← Back to course
Statistics 9-12 / Week 08 / Wednesday
3/6
Week 08 · Margin of Error

Wednesday

Data Lab: draws from the token bag
// A percent with a plus-or-minus bracket
⏱ about 20 min

Wednesday: Data Lab, Draws from the Token Bag

"Lab day," Comet says, shaking a cloth bag. "20 tokens. 12 are marked M for morning."

"So the bag's true proportion is 0.6," Wren says. "Today we pretend we do not know that. We sample."

"Draw 10, one at a time, putting each back," Comet says. "Count the Ms. Then do it again."

Nova logs ten rounds: 6, 7, 5, 8, 6, 7, 6, 4, 7, 6 Ms out of 10.

"Round 2 gave 7 of 10, that is 70%," Wren says. "The rule's margin for n = 10 is 29%. What do you notice?"

"Huge," Comet says. "70% plus or minus 29% covers almost everything."

"Ten draws is a tiny sample," Wren says. "The bracket is honest about that."

Nova glows. "Would you like a hint? Every round's interval should still catch the true 60%."

What you need

  • A cloth bag or a box, and 20 tokens or paper scraps, 12 of them marked M.
  • Your Run Log, a pencil and graph paper for a dot plot from 0 to 10.
  • A flat surface so tokens do not roll away.
Safety first
Running or walking happens on the park loop or a safe path with a grown-up aware, and water is at hand. No one runs on a road.
Cut paper scraps with scissors on paper or cardboard, over the table. Keep small tokens away from younger children.
Nothing heavy is lifted alone. Pick up any dropped token so no one slips.

Run the lab

  1. Mark 12 of your 20 tokens with an M. Put all 20 in the bag and shake.
  2. Draw one token, write M or not, put it back and shake. Repeat 10 times. That is one round.
  3. Count the Ms in the round and write the count in your Run Log. Do 10 rounds in all.
  4. Turn one round into a percent and find its margin with 2√(p(1 - p)/10). Write the interval.
  5. Make a dot plot of your 10 counts from 0 to 10. Mark the true count, 6, on it.
  6. Check each round's interval: does it include the true 60%? Count how many do.

The crew's lab results

The crew's 10 rounds are made up for the Run Log. Yours will differ, and the comparison is the point.

Round12345678910
Ms in 10 draws6758676476
A dot plot of the crew's 10 counts of M, from 4 to 8, bunched around 6.

The crew's mean count is 6.2, near the true 6. Each round's sample percent wandered, and each bracket was wide.

With n = 10, the margin for a 70% sample is 29%. A tiny sample gives an honest but almost useless bracket.

READ THE LAB RESULTS
  • Read the question.
  • Tap your answer.
In round 2, 7 of the 10 draws were M. What percent is that? (Round to 1 place.)
Round 2 gave 70% M in 10 draws. Using the crew's rule 2√(p(1 - p)/n), what is the margin? (Round to 1 place.)
Round 2 gave 70% in 10 draws. With the crew's margin rule, what is the low end of the interval? (Round to 1 place.)
The crew's counts are 6, 7, 5, 8, 6, 7, 6, 4, 7, 6. What is the mean count of Ms? (Round to 1 place.)
The bag holds 20 tokens and 12 are marked M. In 10 draws, what count of M does the true proportion predict? Type the number.
WHY THIS EXERCISEKnowing the true answer in the lab lets you see how often a sample misses it, and by how much.
What is the true proportion of M in the bag, as a decimal? Type the number.
Putting each token back before the next draw is drawing with what? Type one word.
StatementTrue or false?
12 ÷ 20 = 0.6?
A sample of 10 gives a smaller margin than a sample of 50.?
Every one of the crew's 10 rounds gave exactly 6 Ms.?
A sample percent can miss the true percent and still be a fair estimate.?
WHY THIS EXERCISESeeing your own rounds miss the known truth is the fastest way to respect the bracket.
Draw your dot plot of M counts from 0 to 10. Mark the true count and circle the round you turned into an interval.

Good lab work. Tomorrow the same idea estimates a mean: the crew's stretch times and a bracket in seconds.

← Tuesday