What Do Samples Tell Us

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What Do Samples Tell Us?

• MA155
• Chapter 3

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Sampling Basics

• Population
• The entire group of subjects that we want
  information
• Individual
• Any single member of the population
• Sample
• A part of the population from which we actually
  collect information, used to draw conclusions
  about the whole
• Sampling frame
• The list of individuals from which we choose the
  sample
• Variable
• A characteristic of a unit, to be measured for
  those units in the sample

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A Statistics MetaphorThink Ice Cream

• All Ice Cream Flavors is the population
• The menu is the sampling frame
• The individual is one flavor of ice cream
• The sample is the flavors that are selected

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Terminology

• Parameter
• A number that describes the population. It is a
  fixed number, but we do not know what it is.
• Statistic
• A number that describes the sample. The value of
  a statistic is known only after we have taken the
  sample. We use the statistic to estimate the
  parameter.

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Sampling Variability

• Random Samples eliminate bias, but they can still
  be wrong
• Sampling Variability
• If we take many different samples from the same
  population, a statistic will be different for
  every different sample.
• If the variation is too great, we cannot trust
  the results of any one sample.

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Advantages of SRS

• Choosing at random prevents favoritism
• If we take lots of random samples, the variation
  follows a predictable pattern
• Larger samples are less variable than small ones.

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Statistics Lingo

• Bias
• Consistent, repeated deviation from the sample
  statistic from the population parameter in the
  same direction
• Variability
• In repeated sampling, describes how spread out
  the values of the statistic are
• Large variability means that the result of
  sampling is not repeatable

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Bias and Variability
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Proportions

• Proportions are percentages or rates

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Variability and Sampling

• As you increase the sample size, you decrease
  your variability.
• When you decrease variability, you get statistics
  that are more often the parameter.
• Your sampling distribution becomes more narrow,
  and it is easier to see what the parameter is.

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Sampling Variability
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Why?

• The statistic that occurs the most will be the
  population parameter because
• When we take a SRS, the sample is representative
  of the population.
• Taking many more samples will help us to realize
  that there is some differences between samples.
• If a statistic keeps showing up, then the actual
  population must be able to be described by that
  statistic.

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What To Do

• To Reduce Bias
• Use Random Sampling
• To Reduce Variability
• Use a larger sample
• Large random samples usually give an estimate
  that is close to the truth

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Homework

• Problems 2 5
• Read Chapter 3

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What Do Samples Tell Us?

• MA 155
• Chapter 3
• Part II

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Key Concepts So Far

• Know the difference between a statistic and a
  parameter
• Statistic can estimate a parameter
• Different samples can have different statistics
  (sample variability)
• Sample size decreases variability
• Large random samples almost always give an
  estimate that is close to the truth.

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Americans Still Prefer Boys

• If Americans could only have one child
• 42 say they would prefer a boy
• 27 say they would prefer a girl
• 25 say it would not matter to them either way
• The results are based on telephone interviews
  with a randomly selected national sample of 1,026
  adults, 18 years and older, conducted December
  2-4, 2000.

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Americans Still Prefer Boys

• For results based on this sample, one can say
  with 95 percent confidence that the maximum error
  attributable to sampling and other random effects
  is plus or minus 3 percentage points.

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What Does It Mean?

• If we took many samples using the same method we
  used to get this one sample, 95 of the samples
  would give a result within plus or minus 3
  percentage points of the truth about the
  population.

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Now What?

• We do not want to take 1000 SRS, so how can we be
  sure that the statistic we get in one sample is a
  good one?
• The statistic we get may be exactly the
  parameter, close to the parameter, or way off.
• With each statistic, we can compute a confidence
  interval.
• The interval we calculate may or may not hold the
  parameter.

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Confidence Statements

• Two Parts
• Level of Confidence
• Margin of Error
• Margin of Error
• How close the statistic is to the parameter
• Level of Confidence
• The percent of all possible samples that satisfy
  the margin of error

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Increase the Confidence

• If you want to increase the amount of confidence,
  you must increase the size of the interval.
• To increase the size of the interval, increase
  the margin of error.
• We can only increase it so far, we cannot have
  100 confidence.
• Think Soccer Goals!

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Example

• Linda is working for the EPA and is studying
  chemicals in groundwater. She is 90 sure that
  the percentage of harmful toxins in the water is
  5 1. This is well below the state minimum.
• Are her results good enough for you?
• What does this statement actually mean?
• What would happen if she did another sample?

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Example

• Linda is working for the EPA and is studying
  chemicals in groundwater. She is 90 sure that
  the percentage of harmful toxins in the water is
  5 1. This is well below the state minimum.
• Are her results good enough for you?
• What does this statement actually mean?
• What would happen if she did another sample?

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Example

• Aaron is flipping a coin and wonders whether or
  not it is fair. He flips the coin 100 times and
  it lands on heads 62 times. He is 90 confident
  that the proportion of times that he will get
  heads is 62 8.
• What does 90 confident mean?
• Is the coin fair?
• What would happen if he did this experiment
  again?

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Wrap Up

• The conclusion of a confidence statement always
  applies to the population, not the sample.
• Our conclusion about the population is never
  completely certain
• If we want to be more confident, we must accept a
  larger margin of error
• It is usual to report the margin of error for 95
  confidence
• If we want a smaller margin of error with the
  same confidence, we must take a larger sample.

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Size Really Doesnt Matter

• The population size doesnt matter
• The variability of a statistic from a random
  sample does not depend on the size of the
  population, as long as the population is much
  larger than the sample.

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Homework

• Do Problems 11 15
• Read Chapter 4