Sampling, Inferences & Comparing Populations — Challenge
You cannot measure everyone, so you measure some. A random sample lets a small group speak honestly for a large one.
What these problems cover
These questions use 12 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Rate — A ratio of two different units, like miles per hour or dollars per pound.
- Percent — Parts per hundred. 37% = 37100 = 0.37.
- Population — Every member of the group you care about.
- Sample — The part of the population you actually measure.
- Random sample — Every member has an equal chance of being picked — the only kind you can trust.
- Mean — Two kinds of centre: the balance point, and the middle value.
- Median — Two kinds of centre: the balance point, and the middle value.
- MAD — Mean absolute deviation — the average distance from the mean; a measure of spread.
- Scale — What one grid step is worth — and where the axis starts.
- Outlier — A dot far away from the rest of the pattern.
- Mode — The value that appears most often.
- Mean absolute deviation — The average distance of the values from the mean.
Problem 1
What is the mean of 4, 7, 9, 12 and 13?
- 7
- 8
- 9
- 10
- 45
Worked answer
9
Add the values: 4+7+9+12+13 = 45.
Divide by how many there are: 455 = 9.
Problem 2
What is the median of 3, 8, 5, 12, 7 and 9?
- 7
- 7.5
- 8
- 8.5
- 9
Worked answer
7.5
Put the values in order first: 3, 5, 7, 8, 9, 12.
There are six values, an even number, so the median is the mean of the middle two.
The middle two are 7 and 8, so the median is 7+82 = 7.5.
Problem 3
The mean of five numbers is 14. Four of them are 10, 12, 15 and 18. What is the fifth?
- 13
- 14
- 15
- 16
- 17
Worked answer
15
If the mean of five numbers is 14, their total is 5· 14 = 70.
The four known numbers add to 10+12+15+18 = 55.
The fifth is 70-55 = 15.
Problem 4
What is the mode of 2, 3, 3, 5, 7, 3 and 9?
- 2
- 3
- 5
- 7
- 9
Worked answer
3
The mode is the value that appears most often.
3 appears three times and every other value appears once, so the mode is 3.
Problem 5
What is the range of 14, 8, 22, 5 and 19?
- 14
- 16
- 17
- 19
- 22
Worked answer
17
The range is the largest value minus the smallest.
The largest is 22 and the smallest is 5, so the range is 22-5 = 17.
Problem 6
Six numbers have a mean of 20. One number is removed and the remaining five have a mean of 21. What number was removed?
- 10
- 12
- 15
- 18
- 20
Worked answer
15
The six numbers total 6· 20 = 120.
The five that remain total 5· 21 = 105.
The number removed is the difference: 120-105 = 15. It is below the old mean, which is why removing it pulled the mean up.
Problem 7
What is the mean of 6, 6, 9 and 11?
- 6
- 7
- 8
- 9
- 32
Worked answer
8
Add the values: 6+6+9+11 = 32.
Divide by 4: 324 = 8.
Problem 8
What is the median of 4, 9, 1, 7 and 3?
- 3
- 4
- 5
- 7
- 9
Worked answer
4
Order the values: 1, 3, 4, 7, 9.
With five values the median is the middle one, which is 4.
Taking the middle of the unordered list would wrongly give 1.
Problem 9
A student has scores of 80, 85 and 90 on three tests. What score on a fourth test would make the mean of all four equal to 86?
- 86
- 87
- 88
- 89
- 90
Worked answer
89
Four tests with mean 86 must total 4· 86 = 344.
The first three total 80+85+90 = 255.
The fourth score must be 344-255 = 89.
Problem 10
The values 2, 4, 4, 6 and 9 have a mean. If the 9 is replaced by 19, by how much does the mean increase?
- 1
- 2
- 3
- 5
- 10
Worked answer
2
The original total is 2+4+4+6+9 = 25, so the mean is 5.
Replacing 9 by 19 raises the total by 10, to 35, so the new mean is 7.
The mean rises by 2, which is the 10 spread over the five values.