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Statistics 9-12 / Week 08 / Track Day
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Week 08 · Margin of Error

Track Day

Track Day: the family bracket
// A percent with a plus-or-minus bracket
⏱ about 20 min

Track Day: The Family Bracket

Saturday the crew brings a paper bag to the kitchen table. Twenty paper scraps, eight marked with a star.

"Pretend you do not know how many stars," Comet says. "Draw 10, one at a time, putting each back."

The family draws. 3 stars in 10. "30%," Wren says. "The bag is really 40%."

"Off by ten," someone says. "That is bad."

"Check the bracket," Comet says. "The rule gives 29% for ten draws. Forty sits inside."

"So the sample did its job," Wren says. "It missed, but it told us how far it might miss. What do you notice?"

Nova hovers. "Would you like a hint? Try pooling everyone's draws into one big sample."

Four people, forty draws, a tighter bracket. The family sees the square root at work.

  1. A sample percent or mean estimates the whole-group number, and chance moves it.
  2. Simulate many samples; two standard deviations of the estimates is the margin of error.
  3. The crew's rule for a proportion: 2√(p(1 - p)/n). Estimate ± margin is the interval.
  4. Four times the sample halves the margin.
  5. A report needs the sample size, the method and the margin before anyone trusts its number.
◇ FAMILY BAG DRAW
Put 20 paper scraps in a bag, 8 marked with a star. Do not tell the drawer how many stars.
Each person draws 10 with replacement and counts stars. Turn the count into a percent and a margin with the rule.
Write each interval and check whether 40% sits inside. Then pool all the draws into one sample and redo the bracket.
Take turns saying what the bracket means in one plain sentence.
For a grown-up
Expect the ten-draw percentages to miss 40% by a lot. The point is that the bracket is wide enough to catch it.
Pooling the draws shows why the crew wants bigger samples. Help the family notice the bracket shrink.
JUDGE THE FAMILY BRACKET
  • Read the question.
  • Tap your answer.
3 of 10 draws showed a star. What percent is that? (Round to 1 place.)
A family draw gave 30% stars in 10 draws. Using the crew's rule, what is the margin of error? (Round to 1 place.)
Twenty scraps, eight with a star. What is the true percent of stars in the bag? Type the number.
WHY THIS EXERCISEKnowing the true answer at home lets the family see a sample miss and a bracket catch.
Draw each family interval as a bracket on one number line from 0 to 100, with 40 marked.

Track Day done. Next week the crew compares two warm-up routines and asks whether a difference is more than chance.

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