Sampling, Inferences & Comparing Populations — Drills
Sampling, Inferences & Comparing Populations has 10 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills 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.
Mean
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the mean of 22, 8, 19, 24, 5, 60.
Worked answer
23
22 + 8 + 19 + 24 + 5 + 60 = 138
138 ÷ 6 = 23
Problem 2
Find the mean of 13, 10, 6, 17, 59.
Worked answer
21
13 + 10 + 6 + 17 + 59 = 105
105 ÷ 5 = 21
Median
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the median of 26, 31, 24, 21, 40.
Worked answer
26
In order: 21, 24, 26, 31, 40
The middle value is 26.
Problem 2
Find the median of 24, 5, 29, 15, 20, 13.
Worked answer
17.5
In order: 5, 13, 15, 20, 24, 29
Average the two middle values: 15 + 202 = 17.5
Meansum
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Theo's 6 tests average 79. What score on the next test raises the average to 81?
Worked answer
93
Current total: 79 × 6 = 474
Target total: 81 × 7 = 567
567 - 474 = 93
Problem 2
Nina's 4 tests average 77. What score on the next test raises the average to 82?
Worked answer
102
Current total: 77 × 4 = 308
Target total: 82 × 5 = 410
410 - 308 = 102
Mad
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the mean absolute deviation of 4, -1, 19, 14.
Worked answer
7.5
Mean = 9
Distances: 10, 5, 5, 10
(10 + 5 + 5 + 10) ÷ 4 = 7.5
Problem 2
Find the mean absolute deviation of 20, 4, 16, 8.
Worked answer
6
Mean = 12
Distances: 8, 4, 4, 8
(8 + 4 + 4 + 8) ÷ 4 = 6
Sample
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
In a random sample of 50, exactly 12.5 have a trait. Estimate how many of 2,000 have it.
Worked answer
500
12.550 = 0.25
0.25 × 2000 = 500
Problem 2
In a random sample of 100, exactly 20 have a trait. Estimate how many of 1,000 have it.
Worked answer
200
20100 = 0.2
0.2 × 1000 = 200
Mode
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the mode of 13, 15, 5, 16, 13, 13.
Worked answer
13
13 appears three times, more than any other value.
Problem 2
Find the mode of 12, 8, 3, 3, 16, 3.
Worked answer
3
3 appears three times, more than any other value.
Bias
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
You want to find out the most popular sport in town. Which sample is biased?
- 20 students picked at random from each grade
- 60 students drawn at random from the full roster
- Fans leaving a basketball game
- Every 5th name on an alphabetical list of all students
Worked answer
Fans leaving a basketball game
Those people are far more interested in the topic than a typical person.
That over-represents one kind of answer — a biased sample.
The others give everyone a fair chance of being picked.
Problem 2
You want to find out how much television a typical student watches. Which sample is biased?
- 60 students drawn at random from the full roster
- Students waiting outside the school television studio
- 20 students picked at random from each grade
- Every 5th name on an alphabetical list of all students
Worked answer
Students waiting outside the school television studio
Those people are far more interested in the topic than a typical person.
That over-represents one kind of answer — a biased sample.
The others give everyone a fair chance of being picked.
Centre
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A data set is 12, 18, 18, 18, 298. Which measure better describes a typical value?
- The median
- The mean
- They are equally good
Worked answer
The median
Mean = 72.8
Median = 18
One large value dragged the mean away from where the data actually sits.
Problem 2
A data set is 11, 14, 15, 16, 172. Which measure better describes a typical value?
- The median
- They are equally good
- The mean
Worked answer
The median
Mean = 45.6
Median = 15
One large value dragged the mean away from where the data actually sits.
Comparing
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Class A: mean 83, MAD 2. Class B: mean 88, MAD 2. Is the 5-point gap convincing?
- Yes — the gap is larger than the spread
- No — the gap is smaller than the spread
- There is not enough information
Worked answer
Yes — the gap is larger than the spread
Gap between means: 5
Typical variation inside each class: about 2
The gap is bigger than the noise inside the classes, so it means something.
Problem 2
Class A: mean 71, MAD 12. Class B: mean 75, MAD 12. Is the 4-point gap convincing?
- No — the gap is smaller than the spread
- Yes — the gap is larger than the spread
- There is not enough information
Worked answer
No — the gap is smaller than the spread
Gap between means: 4
Typical variation inside each class: about 12
The gap is smaller than the variation inside each class, so it is not convincing.
Capture
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A biologist tags 100 fish and releases them. Later a sample of 45 fish contains 9 tagged ones. Estimate the lake's fish population.
Worked answer
500
Sample fraction tagged: 15
So the 100 tagged fish are about 15 of the lake
100 × 5 = 500 fish
Problem 2
A biologist tags 40 fish and releases them. Later a sample of 44 fish contains 11 tagged ones. Estimate the lake's fish population.
Worked answer
160
Sample fraction tagged: 14
So the 40 tagged fish are about 14 of the lake
40 × 4 = 160 fish