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Statistics 9-12 / Week 08 / Tuesday
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Week 08 · Margin of Error

Tuesday

Where the margin comes from
// A percent with a plus-or-minus bracket
⏱ about 20 min

Tuesday: Where the Margin Comes From

"Suppose the whole bag really is 60% morning," Wren says. "What would samples of 50 look like?"

"Let us find out," Comet says. She fills a bag with tokens, 60% of them marked M, and draws 50.

Nova logs each draw as a percent. Twenty draws later the list reads 44, 48, 52, 54, 54, 56 and on.

"They bunch around 60%," Wren says. "The standard deviation of the twenty percentages is about 7.5. What do you notice?"

"Two of those is 15%," Comet says. "Close to the 13.9% from the rule."

"The rule is a shortcut for this picture," Wren says. "Two standard deviations of the simulated percentages."

Nova glows. "Would you like a hint? Try the rule with a bigger n and watch the bracket shrink."

"Two hundred cards gives 6.9%," Comet says. "Four times the sample, half the margin."

Way 1: simulate the draws

Build a bag that matches the estimate: 60% of the tokens marked M. Draw a sample of 50, count the Ms, and record the percent.

Repeat many times. The dot plot of sample percentages shows how far a random sample of this size wanders.

Two standard deviations of those percentages is the margin of error. About 95% of random samples land inside it.

Simulated sample percentages (20 draws of 50)
44, 48, 52, 54, 54, 56, 58, 58, 60, 6060, 60, 62, 62, 64, 66, 66, 68, 72, 76
A dot plot of the crew's 20 simulated sample percentages, from 44 to 76, bunched near 60%.

Way 2: the crew's rule

Many simulations like this one agree on a shortcut: the standard deviation of sample proportions is close to √(p(1 - p)/n).

Doubling it gives the margin: 2√(p(1 - p)/n). The crew uses this rule when it cannot draw hundreds of samples.

For p = 0.6 and n = 50 the rule gives 13.9%. The simulation gave 15%. Both ways agree closely.

How the margin shrinks

Sample size nMargin for p = 0.6Interval
5013.9%46.1% to 73.9%
1009.8%50.2% to 69.8%
2006.9%53.1% to 66.9%
8003.5%56.5% to 63.5%

The n sits under a square root. Four times the sample makes the margin 2 times smaller, not four times smaller.

SIMULATION AND RULE
  • Read the question.
  • Tap your answer.
The crew's 20 simulated sample percentages are in the table above. What is their standard deviation? (Round to 1 place.)
The crew's 20 simulated sample percentages are in the table above. What is their mean? (Round to 1 place.)
A sample of 100 cards has 60% saying morning. By the crew's rule, what is the margin of error? (Round to 1 place.)
A sample of 200 cards has 60% saying morning. By the crew's rule, what is the margin of error? (Round to 1 place.)
TWO WAYS, ONE IDEA
  • Read the question.
  • Tap your answer.
In the simulation, what does the margin of error measure?
The margin for n = 50 is 13.9% and for n = 200 it is 6.9%. How many times smaller is the second?
The crew's rule for the margin is 2√(p(1 - p)/n). To halve the margin, multiply n by what? Type the number.
Drawing many samples from a known bag to see the spread is called what? Type one word.
StatementTrue or false?
50 × 4 = 200?
Doubling the sample size halves the margin of error.?
About 95% of random samples land within the margin of the whole-group number.?
The simulated margin 15% and the rule's 13.9% are close.?
WHY THIS EXERCISEKnowing how the margin shrinks tells the crew how many cards to draw before the poster goes up.
Try it
Put 10 tokens in a bag, 6 marked M. Draw 10 with replacement (draw, note, return, shake) and record the percent of Ms.
Repeat 5 times and write the five percentages. How far do they wander from 60%?
Draw the dot plot of the 20 simulated sample percentages and shade two standard deviations on each side of 60.

Two ways that agree. Tomorrow is Data Lab: your own bag, your own draws and your own bracket.

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