Probability and inference

The or-rule, independent events, correlation against causation, how a study was run, and how far a sample can be out — the half of high-school statistics that is an argument rather than a calculation.

5 sheets · 30 questions · one answer key · 8 pages

Recommended for grades 9–11

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Union and intersection Sheet 1 of 5

The addition ruleAnd, or

Name: Date:

Watch for AND or OR in each question. Answer as a whole percent.

1

	Renewed	Returned	Total
Fiction	18	12	30
Facts	9	21	30
Total	27	33	60

What percent are Fiction OR Renewed?

%

2

	Won	Lost	Total
Team A	12	8	20
Team B	6	14	20
Total	18	22	40

What percent are Team A OR Won?

%

3

	Finished	Gave up	Total
Indoors	9	1	10
Outdoors	4	6	10
Total	13	7	20

What percent are Indoors OR Finished?

%

4

	Pet	No pet	Total
Grade 7	24	16	40
Grade 8	12	28	40
Total	36	44	80

What percent are Grade 7 OR Pet?

%

5

	Chess	Cards	Total
Juniors	24	16	40
Seniors	12	28	40
Total	36	44	80

What percent are Juniors OR Chess?

%

6

	Helmet	None	Total
Bikes	9	1	10
Scooters	4	6	10
Total	13	7	20

What percent are Bikes OR Helmet?

%

Independent eventsClear and close

Name: Date:

Multiply the two percents. Does the table agree? Circle your answer.

1

	Fruit	No fruit	Total
Packed	56	24	80
Canteen	14	6	20
Total	70	30	100

Are Packed and Fruit independent?

IndependentNot independent

2

	Won	Lost	Total
Team A	37	3	40
Team B	43	17	60
Total	80	20	100

Are Team A and Won independent?

IndependentNot independent

3

	Finished	Gave up	Total
Indoors	19	21	40
Outdoors	41	19	60
Total	60	40	100

Are Indoors and Finished independent?

IndependentNot independent

4

	Sold	Left	Total
Small	42	28	70
Large	18	12	30
Total	60	40	100

Are Small and Sold independent?

IndependentNot independent

5

	Walked	Drove	Total
Sunny	12	28	40
Rainy	18	42	60
Total	30	70	100

Are Sunny and Walked independent?

IndependentNot independent

6

	Chess	Cards	Total
Juniors	49	21	70
Seniors	21	9	30
Total	70	30	100

Are Juniors and Chess independent?

IndependentNot independent

Reading a studyWhy two things move together

Name: Date:

Two things move together. Circle the best explanation for each.

1

People with larger shoe sizes know more words.

First causes secondSecond causes firstSomething else causes both

2

Children with bigger feet tend to read better.

First causes secondSecond causes firstSomething else causes both

3

Hospitals with more staff treat more patients.

First causes secondSecond causes firstSomething else causes both

4

Towns with more police officers report more crime.

First causes secondSecond causes firstSomething else causes both

5

The more it rains, the fuller the reservoir gets.

First causes secondSecond causes firstSomething else causes both

6

The more hours a fridge runs, the more power it uses.

First causes secondSecond causes firstSomething else causes both

Reading a studySurvey, experiment or study

Name: Date:

Read each study and circle which kind it is.

1

A town phones 500 households to ask about a new library.

Sample surveyExperimentObservational study

2

Half the plants, chosen at random, are given the new fertilizer.

Sample surveyExperimentObservational study

3

Half the aisles, picked at random, get the brighter lighting.

Sample surveyExperimentObservational study

4

200 randomly chosen students are asked how they get to school.

Sample surveyExperimentObservational study

5

Runners who already train daily are compared with those who do not.

Sample surveyExperimentObservational study

6

Lung capacity is measured in people who already cycle to work.

Sample surveyExperimentObservational study

Margin of errorMargins and ranges

Name: Date:

Find the margin of error, or use it to say how far out a poll could be.

1

2,500 fans were asked; 48% watched the match, give or take 2%. What is the lowest the true figure could be?

%

2

25 voters were asked about the plan. What is the margin of error?

%

3

400 users were asked; 66% have tried the app, give or take 5%. What is the lowest the true figure could be?

%

4

2,500 shoppers were asked about the new label. What is the margin of error?

%

5

100 users were asked; 30% have tried the app, give or take 10%. What is the lowest the true figure could be?

%

6

100 commuters were asked about the bus. What is the margin of error?

%

Power Pack · Probability and inferenceAnswer key

Union and intersection

1. What percent are Fiction OR Renewed? → 65% (30 + 27 − 18 = 39 of 60)2. What percent are Team A OR Won? → 65% (20 + 18 − 12 = 26 of 40)3. What percent are Indoors OR Finished? → 70% (10 + 13 − 9 = 14 of 20)4. What percent are Grade 7 OR Pet? → 65% (40 + 36 − 24 = 52 of 80)5. What percent are Juniors OR Chess? → 65% (40 + 36 − 24 = 52 of 80)6. What percent are Bikes OR Helmet? → 70% (10 + 13 − 9 = 14 of 20)

Independent events

1. Are Packed and Fruit independent? → Independent — 80% × 70% = 56%, and the table shows 56%2. Are Team A and Won independent? → Not independent — 40% × 80% = 32%, and the table shows 37%3. Are Indoors and Finished independent? → Not independent — 40% × 60% = 24%, and the table shows 19%4. Are Small and Sold independent? → Independent — 70% × 60% = 42%, and the table shows 42%5. Are Sunny and Walked independent? → Independent — 40% × 30% = 12%, and the table shows 12%6. Are Juniors and Chess independent? → Independent — 70% × 70% = 49%, and the table shows 49%

Correlation and causation

1. People with larger shoe sizes know more words. → Something else causes both — Adults have bigger feet and larger vocabularies than children.2. Children with bigger feet tend to read better. → Something else causes both — Older children have bigger feet AND have been taught for longer.3. Hospitals with more staff treat more patients. → Second causes first — A busy hospital hires more staff; the staff do not create patients.4. Towns with more police officers report more crime. → Second causes first — A town hires police BECAUSE it has crime — the crime came first.

Power Pack · Probability and inferenceAnswer key

Correlation and causation

5. The more it rains, the fuller the reservoir gets. → First causes second — Rain runs into the reservoir; the reservoir cannot make it rain.6. The more hours a fridge runs, the more power it uses. → First causes second — Running is what draws the power, not the other way around.

Survey, experiment or study

1. A town phones 500 households to ask about a new library. → Sample survey — Questions to a sample, and no treatment given: a sample survey.2. Half the plants, chosen at random, are given the new fertilizer. → Experiment — The researcher ASSIGNS who gets the treatment, at random: an experiment.3. Half the aisles, picked at random, get the brighter lighting. → Experiment — The aisles were assigned their lighting at random: an experiment.4. 200 randomly chosen students are asked how they get to school. → Sample survey — They are ASKED a question. Nothing is done to anyone: a sample survey.5. Runners who already train daily are compared with those who do not. → Observational study — The groups were ALREADY different; nobody assigned the training.6. Lung capacity is measured in people who already cycle to work. → Observational study — No one was told to cycle; the habit came first: an observational study.

Margin of error

1. 2,500 fans were asked; 48% watched the match, give or take 2%. What is the lowest the true figure could be? → 46% (48% − 2% = 46%)2. 25 voters were asked about the plan. What is the margin of error? → 20% (1 ÷ √25 = 1 ÷ 5 = 20%)3. 400 users were asked; 66% have tried the app, give or take 5%. What is the lowest the true figure could be? → 61% (66% − 5% = 61%)4. 2,500 shoppers were asked about the new label. What is the margin of error? → 2% (1 ÷ √2,500 = 1 ÷ 50 = 2%)

Power Pack · Probability and inferenceAnswer key

Margin of error

5. 100 users were asked; 30% have tried the app, give or take 10%. What is the lowest the true figure could be? → 20% (30% − 10% = 20%)6. 100 commuters were asked about the bus. What is the margin of error? → 10% (1 ÷ √100 = 1 ÷ 10 = 10%)