32. Compare the conditional proportions that meet the criterion and comment on association.
Answer: C
Within Group A, p̂A=50(50+58)=0.463. Within Group B, p̂B=47(47+32)=0.595. The difference is -0.132 (13.2% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice A: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.463 for A and 0.595 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice B: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.463 for A and 0.595 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.463 for A and 0.595 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 8. Two-Way Tables and Conditional Relative Frequency
A two-way table for a grocery cooperative in Pine Ridge during a multiweek validation study has Group A counts 62 meeting and 32 not meeting the checkout time criterion; Group B counts are 62 and 16. Compare the conditional proportions that meet the criterion and comment on association.
Answer: D
Within Group A, p̂A=62(62+32)=0.66. Within Group B, p̂B=62(62+16)=0.795. The difference is -0.135 (13.5% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice A: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.66 for A and 0.795 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice B: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.66 for A and 0.795 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice C: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.66 for A and 0.795 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 9. Two-Way Tables and Conditional Relative Frequency
A two-way table for a county election office in Desert County during a six-week field trial has Group A counts 52 meeting and 29 not meeting the ballot-processing time criterion; Group B counts are 25 and 33. Compare the conditional proportions that meet the criterion and comment on association.
Answer: C
Within Group A, p̂A=52(52+29)=0.642. Within Group B, p̂B=25(25+33)=0.431. The difference is 0.211 (21.1% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice A: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.642 for A and 0.431 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice B: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.642 for A and 0.431 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.642 for A and 0.431 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 10. Two-Way Tables and Conditional Relative Frequency
A two-way table for a digital learning platform in Pacific Northwest during a monthly quality review has Group A counts 69 meeting and 42 not meeting the lesson completion criterion; Group B counts are 24 and 50. Compare the conditional proportions that meet the criterion and comment on association.
Answer: D
Within Group A, p̂A=69(69+42)=0.622. Within Group B, p̂B=24(24+50)=0.324. The difference is 0.297 (29.7% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice A: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.622 for A and 0.324 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice B: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.622 for A and 0.324 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice C: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.622 for A and 0.324 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 11. Two-Way Tables and Conditional Relative Frequency
A two-way table for a food safety laboratory in Capital Region during a two-month observation window has Group A counts 30 meeting and 56 not meeting the sample concentration criterion; Group B counts are 65 and 47. Compare the conditional proportions that meet the criterion and comment on association.
Answer: C
Within Group A, p̂A=30(30+56)=0.349. Within Group B, p̂B=65(65+47)=0.58. The difference is -0.232 (23.2% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice A: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.349 for A and 0.58 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice B: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.349 for A and 0.58 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.349 for A and 0.58 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 12. Two-Way Tables and Conditional Relative Frequency
A two-way table for a food safety laboratory in Prairie District during a baseline measurement week has Group A counts 52 meeting and 45 not meeting the sample concentration criterion; Group B counts are 50 and 57. Compare the conditional proportions that meet the criterion and comment on association.
Answer: D
Within Group A, p̂A=52(52+45)=0.536. Within Group B, p̂B=50(50+57)=0.467. The difference is 0.069 (6.9% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice A: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.536 for A and 0.467 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice B: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.536 for A and 0.467 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice C: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.536 for A and 0.467 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 13. Two-Way Tables and Conditional Relative Frequency
A two-way table for a community college in Pacific Northwest during a multiweek validation study has Group A counts 52 meeting and 22 not meeting the course completion criterion; Group B counts are 33 and 42. Compare the conditional proportions that meet the criterion and comment on association.
Answer: C
Within Group A, p̂A=52(52+22)=0.703. Within Group B, p̂B=33(33+42)=0.44. The difference is 0.263 (26.3% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice A: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.703 for A and 0.44 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice B: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.703 for A and 0.44 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.703 for A and 0.44 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 14. Two-Way Tables and Conditional Relative Frequency
A two-way table for a farm cooperative in Metro East during a weekday operations study has Group A counts 48 meeting and 51 not meeting the crop yield criterion; Group B counts are 66 and 17. Compare the conditional proportions that meet the criterion and comment on association.
Answer: C
Within Group A, p̂A=48(48+51)=0.485. Within Group B, p̂B=66(66+17)=0.795. The difference is -0.31 (31.0% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice A: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.485 for A and 0.795 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice B: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.485 for A and 0.795 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.485 for A and 0.795 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 15. Two-Way Tables and Conditional Relative Frequency
A two-way table for a public health department in Capital Region during a community outreach cycle has Group A counts 52 meeting and 23 not meeting the vaccination appointment completion criterion; Group B counts are 61 and 23. Compare the conditional proportions that meet the criterion and comment on association.
Answer: C
Within Group A, p̂A=52(52+23)=0.693. Within Group B, p̂B=61(61+23)=0.726. The difference is -0.033 (3.3% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice A: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.693 for A and 0.726 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice B: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.693 for A and 0.726 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.693 for A and 0.726 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 16. Two-Way Tables and Conditional Relative Frequency
A two-way table for a city recreation department in Desert County during a spring 2027 pilot has Group A counts 41 meeting and 32 not meeting the program satisfaction criterion; Group B counts are 40 and 28. Compare the conditional proportions that meet the criterion and comment on association.
Answer: A
Within Group A, p̂A=41(41+32)=0.562. Within Group B, p̂B=40(40+28)=0.588. The difference is -0.027 (2.7% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice B: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.562 for A and 0.588 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice C: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.562 for A and 0.588 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.562 for A and 0.588 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 17. Two-Way Tables and Conditional Relative Frequency
A two-way table for a county library in North Valley during a pre-exam training cycle has Group A counts 56 meeting and 19 not meeting the weekly program attendance criterion; Group B counts are 58 and 37. Compare the conditional proportions that meet the criterion and comment on association.
Answer: B
Within Group A, p̂A=56(56+19)=0.747. Within Group B, p̂B=58(58+37)=0.611. The difference is 0.136 (13.6% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice A: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.747 for A and 0.611 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice C: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.747 for A and 0.611 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.747 for A and 0.611 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 18. Two-Way Tables and Conditional Relative Frequency
A two-way table for a public health department in Midwest consortium during a weekday operations study has Group A counts 50 meeting and 60 not meeting the vaccination appointment completion criterion; Group B counts are 59 and 47. Compare the conditional proportions that meet the criterion and comment on association.
Answer: A
Within Group A, p̂A=50(50+60)=0.455. Within Group B, p̂B=59(59+47)=0.557. The difference is -0.102 (10.2% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice B: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.455 for A and 0.557 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice C: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.455 for A and 0.557 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.455 for A and 0.557 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 19. Two-Way Tables and Conditional Relative Frequency
A two-way table for a university advising center in Capital Region during a regional benchmarking study has Group A counts 24 meeting and 49 not meeting the appointment wait time criterion; Group B counts are 48 and 50. Compare the conditional proportions that meet the criterion and comment on association.
Answer: A
Within Group A, p̂A=24(24+49)=0.329. Within Group B, p̂B=48(48+50)=0.49. The difference is -0.161 (16.1% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice B: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.329 for A and 0.49 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice C: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.329 for A and 0.49 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.329 for A and 0.49 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 20. Two-Way Tables and Conditional Relative Frequency
A two-way table for a regional airport authority in Mountain Region during a service-improvement study has Group A counts 58 meeting and 55 not meeting the security wait time criterion; Group B counts are 40 and 24. Compare the conditional proportions that meet the criterion and comment on association.
Answer: B
Within Group A, p̂A=58(58+55)=0.513. Within Group B, p̂B=40(40+24)=0.625. The difference is -0.112 (11.2% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice A: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.513 for A and 0.625 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice C: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.513 for A and 0.625 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.513 for A and 0.625 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 21. Two-Way Tables and Conditional Relative Frequency
A two-way table for a public health department in Coastal Plains during a monthly quality review has Group A counts 70 meeting and 21 not meeting the vaccination appointment completion criterion; Group B counts are 59 and 53. Compare the conditional proportions that meet the criterion and comment on association.
Answer: A
Within Group A, p̂A=70(70+21)=0.769. Within Group B, p̂B=59(59+53)=0.527. The difference is 0.242 (24.2% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice B: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.769 for A and 0.527 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice C: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.769 for A and 0.527 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.769 for A and 0.527 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 22. Two-Way Tables and Conditional Relative Frequency
A two-way table for a grocery cooperative in Pacific Northwest during a weekday operations study has Group A counts 30 meeting and 45 not meeting the checkout time criterion; Group B counts are 32 and 24. Compare the conditional proportions that meet the criterion and comment on association.
Answer: B
Within Group A, p̂A=30(30+45)=0.4. Within Group B, p̂B=32(32+24)=0.571. The difference is -0.171 (17.1% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice A: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.4 for A and 0.571 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice C: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.4 for A and 0.571 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.4 for A and 0.571 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 23. Two-Way Tables and Conditional Relative Frequency
A two-way table for a university advising center in Coastal Plains during a baseline measurement week has Group A counts 51 meeting and 57 not meeting the appointment wait time criterion; Group B counts are 22 and 19. Compare the conditional proportions that meet the criterion and comment on association.
Answer: A
Within Group A, p̂A=51(51+57)=0.472. Within Group B, p̂B=22(22+19)=0.537. The difference is -0.064 (6.4% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice B: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.472 for A and 0.537 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice C: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.472 for A and 0.537 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.472 for A and 0.537 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 24. Two-Way Tables and Conditional Relative Frequency
A two-way table for a city transit agency in Pine Ridge during a service-improvement study has Group A counts 34 meeting and 41 not meeting the on-time arrival criterion; Group B counts are 39 and 58. Compare the conditional proportions that meet the criterion and comment on association.
Answer: A
Within Group A, p̂A=34(34+41)=0.453. Within Group B, p̂B=39(39+58)=0.402. The difference is 0.051 (5.1% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice B: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.453 for A and 0.402 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice C: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.453 for A and 0.402 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.453 for A and 0.402 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 25. Two-Way Tables and Conditional Relative Frequency
A two-way table for a university advising center in Capital Region during a multiweek validation study has Group A counts 64 meeting and 38 not meeting the appointment wait time criterion; Group B counts are 24 and 28. Compare the conditional proportions that meet the criterion and comment on association.
Answer: D
Within Group A, p̂A=64(64+38)=0.627. Within Group B, p̂B=24(24+28)=0.462. The difference is 0.166 (16.6% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice A: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.627 for A and 0.462 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice B: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.627 for A and 0.462 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice C: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.627 for A and 0.462 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 26. Two-Way Tables and Conditional Relative Frequency
A two-way table for a grocery cooperative in Sunbelt district during a summer implementation review has Group A counts 26 meeting and 59 not meeting the checkout time criterion; Group B counts are 24 and 61. Compare the conditional proportions that meet the criterion and comment on association.
Answer: D
Within Group A, p̂A=26(26+59)=0.306. Within Group B, p̂B=24(24+61)=0.282. The difference is 0.024 (2.4% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice A: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.306 for A and 0.282 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice B: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.306 for A and 0.282 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice C: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.306 for A and 0.282 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 27. Two-Way Tables and Conditional Relative Frequency
A two-way table for a solar installer in North Valley during a service-improvement study has Group A counts 30 meeting and 62 not meeting the daily energy output criterion; Group B counts are 57 and 49. Compare the conditional proportions that meet the criterion and comment on association.
Answer: A
Within Group A, p̂A=30(30+62)=0.326. Within Group B, p̂B=57(57+49)=0.538. The difference is -0.212 (21.2% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice B: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.326 for A and 0.538 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice C: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.326 for A and 0.538 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.326 for A and 0.538 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 28. Two-Way Tables and Conditional Relative Frequency
A two-way table for a regional hospital in South Harbor during a winter readiness review has Group A counts 45 meeting and 62 not meeting the appointment completion criterion; Group B counts are 28 and 22. Compare the conditional proportions that meet the criterion and comment on association.
Answer: C
Within Group A, p̂A=45(45+62)=0.421. Within Group B, p̂B=28(28+22)=0.56. The difference is -0.139 (13.9% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice A: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.421 for A and 0.56 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice B: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.421 for A and 0.56 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.421 for A and 0.56 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 29. Two-Way Tables and Conditional Relative Frequency
A two-way table for a farm cooperative in Lakeside district during a monthly quality review has Group A counts 44 meeting and 23 not meeting the crop yield criterion; Group B counts are 22 and 40. Compare the conditional proportions that meet the criterion and comment on association.
Answer: C
Within Group A, p̂A=44(44+23)=0.657. Within Group B, p̂B=22(22+40)=0.355. The difference is 0.302 (30.2% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice A: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.657 for A and 0.355 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice B: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.657 for A and 0.355 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.657 for A and 0.355 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 30. Two-Way Tables and Conditional Relative Frequency
A two-way table for a food safety laboratory in Sunbelt district during a semester-long cohort study has Group A counts 59 meeting and 45 not meeting the sample concentration criterion; Group B counts are 34 and 27. Compare the conditional proportions that meet the criterion and comment on association.
Answer: A
Within Group A, p̂A=59(59+45)=0.567. Within Group B, p̂B=34(34+27)=0.557. The difference is 0.01 (1.0% percentage points in magnitude). The conditional proportions are very similar, so the table gives little evidence of association.
Why the other choices fail
- Choice B: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.567 for A and 0.557 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice C: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.567 for A and 0.557 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.567 for A and 0.557 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 31. Two-Way Tables and Conditional Relative Frequency
A two-way table for a city recreation department in Riverbend during a regional benchmarking study has Group A counts 51 meeting and 37 not meeting the program satisfaction criterion; Group B counts are 38 and 50. Compare the conditional proportions that meet the criterion and comment on association.
Answer: C
Within Group A, p̂A=51(51+37)=0.58. Within Group B, p̂B=38(38+50)=0.432. The difference is 0.148 (14.8% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice A: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.58 for A and 0.432 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice B: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.58 for A and 0.432 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.58 for A and 0.432 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 32. Two-Way Tables and Conditional Relative Frequency
A two-way table for a farm cooperative in Pacific Northwest during a weekday operations study has Group A counts 30 meeting and 59 not meeting the crop yield criterion; Group B counts are 48 and 45. Compare the conditional proportions that meet the criterion and comment on association.
Answer: D
Within Group A, p̂A=30(30+59)=0.337. Within Group B, p̂B=48(48+45)=0.516. The difference is -0.179 (17.9% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice A: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.337 for A and 0.516 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice B: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.337 for A and 0.516 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice C: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.337 for A and 0.516 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 33. Two-Way Tables and Conditional Relative Frequency
A two-way table for a digital learning platform in Great Lakes during a monthly quality review has Group A counts 61 meeting and 40 not meeting the lesson completion criterion; Group B counts are 57 and 49. Compare the conditional proportions that meet the criterion and comment on association.
Answer: A
Within Group A, p̂A=61(61+40)=0.604. Within Group B, p̂B=57(57+49)=0.538. The difference is 0.066 (6.6% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice B: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.604 for A and 0.538 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice C: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.604 for A and 0.538 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.604 for A and 0.538 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 34. Two-Way Tables and Conditional Relative Frequency
A two-way table for a regional airport authority in Metro East during a fall 2026 audit has Group A counts 40 meeting and 27 not meeting the security wait time criterion; Group B counts are 54 and 47. Compare the conditional proportions that meet the criterion and comment on association.
Answer: C
Within Group A, p̂A=40(40+27)=0.597. Within Group B, p̂B=54(54+47)=0.535. The difference is 0.062 (6.2% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice A: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.597 for A and 0.535 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice B: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.597 for A and 0.535 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.597 for A and 0.535 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 35. Two-Way Tables and Conditional Relative Frequency
A two-way table for a city transit agency in New England network during a fall 2026 audit has Group A counts 56 meeting and 57 not meeting the on-time arrival criterion; Group B counts are 64 and 39. Compare the conditional proportions that meet the criterion and comment on association.
Answer: B
Within Group A, p̂A=56(56+57)=0.496. Within Group B, p̂B=64(64+39)=0.621. The difference is -0.126 (12.6% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice A: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.496 for A and 0.621 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice C: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.496 for A and 0.621 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.496 for A and 0.621 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Question 36. Two-Way Tables and Conditional Relative Frequency
A two-way table for a food safety laboratory in Central County during a quarterly performance study has Group A counts 51 meeting and 43 not meeting the sample concentration criterion; Group B counts are 55 and 51. Compare the conditional proportions that meet the criterion and comment on association.
Answer: C
Within Group A, p̂A=51(51+43)=0.543. Within Group B, p̂B=55(55+51)=0.519. The difference is 0.024 (2.4% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
Why the other choices fail
- Choice A: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.543 for A and 0.519 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice B: It makes a conclusion that the data do not support. The correct comparison or result is: Use row denominators: 0.543 for A and 0.519 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
- Choice D: It uses the wrong denominator or omits a required scale adjustment. The correct comparison or result is: Use row denominators: 0.543 for A and 0.519 for B; compare those conditional proportions. Key check: Compare conditional distributions to assess association between categorical variables.
Two-Way Tables and Conditional Relative Frequency free-response practice
| Evidence | Points |
|---|---|
| Correct target, notation, direction, or group order | 2 |
| Correct method/model and defensible conditions | 2 |
| Correct setup and execution | 3 |
| Contextual interpretation and scope/limitation | 2 |
| Clear communication with units and labels | 1 |
FRQ set 1: Two-Way Tables and Conditional Relative Frequency
Scenario. A two-way table for a solar installer in Pacific Northwest during a six-week field trial has Group A counts 41 meeting and 61 not meeting the daily energy output criterion; Group B counts are 48 and 49. Compare the conditional proportions that meet the criterion and comment on association.
- Define the variable, units, groups, and requested distribution or model feature.
- Show the required calculation or graphical/model reasoning with labeled quantities.
- Interpret the numerical result in the context of the data rather than as an isolated number.
- Identify an unusual feature, limitation, or condition that affects the conclusion.
Model response
Within Group A, p̂A=41(41+61)=0.402. Within Group B, p̂B=48(48+49)=0.495. The difference is -0.093 (9.3% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
FRQ set 2: Two-Way Tables and Conditional Relative Frequency
Scenario. A two-way table for a community college in Lakeside district during a baseline measurement week has Group A counts 30 meeting and 27 not meeting the course completion criterion; Group B counts are 31 and 32. Compare the conditional proportions that meet the criterion and comment on association.
- Define the variable, units, groups, and requested distribution or model feature.
- Show the required calculation or graphical/model reasoning with labeled quantities.
- Interpret the numerical result in the context of the data rather than as an isolated number.
- Identify an unusual feature, limitation, or condition that affects the conclusion.
Model response
Within Group A, p̂A=30(30+27)=0.526. Within Group B, p̂B=31(31+32)=0.492. The difference is 0.034 (3.4% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
FRQ set 3: Two-Way Tables and Conditional Relative Frequency
Scenario. A two-way table for a housing authority in Capital Region during a fall 2026 audit has Group A counts 27 meeting and 56 not meeting the application processing time criterion; Group B counts are 42 and 50. Compare the conditional proportions that meet the criterion and comment on association.
- Define the variable, units, groups, and requested distribution or model feature.
- Show the required calculation or graphical/model reasoning with labeled quantities.
- Interpret the numerical result in the context of the data rather than as an isolated number.
- Identify an unusual feature, limitation, or condition that affects the conclusion.
Model response
Within Group A, p̂A=27(27+56)=0.325. Within Group B, p̂B=42(42+50)=0.457. The difference is -0.131 (13.1% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
FRQ set 4: Two-Way Tables and Conditional Relative Frequency
Scenario. A two-way table for a public high school in Cedar Grove during a winter readiness review has Group A counts 45 meeting and 32 not meeting the algebra benchmark completion criterion; Group B counts are 53 and 48. Compare the conditional proportions that meet the criterion and comment on association.
- Define the variable, units, groups, and requested distribution or model feature.
- Show the required calculation or graphical/model reasoning with labeled quantities.
- Interpret the numerical result in the context of the data rather than as an isolated number.
- Identify an unusual feature, limitation, or condition that affects the conclusion.
Model response
Within Group A, p̂A=45(45+32)=0.584. Within Group B, p̂B=53(53+48)=0.525. The difference is 0.06 (6.0% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
FRQ set 5: Two-Way Tables and Conditional Relative Frequency
Scenario. A two-way table for a food safety laboratory in South Harbor during a monthly quality review has Group A counts 51 meeting and 20 not meeting the sample concentration criterion; Group B counts are 43 and 56. Compare the conditional proportions that meet the criterion and comment on association.
- Define the variable, units, groups, and requested distribution or model feature.
- Show the required calculation or graphical/model reasoning with labeled quantities.
- Interpret the numerical result in the context of the data rather than as an isolated number.
- Identify an unusual feature, limitation, or condition that affects the conclusion.
Model response
Within Group A, p̂A=51(51+20)=0.718. Within Group B, p̂B=43(43+56)=0.434. The difference is 0.284 (28.4% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
FRQ set 6: Two-Way Tables and Conditional Relative Frequency
Scenario. A two-way table for a food safety laboratory in Coastal Plains during a pre-exam training cycle has Group A counts 29 meeting and 54 not meeting the sample concentration criterion; Group B counts are 62 and 56. Compare the conditional proportions that meet the criterion and comment on association.
- Define the variable, units, groups, and requested distribution or model feature.
- Show the required calculation or graphical/model reasoning with labeled quantities.
- Interpret the numerical result in the context of the data rather than as an isolated number.
- Identify an unusual feature, limitation, or condition that affects the conclusion.
Model response
Within Group A, p̂A=29(29+54)=0.349. Within Group B, p̂B=62(62+56)=0.525. The difference is -0.176 (17.6% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
FRQ set 7: Two-Way Tables and Conditional Relative Frequency
Scenario. A two-way table for a regional manufacturer in Lakeside district during a service-improvement study has Group A counts 35 meeting and 54 not meeting the part diameter criterion; Group B counts are 63 and 34. Compare the conditional proportions that meet the criterion and comment on association.
- Define the variable, units, groups, and requested distribution or model feature.
- Show the required calculation or graphical/model reasoning with labeled quantities.
- Interpret the numerical result in the context of the data rather than as an isolated number.
- Identify an unusual feature, limitation, or condition that affects the conclusion.
Model response
Within Group A, p̂A=35(35+54)=0.393. Within Group B, p̂B=63(63+34)=0.649. The difference is -0.256 (25.6% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
FRQ set 8: Two-Way Tables and Conditional Relative Frequency
Scenario. A two-way table for a regional hospital in Lakeside district during a multiweek validation study has Group A counts 41 meeting and 41 not meeting the appointment completion criterion; Group B counts are 29 and 46. Compare the conditional proportions that meet the criterion and comment on association.
- Define the variable, units, groups, and requested distribution or model feature.
- Show the required calculation or graphical/model reasoning with labeled quantities.
- Interpret the numerical result in the context of the data rather than as an isolated number.
- Identify an unusual feature, limitation, or condition that affects the conclusion.
Model response
Within Group A, p̂A=41(41+41)=0.5. Within Group B, p̂B=29(29+46)=0.387. The difference is 0.113 (11.3% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
FRQ set 9: Two-Way Tables and Conditional Relative Frequency
Scenario. A two-way table for a wildlife clinic in Pacific Northwest during a community outreach cycle has Group A counts 49 meeting and 39 not meeting the recovery time criterion; Group B counts are 66 and 38. Compare the conditional proportions that meet the criterion and comment on association.
- Define the variable, units, groups, and requested distribution or model feature.
- Show the required calculation or graphical/model reasoning with labeled quantities.
- Interpret the numerical result in the context of the data rather than as an isolated number.
- Identify an unusual feature, limitation, or condition that affects the conclusion.
Model response
Within Group A, p̂A=49(49+39)=0.557. Within Group B, p̂B=66(66+38)=0.635. The difference is -0.078 (7.8% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
FRQ set 10: Two-Way Tables and Conditional Relative Frequency
Scenario. A two-way table for a regional hospital in South Harbor during a yearly program evaluation has Group A counts 64 meeting and 61 not meeting the appointment completion criterion; Group B counts are 31 and 30. Compare the conditional proportions that meet the criterion and comment on association.
- Define the variable, units, groups, and requested distribution or model feature.
- Show the required calculation or graphical/model reasoning with labeled quantities.
- Interpret the numerical result in the context of the data rather than as an isolated number.
- Identify an unusual feature, limitation, or condition that affects the conclusion.
Model response
Within Group A, p̂A=64(64+61)=0.512. Within Group B, p̂B=31(31+30)=0.508. The difference is 0.004 (0.4% percentage points in magnitude). The conditional proportions are very similar, so the table gives little evidence of association.
FRQ set 11: Two-Way Tables and Conditional Relative Frequency
Scenario. A two-way table for a city transit agency in Great Lakes during a regional benchmarking study has Group A counts 48 meeting and 28 not meeting the on-time arrival criterion; Group B counts are 44 and 36. Compare the conditional proportions that meet the criterion and comment on association.
- Define the variable, units, groups, and requested distribution or model feature.
- Show the required calculation or graphical/model reasoning with labeled quantities.
- Interpret the numerical result in the context of the data rather than as an isolated number.
- Identify an unusual feature, limitation, or condition that affects the conclusion.
Model response
Within Group A, p̂A=48(48+28)=0.632. Within Group B, p̂B=44(44+36)=0.55. The difference is 0.082 (8.2% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
FRQ set 12: Two-Way Tables and Conditional Relative Frequency
Scenario. A two-way table for a city transit agency in Prairie District during a school-year data collection has Group A counts 60 meeting and 56 not meeting the on-time arrival criterion; Group B counts are 47 and 20. Compare the conditional proportions that meet the criterion and comment on association.
- Define the variable, units, groups, and requested distribution or model feature.
- Show the required calculation or graphical/model reasoning with labeled quantities.
- Interpret the numerical result in the context of the data rather than as an isolated number.
- Identify an unusual feature, limitation, or condition that affects the conclusion.
Model response
Within Group A, p̂A=60(60+56)=0.517. Within Group B, p̂B=47(47+20)=0.701. The difference is -0.184 (18.4% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
FRQ set 13: Two-Way Tables and Conditional Relative Frequency
Scenario. A two-way table for a wildlife clinic in Westview during a service-improvement study has Group A counts 65 meeting and 39 not meeting the recovery time criterion; Group B counts are 29 and 62. Compare the conditional proportions that meet the criterion and comment on association.
- Define the variable, units, groups, and requested distribution or model feature.
- Show the required calculation or graphical/model reasoning with labeled quantities.
- Interpret the numerical result in the context of the data rather than as an isolated number.
- Identify an unusual feature, limitation, or condition that affects the conclusion.
Model response
Within Group A, p̂A=65(65+39)=0.625. Within Group B, p̂B=29(29+62)=0.319. The difference is 0.306 (30.6% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
FRQ set 14: Two-Way Tables and Conditional Relative Frequency
Scenario. A two-way table for a school district in Atlantic Corridor during a six-week field trial has Group A counts 67 meeting and 62 not meeting the lunch-program participation criterion; Group B counts are 46 and 21. Compare the conditional proportions that meet the criterion and comment on association.
- Define the variable, units, groups, and requested distribution or model feature.
- Show the required calculation or graphical/model reasoning with labeled quantities.
- Interpret the numerical result in the context of the data rather than as an isolated number.
- Identify an unusual feature, limitation, or condition that affects the conclusion.
Model response
Within Group A, p̂A=67(67+62)=0.519. Within Group B, p̂B=46(46+21)=0.687. The difference is -0.167 (16.7% percentage points in magnitude). Because the conditional proportions differ, the table suggests an association between group and outcome.
