Mathematics 9709/63 — May/June 2014
Cambridge AS Level · Probability & Statistics 1 · worked solutions for every part, with the mark scheme
Topics Discrete Random Variables · Representation of Data · The Normal Distribution · Permutations and Combinations · Probability
Some adults and some children each tried to estimate, without using a watch, the number of seconds that had elapsed in a fixed time-interval. Their estimates are shown below.
Adults: 55 58 67 74 63 61 63 71 56 53 54 78 73 64 62
Children: 86 95 89 72 61 84 77 92 81 54 43 68 62 67 83
Draw a back-to-back stem-and-leaf diagram to represent the data.
Approach
Split each data set by tens digit to form the stem. Put adult leaves on the left of the stem and children leaves on the right, each set in increasing order away from the stem. Add a key so the numbers can be read as seconds.
Working
For adults the tens digits are 5, 6 and 7; for children the tens digits are 4, 5, 6, 7, 8 and 9. Ordering the leaves gives:
| Adults | Stem | Children |
|---|---|---|
| 4 | 3 | |
| 8 6 5 4 3 | 5 | 4 |
| 7 4 3 3 2 1 | 6 | 1 2 7 8 |
| 8 4 3 1 | 7 | 2 7 |
| 8 | 1 3 4 6 9 | |
| 9 | 2 5 |
Key: 3 | 5 | 4 represents 53 seconds for an adult and 54 seconds for a child.
Answer
The back-to-back stem-and-leaf diagram above, with the key, fully represents both data sets.
Back-to-back stem-and-leaf diagram drawn with key 3 | 5 | 4 representing 53 seconds for adults and 54 seconds for children.
Walkthrough
Start by taking the tens digit of each estimate as the stem. Since the estimates range from 43 to 95, the stems are 4, 5, 6, 7, 8 and 9. Write adult units on the left of the stem and children units on the right. The leaves on each side must be written in increasing order away from the stem. For example, the adult values 53, 54, 55, 56 and 58 have leaves 3, 4, 5, 6 and 8 moving outward, so they are printed as 8 6 5 4 3 in the left-hand column. The child values 61, 62, 67 and 68 have leaves 1, 2, 7 and 8 moving outward, printed as 1 2 7 8 on the right. Always include a key such as 3 | 5 | 4, because otherwise the reader could not tell that stems are tens and leaves are units. The completed diagram is shown below.
Key Takeaways
A stem-and-leaf diagram preserves all the original data values while giving the shape of a distribution. A back-to-back version uses one common stem for two data sets, making it easy to compare their centres, spreads and shapes.
Common Mistakes
A common mistake is writing leaves on the left in the wrong order. They must increase as they go away from the stem. Another common mistake is forgetting the key, or using stems that do not cover all the data. Mixing adult and child leaves on the same side also loses the comparison.
Things to Be Careful About
Make sure every data value appears exactly once. Use tens as the stem and units as the leaves, with no rounding. For the left-hand side, remember that the final answer must be read from the stem outward, not from left to right. The key is essential and must state what 3 | 5 | 4 represents.
Make two comparisons between the estimates of the adults and the children.
Approach
Use the completed back-to-back stem-and-leaf diagram to compare the general position, the spread and the shape of the two distributions.
Working
From the diagram, the adults' estimates cluster on lower stems, with a median around 63 seconds, while the children's estimates are centred higher, with a median around 77 seconds. The adults' range is 78 - 53 = 25 seconds, while the children's range is 95 - 43 = 52 seconds. The children's estimates are therefore much more spread out. Also, the adult leaves are fairly balanced on both sides of the stem, whereas the children's leaves have a long tail towards stem 4, so the adults are roughly symmetrical while the children are skewed.
Answer
Two valid comparisons are:
- Children's estimates are more spread out than adults' estimates.
- Adults' estimates are generally lower than children's estimates.
Children's estimates are more spread out than adults' estimates; adults' estimates are generally lower than children's estimates.
Walkthrough
The back-to-back diagram from part (i) makes comparisons easy. First compare centres: the adults' leaves are mostly on stems 5, 6 and 7, while the children's leaves extend up to stems 8 and 9, so adults' estimates are lower. Next compare spread: the adults have no values below 53 or above 78, while the children range from 43 to 95, so the children's estimates are considerably more spread out. A third comparison is shape: the adult leaves are quite balanced around the central stems, whereas the children's distribution has a longer tail towards the low values, so it is skewed. Any two of these statements would be accepted.
Key Takeaways
A back-to-back stem-and-leaf diagram supports quick comparisons of centre, spread and shape. When asked to make comparisons, use the visual structure of the diagram rather than trying to recalculate everything from raw data, though range and median are useful supporting measures.
Common Mistakes
Giving only a restatement of individual data values is not a valid comparison. Avoid vague statements such as 'children are higher' without connecting them to the data. Also, comparing means is not required here; comparing medians or the overall position of the leaves is enough.
Things to Be Careful About
Make sure the two comparisons are distinct and clearly different aspects, such as location and spread. If you write three comparisons, the first two are usually the ones that matter. Do not confuse the shape of the children's distribution: the low value 43 makes it skewed even though most of the children's values are high.
The rest of this paper
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- Q3Permutations and Combinations · Probability · Discrete Random Variables6M
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- Q5The Normal Distribution · Discrete Random Variables8M
- Q6Probability8M
- Q7Permutations and Combinations11M