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IB Maths: The Normal Distribution and z-Scores (Paper 3: challenge)
IB Maths: The Normal Distribution and z-Scores (Paper 3: challenge)
IB maths normal distribution paper 3, challenge: standard error, sample means, critical values and confidence intervals.
Direct answers Answers in order: 2, 1, from 3 to 1.5, 1.645, 0.4332, 15, about 228 parts, 2.28 percent, 48.08 to 55.92, and the standard error of 5.
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)
- FoundationThe standard deviation is 12 and the sample size 36. What is the standard error of the sample mean?
- FoundationWith mean 50, standard deviation 12 and sample size 36, find the z-score of a sample mean of 52.
- FoundationWith the standard deviation fixed at 12, the sample size rises from 16 to 64. How does the standard error change?
- IntermediateThe critical z for a two-tailed 5 percent test is 1.96. What is it for a one-tailed 5 percent test?
- IntermediateFind the probability that Z lies between 0 and 1.5 given the table value 0.9332.
- IntermediateThe mean is 100 and the probability of a value below 115 is 0.8413. Find the standard deviation.
- ChallengeA part has mean length 100 and standard deviation 15, with 2.28 percent below the lower limit. About how many of 10 000 parts are rejected?
- ChallengeWith mean 500, standard deviation 50 and sample size 100, what proportion of sample means exceeds 510 given the table value at 2 is 0.9772?
- ChallengeA sample mean is 52 with a standard error of 2. Give the 95 percent confidence interval.
- Error redoA student uses the standard deviation of 50 directly when standardising a sample mean with 100 observations. What is wrong?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | 2 | The standard error divides the standard deviation by the square root of the sample size, so 12 over 6 gives 2. |
| 2 | 1 | The standard error is 2, so a difference of 2 divided by 2 gives 1. |
| 3 | From 3 down to 1.5 | The square root of the sample size rises from 4 to 8, so the standard error halves. |
| 4 | 1.645 | A one-tailed test puts the whole 5 percent in one tail, so the cut-off is smaller. |
| 5 | 0.4332 | Measuring from zero means taking away 0.5, so 0.9332 minus 0.5 gives 0.4332. |
| 6 | 15 | A probability of 0.8413 corresponds to z = 1, so 15 equals one standard deviation. |
| 7 | About 228 parts | 10 000 times 0.0228 gives 228. |
| 8 | 2.28 percent | The standard error is 5, so z is 2, and exceeding needs 1 minus 0.9772. |
| 9 | 48.08 to 55.92 | 1.96 times 2 is 3.92, so the interval is 52 plus or minus 3.92. |
| 10 | The standard error of 5 should be used, otherwise z is understated five times | A sample mean is standardised with the standard deviation divided by the square root of the sample size, not the raw standard deviation. |
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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