Section%201.3%20

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Section 1.3 Things Vary, and Small Things Vary
the Most
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Population vs. Sample

• A sample is a part of the population that we
  actually examine to gather information about the
  whole.

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Example

• A sociologist is interested in determining the
  percentage of college students who have smoked
  marijuana so she randomly samples 50 college
  students and finds that 60 of the students in
  the sample respond yes.
• Population All college students.
• Sample 50 college students surveyed.

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Sampling Variability

• How confident can we be that the sociologists
  estimate reflects ALL college students?
• _____________________________________
• Assume the sample is randomly selected .
• The main question we need to understand is how
  does the sample proportion vary in repeated
  samples?

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Well Use Simulation to Answer this Question

• Suppose we are interested in estimating the
  proportion of heads that would land if a penny
  was flipped a large number of times.
• Population All pennies
• Population proportion of interest percentage of
  coins that lands heads up.

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How do we estimate this quantity?

• Randomly sample 25 pennies from the population of
  all pennies.
• Compute the proportion of heads, ,
• in the sample.

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Student Activity

• Each person will flip a coin 25 times, keeping
  track of the proportion of heads in 25 tosses.
• Before doing the experiment, can you say for
• certain what your will be? What can you
  say about the class data?
• Consider the following example

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Individual Student Worksheet
Flip number (n) Head landed? number of heads in n tosses Proportion of heads in n tosses
1 Yes 1 1.00
2 Yes 2 1.00
3 No 2 0.67

25 Yes
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Percentage of Heads in n tosses
100
of heads in n tosses
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25
n (number of tosses)
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Law of Large Numbers

• As the sample gets larger, the proportion of each
  type of event tends to move toward the
  probability of that type of event occurring on
  each trial.
• What does this mean in terms of the penny example?

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Gamblers Fallacy

• Common misconception about the Law of Large
  Numbers
• Gamblers Fallacy After several appearances of
  a chance outcome, that outcome is less likely.
• ex. Flip a coin 4 times and get T, T, T, T.
  Gamblers Fallacy says that youre more likely to
  get a H on the next flip.

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Final Points

• Small samples vary more.
• Extreme samples tend to be small samples.
• Things vary.