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Statistics 9-12 / Week 03 / Thursday
4/6
Week 03 · Two-Way Tables

Thursday

Association, and the table as a sample space
// Sign-up cards sorted two ways at once
⏱ about 20 min

Thursday: Association, and the Table as a Sample Space

"Finished given morning 75%, given evening 50%," Wren says. "A 25-point gap. Morning and finishing are associated."

"Associated means knowing the slot changes the finishing percent," Comet says. "Not that mornings cause finishing."

Wren pins up a second table. "Stretched or not, against finished or stopped. 12, 8, 6, 4."

"Finished given stretched is 3/5," Comet says. "Given no stretch, 3/5. The same. What do you notice?"

"No association," Wren says. "Knowing whether a runner stretched tells you nothing about finishing, in our cards."

Nova lights one card on the grid. "Would you like a hint? Pick a card at random. The table is now a sample space."

"P(finished) is 30 of 48," Comet says. "P(finished given morning) is 18 of 24. Same fractions, new names."

Judging an association

Compare the conditional percentages across the rows. If they differ by 10 points or more, the crew says the two questions are associated.

Sign-up cards: 75% against 50%, a gap of 25 points. Associated.

Stretch cards: 60% against 60%. No gap at all. Not associated.

An association is a pattern in the table. It is not proof that one thing caused the other. The slot was chosen, not assigned.

FinishedStoppedTotal
Stretched12820
No stretch6410
Total181230
The stretch table: stretched or not against finished or stopped, 30 cards, built so stretching makes no difference.

The table as a sample space

Pick one sign-up card at random. Every card is one outcome, so the whole table is the sample space.

P(finished) is the finished total over the grand total: 30 ÷ 48 = 5/8. That is the marginal percent as a fraction.

P(finished given morning) is the cell over the morning total: 18 ÷ 24 = 3/4. That is the conditional percent as a fraction.

Independence test: P(finished and morning) = 3/8, but P(finished) × P(morning) = 5/16. Not equal, so not independent.

Stretch table: P(finished and stretched) = 2/5 and P(finished) × P(stretched) = 2/5. Equal, so independent.

Why a conditional percent uses the group total

  1. Read the "given" group. It names one row or one column.
  2. Only the cards in that group are possible now, so that group is the new whole.
  3. Count the cards in the group that also have the second feature.
  4. Divide that count by the group total, not by the grand total.
  5. Multiply by 100 to write it as a percent of the group.
THE CONDITIONAL PERCENT, IN ORDER
  • ?Read the "given" group
  • ?That group becomes the new whole
  • ?Count the cards in the group with the second feature
  • ?Divide by the group total, not the grand total
  • ?Multiply by 100
WHY THIS EXERCISEDividing by the grand total instead would mix in cards that the "given" already ruled out.
ASSOCIATION OR NOT?
  • Read the question.
  • Tap your answer.
A two-way table: rows Morning and Evening, columns Finished and Stopped, counts 18 and 6; 12 and 12, total 48.Sign-up cards: finished given morning is 75% and given evening is 50%. Does the table suggest an association?
A two-way table: rows Stretched and No stretch, columns Finished and Stopped, counts 12 and 8; 6 and 4, total 30.Stretch cards: finished given stretched is 60% and given no stretch is 60%. Does the table suggest an association?
Morning slot and finishing are associated in the crew's cards. Which conclusion is justified?
THE TABLE AS A SAMPLE SPACE
  • Read the question.
  • Tap your answer.
A two-way table: rows Morning and Evening, columns Finished and Stopped, counts 18 and 6; 12 and 12, total 48.One of the 48 sign-up cards is picked at random. What is the probability it says finished?
A two-way table: rows Morning and Evening, columns Finished and Stopped, counts 18 and 6; 12 and 12, total 48.Given that the picked card is a morning card, what is the probability it says finished?
P(finished) = 3/5 and P(finished given stretched) = 3/5. Are "finished" and "stretched" independent?
P(finished) = 5/8 and P(finished given morning) = 3/4. Are "finished" and "morning" independent?
A percent that uses one group's total as the whole is called what? Type one word.
When the conditional percentages differ across rows, the two questions show an what? Type one word.
Draw the stretch table as a row-percent table and circle the two matching finished percentages.

Careful thinking. Tomorrow you read two-way tables in everyday sentences and review the week.

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