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

Monday

A percent and its bracket
// A percent with a plus-or-minus bracket
⏱ about 20 min

Monday: A Percent and Its Bracket

The community center lobby has a big poster. In chalk it says "60% want a morning start" with a bracket drawn under it.

"We asked 50 runners, drawn from the bag of 240 sign-up cards," Comet says. "30 said morning."

"60% of the sample," Wren says. "But not exactly 60% of all 240. What do you notice about the bracket?"

Nova lights up the bracket. "Would you like a hint? A different draw of fifty would give a different percent."

"So the bracket says how far off we might be," Comet says. "Plus or minus 13.9%."

"From 46.1% to 73.9%," Wren says. "The whole group is most likely somewhere in there."

"That is a wide bracket," Comet says, frowning.

"Fifty cards is a small sample," Wren says. "A bigger sample would draw a tighter one. Let us learn the rule."

A lobby poster with a chalk percent and a plus-or-minus bracket; Comet draws, Wren checks a tally.

From a sample to the whole group

The population is every sign-up card, 240 of them. The sample is the 50 cards the crew drew at random.

The sample proportion p = 30/50 = 0.6 is a statistic. The proportion of all 240 cards is the parameter the crew wants.

A random sample gives a fair estimate, but chance moves it. The margin of error says how far chance usually moves it.

Every card and count in this course is the crew's own made-up reading from Nova's Run Log.

A solved problem to study

Here is how Wren puts the bracket on the poster, with the crew's rule 2√(p(1 - p)/n).

  1. Find the sample proportion: p = 30 ÷ 50 = 0.6, or 60%.
  2. Find p(1 - p): 0.6 × 0.4 = 0.24.
  3. Divide by n: 0.24 ÷ 50 = 0.0048.
  4. Take the square root: √0.0048 ≈ 0.0693. Double it: ≈ 0.139, or 13.9%.
  5. The interval: 60% - 13.9% = 46.1% up to 60% + 13.9% = 73.9%.

The crew reads it: "about 60% of all sign-ups want a morning start, give or take 13.9%."

Sample size nSample percentMargin 2√(p(1 - p)/n)Interval
5060%13.9%46.1% to 73.9%
THE POSTER'S BRACKET
  • Read the question.
  • Tap your answer.
30 of the 50 runners in the crew's sample said "morning". What percent is that? (Round to 1 place.)
In a sample of 50 cards, 60% said morning. Using the crew's rule 2√(p(1 - p)/n), what is the margin of error? (Round to 1 place.)
In a sample of 50, 60% said morning. With the crew's margin rule, what is the low end of the interval? (Round to 1 place.)
In a sample of 50, 60% said morning. With the crew's margin rule, what is the high end of the interval? (Round to 1 place.)
PARAMETER OR STATISTIC?
  • Read the question.
  • Tap your answer.
60% of the 50 sampled cards said morning. Is that number a parameter or a statistic?
The percent of all 240 sign-up cards that say morning. Is that number a parameter or a statistic?
How many of the 50 sampled cards said "morning"? Type the number.
WHY THIS EXERCISEEvery margin starts with a count over a sample size. The count is the fact; the percent is the estimate.
StatementTrue or false?
30 ÷ 50 = 0.6?
The sample percent 60% is exactly the percent of all 240 cards.?
The whole-group percent is most likely between 46.1% and 73.9%.?
The margin of error says how far a random sample usually lands from the whole-group number.?
WHY THIS EXERCISEReading a percent without its bracket is reading half the poster.
Try it
Write 60% on an index card. Under it draw a bracket from 46.1% to 73.9%.
Say out loud what the bracket means. Practice until it comes out in one sentence.
Draw the poster: the percent, the bracket from low end to high end, and the sample size underneath.

A strong start. Tomorrow you see where the rule comes from: a bag, many draws and a dot plot of percentages.