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IB Maths: The Normal Distribution and z-Scores (Paper 2: intermediate)
IB Maths: The Normal Distribution and z-Scores (Paper 2: intermediate)
IB maths normal distribution paper 2, intermediate: inverse values, middle probabilities, cross-class comparison and percentile cut-offs.
Direct answers Answers in order: 50, z of 2, 0.6826, the class A student with z 1.4, 6.68 percent, 1 180.4 to 1 219.6, about 564, about 63 people, 1 and 0.5, and 0.1587.
01 Paper details (check answers afterwards)
| Item | Detail |
|---|---|
| Questions | 10 |
| Suggested time | 20 |
| Difficulty mix | Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1 |
| Chapter | Chapter: IB Maths: The Normal Distribution and z-Scores |
| Level | IB |
02 Questions (10 items)
- FoundationWith mean 40 and standard deviation 8, which score has z = 1.25?
- FoundationThe table gives 0.9772 for some z. Find z.
- FoundationFind the probability that Z lies between -1 and 1.
- IntermediateClass A has mean 65 and standard deviation 5; class B has mean 70 and standard deviation 8. One student scores 72 in A and another 80 in B. Who did better relative to the class?
- IntermediateWith mean 200 and standard deviation 20, what proportion exceeds 230 given that the table value at 1.5 is 0.9332?
- IntermediateWith mean 1 200 and standard deviation 10, give the middle 95 percent range using 1.96 standard deviations.
- ChallengeWith mean 500 and standard deviation 50, find the cut-off for the top 10 percent given that the table value at 1.28 is 0.8997.
- ChallengeGiven that the table value at -1 is 0.1587, about how many of 400 people lie more than one standard deviation below the mean?
- ChallengeThe mean is 60 and the score 75. What is z when the standard deviation is 15, and what is it when the standard deviation is 30?
- Error redoA student claims the probability that Z exceeds 1 is 0.8413. What is wrong?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | 50 | 1.25 times 8 is 10, and 40 plus 10 gives 50. |
| 2 | 2 | 0.9772 is the standard normal table entry at 2, so z is 2. |
| 3 | 0.6826 | A between question subtracts two table values, 0.8413 minus 0.1587 giving 0.6826. |
| 4 | The class A student, whose z of 1.4 beats 1.25 | Standardise both: 7 over 5 gives 1.4 for the first and 10 over 8 gives 1.25 for the second. |
| 5 | 6.68 percent | The z is 1.5, and exceeding means 1 minus 0.9332, which is 0.0668. |
| 6 | 1 180.4 to 1 219.6 | 1.96 times 10 is 19.6, so the limits are 1 200 plus or minus 19.6. |
| 7 | About 564 | 1.28 times 50 is 64, and 500 plus 64 gives 564. |
| 8 | About 63 people | 400 times 0.1587 is 63.48, which is about 63 people. |
| 9 | 1 and 0.5 | The numerator stays at 15, so doubling the denominator halves the z-score. |
| 10 | It is 0.1587, since 0.8413 is the proportion below 1 | The table gives the area below z, so the opposite direction needs one minus that value. |
04 What to do after marking
- After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
- Repeat the same item two days later; move to the next paper only when both attempts are correct.
- Parents only need the answer and note columns; there is no need to explain every item.
Always subtract the mean and divide by the standard deviation first, then read the standard normal table; to go backwards, find z first and multiply by the standard deviation before adding the mean.Tutor Registration →
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