Empirical Rule for Normal Distributions: 68–95–99.7
Empirical Rule for Normal Distributions: 68–95–99.7 practice bank: Solve the visible multiple-choice and free-response questions, then compare every step with the worked answers.
Empirical Rule for Normal Distributions: 68–95–99.7: formulas and targets
| Within 1 SD | about 68% |
|---|---|
| Within 2 SD | about 95% |
| Within 3 SD | about 99.7% |
Empirical Rule multiple-choice practice
Question 1. Empirical Rule
An approximately normal distribution of application processing time at a housing authority in Lakeside district during a two-month observation window has mean 72 and SD 8. Use the empirical rule for the interval from 56 to 88.
Answer: D
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 1. Empirical Rule: The endpoints are μ±2σ. The empirical rule places about 95.0% within 2 standard deviations of the mean.
Why the other choices fail
- Choice A: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 56 and 88. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice B: It makes a conclusion that the data do not support. The correct comparison or result is: About 95.0% lies between 56 and 88. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 56 and 88. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 2. Empirical Rule
An approximately normal distribution of appointment wait time at a university advising center in Pacific Northwest during a school-year data collection has mean 55 and SD 8. Use the empirical rule for the interval from 31 to 79.
Answer: A
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 2. Empirical Rule: The endpoints are μ±3σ. The empirical rule places about 99.7% within 3 standard deviations of the mean.
Why the other choices fail
- Choice B: It makes a conclusion that the data do not support. The correct comparison or result is: About 99.7% lies between 31 and 79. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It does not match the requested calculation. The correct comparison or result is: About 99.7% lies between 31 and 79. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It does not match the requested calculation. The correct comparison or result is: About 99.7% lies between 31 and 79. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 3. Empirical Rule
An approximately normal distribution of algebra benchmark completion at a public high school in Pine Ridge during a six-week field trial has mean 85 and SD 4. Use the empirical rule for the interval from 73 to 97.
Answer: A
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 3. Empirical Rule: The endpoints are μ±3σ. The empirical rule places about 99.7% within 3 standard deviations of the mean.
Why the other choices fail
- Choice B: It makes a conclusion that the data do not support. The correct comparison or result is: About 99.7% lies between 73 and 97. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It does not match the requested calculation. The correct comparison or result is: About 99.7% lies between 73 and 97. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It does not match the requested calculation. The correct comparison or result is: About 99.7% lies between 73 and 97. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 4. Empirical Rule
An approximately normal distribution of weekly program attendance at a county library in Central County during a weekday operations study has mean 79 and SD 12. Use the empirical rule for the interval from 43 to 115.
Answer: D
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 4. Empirical Rule: The endpoints are μ±3σ. The empirical rule places about 99.7% within 3 standard deviations of the mean.
Why the other choices fail
- Choice A: It does not match the requested calculation. The correct comparison or result is: About 99.7% lies between 43 and 115. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice B: It does not match the requested calculation. The correct comparison or result is: About 99.7% lies between 43 and 115. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It makes a conclusion that the data do not support. The correct comparison or result is: About 99.7% lies between 43 and 115. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 5. Empirical Rule
An approximately normal distribution of on-time arrival at a city transit agency in Midwest consortium during a semester-long cohort study has mean 40 and SD 15. Use the empirical rule for the interval from 10 to 70.
Answer: C
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 5. Empirical Rule: The endpoints are μ±2σ. The empirical rule places about 95.0% within 2 standard deviations of the mean.
Why the other choices fail
- Choice A: It makes a conclusion that the data do not support. The correct comparison or result is: About 95.0% lies between 10 and 70. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice B: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 10 and 70. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 10 and 70. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 6. Empirical Rule
An approximately normal distribution of weekly material weight at a recycling program in Metro East during a service-improvement study has mean 96 and SD 4. Use the empirical rule for the interval from 88 to 104.
Answer: A
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 6. Empirical Rule: The endpoints are μ±2σ. The empirical rule places about 95.0% within 2 standard deviations of the mean.
Why the other choices fail
- Choice B: It makes a conclusion that the data do not support. The correct comparison or result is: About 95.0% lies between 88 and 104. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 88 and 104. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 88 and 104. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 7. Empirical Rule
An approximately normal distribution of security wait time at a regional airport authority in Desert County during a randomized pilot period has mean 83 and SD 5. Use the empirical rule for the interval from 78 to 88.
Answer: D
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 7. Empirical Rule: The endpoints are μ±1σ. The empirical rule places about 68.0% within 1 standard deviation of the mean.
Why the other choices fail
- Choice A: It makes a conclusion that the data do not support. The correct comparison or result is: About 68.0% lies between 78 and 88. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice B: It does not match the requested calculation. The correct comparison or result is: About 68.0% lies between 78 and 88. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It does not match the requested calculation. The correct comparison or result is: About 68.0% lies between 78 and 88. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 8. Empirical Rule
An approximately normal distribution of security wait time at a regional airport authority in Pine Ridge during a pre-exam training cycle has mean 62 and SD 4. Use the empirical rule for the interval from 50 to 74.
Answer: D
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 8. Empirical Rule: The endpoints are μ±3σ. The empirical rule places about 99.7% within 3 standard deviations of the mean.
Why the other choices fail
- Choice A: It makes a conclusion that the data do not support. The correct comparison or result is: About 99.7% lies between 50 and 74. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice B: It does not match the requested calculation. The correct comparison or result is: About 99.7% lies between 50 and 74. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It does not match the requested calculation. The correct comparison or result is: About 99.7% lies between 50 and 74. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 9. Empirical Rule
An approximately normal distribution of weekly material weight at a recycling program in New England network during a weekday operations study has mean 51 and SD 10. Use the empirical rule for the interval from 31 to 71.
Answer: B
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 9. Empirical Rule: The endpoints are μ±2σ. The empirical rule places about 95.0% within 2 standard deviations of the mean.
Why the other choices fail
- Choice A: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 31 and 71. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It makes a conclusion that the data do not support. The correct comparison or result is: About 95.0% lies between 31 and 71. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 31 and 71. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 10. Empirical Rule
An approximately normal distribution of ballot-processing time at a county election office in Mountain Region during a baseline measurement week has mean 50 and SD 13. Use the empirical rule for the interval from 24 to 76.
Answer: B
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 10. Empirical Rule: The endpoints are μ±2σ. The empirical rule places about 95.0% within 2 standard deviations of the mean.
Why the other choices fail
- Choice A: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 24 and 76. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 24 and 76. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It makes a conclusion that the data do not support. The correct comparison or result is: About 95.0% lies between 24 and 76. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 11. Empirical Rule
An approximately normal distribution of appointment wait time at a university advising center in Sunbelt district during a monthly quality review has mean 51 and SD 14. Use the empirical rule for the interval from 37 to 65.
Answer: D
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 11. Empirical Rule: The endpoints are μ±1σ. The empirical rule places about 68.0% within 1 standard deviation of the mean.
Why the other choices fail
- Choice A: It makes a conclusion that the data do not support. The correct comparison or result is: About 68.0% lies between 37 and 65. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice B: It does not match the requested calculation. The correct comparison or result is: About 68.0% lies between 37 and 65. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It does not match the requested calculation. The correct comparison or result is: About 68.0% lies between 37 and 65. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 12. Empirical Rule
An approximately normal distribution of monthly household use at a municipal water office in Pine Ridge during a six-week field trial has mean 45 and SD 7. Use the empirical rule for the interval from 31 to 59.
Answer: B
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 12. Empirical Rule: The endpoints are μ±2σ. The empirical rule places about 95.0% within 2 standard deviations of the mean.
Why the other choices fail
- Choice A: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 31 and 59. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 31 and 59. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It makes a conclusion that the data do not support. The correct comparison or result is: About 95.0% lies between 31 and 59. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 13. Empirical Rule
An approximately normal distribution of trail-use duration at a state park in Great Lakes during a baseline measurement week has mean 99 and SD 15. Use the empirical rule for the interval from 69 to 129.
Answer: A
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 13. Empirical Rule: The endpoints are μ±2σ. The empirical rule places about 95.0% within 2 standard deviations of the mean.
Why the other choices fail
- Choice B: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 69 and 129. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 69 and 129. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It makes a conclusion that the data do not support. The correct comparison or result is: About 95.0% lies between 69 and 129. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 14. Empirical Rule
An approximately normal distribution of security wait time at a regional airport authority in Desert County during a semester-long cohort study has mean 43 and SD 16. Use the empirical rule for the interval from 27 to 59.
Answer: D
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 14. Empirical Rule: The endpoints are μ±1σ. The empirical rule places about 68.0% within 1 standard deviation of the mean.
Why the other choices fail
- Choice A: It does not match the requested calculation. The correct comparison or result is: About 68.0% lies between 27 and 59. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice B: It makes a conclusion that the data do not support. The correct comparison or result is: About 68.0% lies between 27 and 59. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It does not match the requested calculation. The correct comparison or result is: About 68.0% lies between 27 and 59. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 15. Empirical Rule
An approximately normal distribution of security wait time at a regional airport authority in Pine Ridge during a monthly quality review has mean 88 and SD 5. Use the empirical rule for the interval from 78 to 98.
Answer: A
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 15. Empirical Rule: The endpoints are μ±2σ. The empirical rule places about 95.0% within 2 standard deviations of the mean.
Why the other choices fail
- Choice B: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 78 and 98. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 78 and 98. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It makes a conclusion that the data do not support. The correct comparison or result is: About 95.0% lies between 78 and 98. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 16. Empirical Rule
An approximately normal distribution of on-time arrival at a city transit agency in Metro East during a multiweek validation study has mean 90 and SD 8. Use the empirical rule for the interval from 66 to 114.
Answer: A
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 16. Empirical Rule: The endpoints are μ±3σ. The empirical rule places about 99.7% within 3 standard deviations of the mean.
Why the other choices fail
- Choice B: It makes a conclusion that the data do not support. The correct comparison or result is: About 99.7% lies between 66 and 114. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It does not match the requested calculation. The correct comparison or result is: About 99.7% lies between 66 and 114. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It does not match the requested calculation. The correct comparison or result is: About 99.7% lies between 66 and 114. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 17. Empirical Rule
An approximately normal distribution of checkout time at a grocery cooperative in Midwest consortium during a pre-exam training cycle has mean 98 and SD 14. Use the empirical rule for the interval from 56 to 140.
Answer: B
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 17. Empirical Rule: The endpoints are μ±3σ. The empirical rule places about 99.7% within 3 standard deviations of the mean.
Why the other choices fail
- Choice A: It does not match the requested calculation. The correct comparison or result is: About 99.7% lies between 56 and 140. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It makes a conclusion that the data do not support. The correct comparison or result is: About 99.7% lies between 56 and 140. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It does not match the requested calculation. The correct comparison or result is: About 99.7% lies between 56 and 140. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 18. Empirical Rule
An approximately normal distribution of weekly material weight at a recycling program in Coastal Plains during a summer implementation review has mean 41 and SD 10. Use the empirical rule for the interval from 21 to 61.
Answer: D
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 18. Empirical Rule: The endpoints are μ±2σ. The empirical rule places about 95.0% within 2 standard deviations of the mean.
Why the other choices fail
- Choice A: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 21 and 61. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice B: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 21 and 61. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It makes a conclusion that the data do not support. The correct comparison or result is: About 95.0% lies between 21 and 61. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 19. Empirical Rule
An approximately normal distribution of ballot-processing time at a county election office in Prairie District during a fall 2026 audit has mean 66 and SD 14. Use the empirical rule for the interval from 24 to 108.
Answer: D
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 19. Empirical Rule: The endpoints are μ±3σ. The empirical rule places about 99.7% within 3 standard deviations of the mean.
Why the other choices fail
- Choice A: It does not match the requested calculation. The correct comparison or result is: About 99.7% lies between 24 and 108. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice B: It does not match the requested calculation. The correct comparison or result is: About 99.7% lies between 24 and 108. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It makes a conclusion that the data do not support. The correct comparison or result is: About 99.7% lies between 24 and 108. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 20. Empirical Rule
An approximately normal distribution of appointment wait time at a university advising center in Pacific Northwest during a spring 2027 pilot has mean 62 and SD 14. Use the empirical rule for the interval from 34 to 90.
Answer: B
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 20. Empirical Rule: The endpoints are μ±2σ. The empirical rule places about 95.0% within 2 standard deviations of the mean.
Why the other choices fail
- Choice A: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 34 and 90. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 34 and 90. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It makes a conclusion that the data do not support. The correct comparison or result is: About 95.0% lies between 34 and 90. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 21. Empirical Rule
An approximately normal distribution of crop yield at a farm cooperative in Mountain Region during a quarterly performance study has mean 50 and SD 9. Use the empirical rule for the interval from 32 to 68.
Answer: D
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 21. Empirical Rule: The endpoints are μ±2σ. The empirical rule places about 95.0% within 2 standard deviations of the mean.
Why the other choices fail
- Choice A: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 32 and 68. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice B: It makes a conclusion that the data do not support. The correct comparison or result is: About 95.0% lies between 32 and 68. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 32 and 68. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 22. Empirical Rule
An approximately normal distribution of sample concentration at a food safety laboratory in Great Lakes during a semester-long cohort study has mean 93 and SD 6. Use the empirical rule for the interval from 87 to 99.
Answer: D
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 22. Empirical Rule: The endpoints are μ±1σ. The empirical rule places about 68.0% within 1 standard deviation of the mean.
Why the other choices fail
- Choice A: It does not match the requested calculation. The correct comparison or result is: About 68.0% lies between 87 and 99. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice B: It does not match the requested calculation. The correct comparison or result is: About 68.0% lies between 87 and 99. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It makes a conclusion that the data do not support. The correct comparison or result is: About 68.0% lies between 87 and 99. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 23. Empirical Rule
An approximately normal distribution of vaccination appointment completion at a public health department in South Harbor during a weekday operations study has mean 49 and SD 6. Use the empirical rule for the interval from 37 to 61.
Answer: C
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 23. Empirical Rule: The endpoints are μ±2σ. The empirical rule places about 95.0% within 2 standard deviations of the mean.
Why the other choices fail
- Choice A: It makes a conclusion that the data do not support. The correct comparison or result is: About 95.0% lies between 37 and 61. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice B: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 37 and 61. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 37 and 61. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 24. Empirical Rule
An approximately normal distribution of application processing time at a housing authority in Lakeside district during a summer implementation review has mean 66 and SD 6. Use the empirical rule for the interval from 54 to 78.
Answer: A
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 24. Empirical Rule: The endpoints are μ±2σ. The empirical rule places about 95.0% within 2 standard deviations of the mean.
Why the other choices fail
- Choice B: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 54 and 78. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It makes a conclusion that the data do not support. The correct comparison or result is: About 95.0% lies between 54 and 78. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 54 and 78. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 25. Empirical Rule
An approximately normal distribution of on-time arrival at a city transit agency in Pine Ridge during a six-week field trial has mean 47 and SD 8. Use the empirical rule for the interval from 31 to 63.
Answer: B
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 25. Empirical Rule: The endpoints are μ±2σ. The empirical rule places about 95.0% within 2 standard deviations of the mean.
Why the other choices fail
- Choice A: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 31 and 63. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It makes a conclusion that the data do not support. The correct comparison or result is: About 95.0% lies between 31 and 63. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 31 and 63. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 26. Empirical Rule
An approximately normal distribution of weekly material weight at a recycling program in South Harbor during a follow-up evaluation period has mean 76 and SD 13. Use the empirical rule for the interval from 37 to 115.
Answer: B
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 26. Empirical Rule: The endpoints are μ±3σ. The empirical rule places about 99.7% within 3 standard deviations of the mean.
Why the other choices fail
- Choice A: It does not match the requested calculation. The correct comparison or result is: About 99.7% lies between 37 and 115. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It makes a conclusion that the data do not support. The correct comparison or result is: About 99.7% lies between 37 and 115. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It does not match the requested calculation. The correct comparison or result is: About 99.7% lies between 37 and 115. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 27. Empirical Rule
An approximately normal distribution of response time at a municipal emergency dispatch center in Sunbelt district during a summer implementation review has mean 56 and SD 4. Use the empirical rule for the interval from 48 to 64.
Answer: B
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 27. Empirical Rule: The endpoints are μ±2σ. The empirical rule places about 95.0% within 2 standard deviations of the mean.
Why the other choices fail
- Choice A: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 48 and 64. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It makes a conclusion that the data do not support. The correct comparison or result is: About 95.0% lies between 48 and 64. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 48 and 64. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 28. Empirical Rule
An approximately normal distribution of application processing time at a housing authority in South Harbor during a spring 2027 pilot has mean 72 and SD 15. Use the empirical rule for the interval from 57 to 87.
Answer: C
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 28. Empirical Rule: The endpoints are μ±1σ. The empirical rule places about 68.0% within 1 standard deviation of the mean.
Why the other choices fail
- Choice A: It does not match the requested calculation. The correct comparison or result is: About 68.0% lies between 57 and 87. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice B: It does not match the requested calculation. The correct comparison or result is: About 68.0% lies between 57 and 87. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It makes a conclusion that the data do not support. The correct comparison or result is: About 68.0% lies between 57 and 87. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 29. Empirical Rule
An approximately normal distribution of recovery time at a wildlife clinic in Great Lakes during a spring 2027 pilot has mean 76 and SD 12. Use the empirical rule for the interval from 64 to 88.
Answer: B
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 29. Empirical Rule: The endpoints are μ±1σ. The empirical rule places about 68.0% within 1 standard deviation of the mean.
Why the other choices fail
- Choice A: It makes a conclusion that the data do not support. The correct comparison or result is: About 68.0% lies between 64 and 88. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It does not match the requested calculation. The correct comparison or result is: About 68.0% lies between 64 and 88. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It does not match the requested calculation. The correct comparison or result is: About 68.0% lies between 64 and 88. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 30. Empirical Rule
An approximately normal distribution of sample concentration at a food safety laboratory in Pine Ridge during a spring 2027 pilot has mean 72 and SD 12. Use the empirical rule for the interval from 48 to 96.
Answer: C
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 30. Empirical Rule: The endpoints are μ±2σ. The empirical rule places about 95.0% within 2 standard deviations of the mean.
Why the other choices fail
- Choice A: It makes a conclusion that the data do not support. The correct comparison or result is: About 95.0% lies between 48 and 96. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice B: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 48 and 96. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It does not match the requested calculation. The correct comparison or result is: About 95.0% lies between 48 and 96. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 31. Empirical Rule
An approximately normal distribution of course completion at a community college in Cedar Grove during a weekday operations study has mean 48 and SD 10. Use the empirical rule for the interval from 18 to 78.
Answer: B
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 31. Empirical Rule: The endpoints are μ±3σ. The empirical rule places about 99.7% within 3 standard deviations of the mean.
Why the other choices fail
- Choice A: It makes a conclusion that the data do not support. The correct comparison or result is: About 99.7% lies between 18 and 78. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It does not match the requested calculation. The correct comparison or result is: About 99.7% lies between 18 and 78. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It does not match the requested calculation. The correct comparison or result is: About 99.7% lies between 18 and 78. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 32. Empirical Rule
An approximately normal distribution of part diameter at a regional manufacturer in Cedar Grove during a semester-long cohort study has mean 67 and SD 5. Use the empirical rule for the interval from 52 to 82.
Answer: C
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 32. Empirical Rule: The endpoints are μ±3σ. The empirical rule places about 99.7% within 3 standard deviations of the mean.
Why the other choices fail
- Choice A: It makes a conclusion that the data do not support. The correct comparison or result is: About 99.7% lies between 52 and 82. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice B: It does not match the requested calculation. The correct comparison or result is: About 99.7% lies between 52 and 82. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It does not match the requested calculation. The correct comparison or result is: About 99.7% lies between 52 and 82. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 33. Empirical Rule
An approximately normal distribution of monthly household use at a municipal water office in South Harbor during a yearly program evaluation has mean 47 and SD 13. Use the empirical rule for the interval from 34 to 60.
Answer: D
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 33. Empirical Rule: The endpoints are μ±1σ. The empirical rule places about 68.0% within 1 standard deviation of the mean.
Why the other choices fail
- Choice A: It does not match the requested calculation. The correct comparison or result is: About 68.0% lies between 34 and 60. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice B: It does not match the requested calculation. The correct comparison or result is: About 68.0% lies between 34 and 60. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It makes a conclusion that the data do not support. The correct comparison or result is: About 68.0% lies between 34 and 60. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 34. Empirical Rule
An approximately normal distribution of security wait time at a regional airport authority in South Harbor during a follow-up evaluation period has mean 86 and SD 6. Use the empirical rule for the interval from 80 to 92.
Answer: C
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 34. Empirical Rule: The endpoints are μ±1σ. The empirical rule places about 68.0% within 1 standard deviation of the mean.
Why the other choices fail
- Choice A: It does not match the requested calculation. The correct comparison or result is: About 68.0% lies between 80 and 92. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice B: It makes a conclusion that the data do not support. The correct comparison or result is: About 68.0% lies between 80 and 92. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It does not match the requested calculation. The correct comparison or result is: About 68.0% lies between 80 and 92. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 35. Empirical Rule
An approximately normal distribution of course completion at a community college in North Valley during a quarterly performance study has mean 86 and SD 14. Use the empirical rule for the interval from 44 to 128.
Answer: B
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 35. Empirical Rule: The endpoints are μ±3σ. The empirical rule places about 99.7% within 3 standard deviations of the mean.
Why the other choices fail
- Choice A: It makes a conclusion that the data do not support. The correct comparison or result is: About 99.7% lies between 44 and 128. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It does not match the requested calculation. The correct comparison or result is: About 99.7% lies between 44 and 128. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It does not match the requested calculation. The correct comparison or result is: About 99.7% lies between 44 and 128. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Question 36. Empirical Rule
An approximately normal distribution of sample concentration at a food safety laboratory in Mountain Region during a spring 2027 pilot has mean 80 and SD 7. Use the empirical rule for the interval from 73 to 87.
Answer: B
Empirical Rule for Normal Distributions: 68–95–99.7 — Question 36. Empirical Rule: The endpoints are μ±1σ. The empirical rule places about 68.0% within 1 standard deviation of the mean.
Why the other choices fail
- Choice A: It makes a conclusion that the data do not support. The correct comparison or result is: About 68.0% lies between 73 and 87. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice C: It does not match the requested calculation. The correct comparison or result is: About 68.0% lies between 73 and 87. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
- Choice D: It does not match the requested calculation. The correct comparison or result is: About 68.0% lies between 73 and 87. Key check: About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.
Empirical Rule 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: Empirical Rule
Scenario. An approximately normal distribution of sample concentration at a food safety laboratory in Lakeside district during a spring 2027 pilot has mean 69 and SD 14. Use the empirical rule for the interval from 41 to 97.
- 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
Empirical Rule for Normal Distributions: 68–95–99.7 — FRQ set 1: Empirical Rule: The endpoints are μ±2σ. The empirical rule places about 95.0% within 2 standard deviations of the mean.
FRQ set 2: Empirical Rule
Scenario. An approximately normal distribution of security wait time at a regional airport authority in Midwest consortium during a follow-up evaluation period has mean 60 and SD 14. Use the empirical rule for the interval from 18 to 102.
- 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
Empirical Rule for Normal Distributions: 68–95–99.7 — FRQ set 2: Empirical Rule: The endpoints are μ±3σ. The empirical rule places about 99.7% within 3 standard deviations of the mean.
FRQ set 3: Empirical Rule
Scenario. An approximately normal distribution of trail-use duration at a state park in Riverbend during a winter readiness review has mean 97 and SD 4. Use the empirical rule for the interval from 85 to 109.
- 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
Empirical Rule for Normal Distributions: 68–95–99.7 — FRQ set 3: Empirical Rule: The endpoints are μ±3σ. The empirical rule places about 99.7% within 3 standard deviations of the mean.
FRQ set 4: Empirical Rule
Scenario. An approximately normal distribution of monthly household use at a municipal water office in Metro East during a school-year data collection has mean 97 and SD 10. Use the empirical rule for the interval from 67 to 127.
- 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
Empirical Rule for Normal Distributions: 68–95–99.7 — FRQ set 4: Empirical Rule: The endpoints are μ±3σ. The empirical rule places about 99.7% within 3 standard deviations of the mean.
FRQ set 5: Empirical Rule
Scenario. An approximately normal distribution of mobile-deposit adoption at a community bank in Cedar Grove during a pre-exam training cycle has mean 50 and SD 13. Use the empirical rule for the interval from 37 to 63.
- 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
Empirical Rule for Normal Distributions: 68–95–99.7 — FRQ set 5: Empirical Rule: The endpoints are μ±1σ. The empirical rule places about 68.0% within 1 standard deviation of the mean.
FRQ set 6: Empirical Rule
Scenario. An approximately normal distribution of mobile-deposit adoption at a community bank in Cedar Grove during a yearly program evaluation has mean 60 and SD 13. Use the empirical rule for the interval from 21 to 99.
- 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
Empirical Rule for Normal Distributions: 68–95–99.7 — FRQ set 6: Empirical Rule: The endpoints are μ±3σ. The empirical rule places about 99.7% within 3 standard deviations of the mean.
FRQ set 7: Empirical Rule
Scenario. An approximately normal distribution of course completion at a community college in Sunbelt district during a semester-long cohort study has mean 55 and SD 15. Use the empirical rule for the interval from 25 to 85.
- 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
Empirical Rule for Normal Distributions: 68–95–99.7 — FRQ set 7: Empirical Rule: The endpoints are μ±2σ. The empirical rule places about 95.0% within 2 standard deviations of the mean.
FRQ set 8: Empirical Rule
Scenario. An approximately normal distribution of daily energy output at a solar installer in Pacific Northwest during a community outreach cycle has mean 42 and SD 12. Use the empirical rule for the interval from 6 to 78.
- 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
Empirical Rule for Normal Distributions: 68–95–99.7 — FRQ set 8: Empirical Rule: The endpoints are μ±3σ. The empirical rule places about 99.7% within 3 standard deviations of the mean.
FRQ set 9: Empirical Rule
Scenario. An approximately normal distribution of appointment completion at a regional hospital in Sunbelt district during a multiweek validation study has mean 90 and SD 6. Use the empirical rule for the interval from 72 to 108.
- 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
Empirical Rule for Normal Distributions: 68–95–99.7 — FRQ set 9: Empirical Rule: The endpoints are μ±3σ. The empirical rule places about 99.7% within 3 standard deviations of the mean.
FRQ set 10: Empirical Rule
Scenario. An approximately normal distribution of mobile-deposit adoption at a community bank in Coastal Plains during a multiweek validation study has mean 63 and SD 7. Use the empirical rule for the interval from 42 to 84.
- 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
Empirical Rule for Normal Distributions: 68–95–99.7 — FRQ set 10: Empirical Rule: The endpoints are μ±3σ. The empirical rule places about 99.7% within 3 standard deviations of the mean.
FRQ set 11: Empirical Rule
Scenario. An approximately normal distribution of recovery time at a wildlife clinic in Sunbelt district during a weekday operations study has mean 54 and SD 10. Use the empirical rule for the interval from 44 to 64.
- 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
Empirical Rule for Normal Distributions: 68–95–99.7 — FRQ set 11: Empirical Rule: The endpoints are μ±1σ. The empirical rule places about 68.0% within 1 standard deviation of the mean.
FRQ set 12: Empirical Rule
Scenario. An approximately normal distribution of course completion at a community college in Riverbend during a yearly program evaluation has mean 60 and SD 16. Use the empirical rule for the interval from 44 to 76.
- 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
Empirical Rule for Normal Distributions: 68–95–99.7 — FRQ set 12: Empirical Rule: The endpoints are μ±1σ. The empirical rule places about 68.0% within 1 standard deviation of the mean.
FRQ set 13: Empirical Rule
Scenario. An approximately normal distribution of on-time arrival at a city transit agency in Pine Ridge during a yearly program evaluation has mean 59 and SD 13. Use the empirical rule for the interval from 46 to 72.
- 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
Empirical Rule for Normal Distributions: 68–95–99.7 — FRQ set 13: Empirical Rule: The endpoints are μ±1σ. The empirical rule places about 68.0% within 1 standard deviation of the mean.
FRQ set 14: Empirical Rule
Scenario. An approximately normal distribution of mobile-deposit adoption at a community bank in Atlantic Corridor during a weekday operations study has mean 51 and SD 10. Use the empirical rule for the interval from 21 to 81.
- 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
Empirical Rule for Normal Distributions: 68–95–99.7 — FRQ set 14: Empirical Rule: The endpoints are μ±3σ. The empirical rule places about 99.7% within 3 standard deviations of the mean.
