Section 6.3: How to See the Future

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Section 6.3 How to See the Future

• Goal To understand how sample means vary in
  repeated samples.

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How do sample means vary?

• Heres a graph of the survival time of 72 guinea
  pigs, each injected with a drug.
• Note The mean and std. dev. for the population
  are

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Understanding the POPULATION Data

• The survival times for the population of guinea
  pigs is skewed to the right.
• The mean, , is 109.2
• The standard deviation, , is large!

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Distribution of Sample Means

• Suppose that lots and lots of us each randomly
  sampled 12 guinea pigs and found the average
  survival time for each set of 12 guinea pigs. We
  would each have a sample mean, .
• Would all of our values be the same?
• Use yesterdays simulation data (consisting of
  106 sample means) to see if there are any
  patterns.

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Graph of 106 sample means
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Graph of the Sample Means

• What is the mean of the sample means,
  ?
• What is the standard deviation of the sample
  means, ?
• What is the shape of the graph of the sample
  means?

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Graph of the Sample Means

• What is the mean of the sample means,
  ? 142.01
• What is the standard deviation of the sample
  means, ? 27.94
• What is the shape of the graph of the sample
  means? Roughly bell-shaped

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Comparing Population Data with Sample Mean Data

• What does this say in
  words?
• What does this say in
  words?
• The graph of the population data is skewed but
  the graph of the sample mean data is bell-shaped.

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Standard Error

• Since standard deviation of the sample means is
  a mouthful, well instead call this quantity
  standard error.
• Remember, we have two standard deviations
  floating around the first is the population
  standard deviation and the second is the standard
  error.
• The first describes how much spread there is in
  the population. The second describes how much
  spread there is in the sample means.

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Understanding Standard Error

• How does the population standard deviation relate
  to the standard error?
• What does this formula say?
• When the sample size is large, the standard error
  is small.
• When the sample size is small, the standard error
  is large.
• When n1, the two values are equal. Why?

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Ex 2 Coin Problem

• Imagine you go home, collect all of the coins in
  your home, and make a graph of the age of each
  coin.
• This graph represents the graph of the population
  data.
• What do you expect its shape to be?

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Sample means of coins

• Take repeated samples, each of size 5 coins, and
  find the mean age of the coins. If you were to
  make a graph of the sample means, what would you
  expect it to look like?
• How about if instead you took samples of size 10?
  Or of size 25?

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Guinea Pigs and Coins

• In both situations the population graphs were
  severely skewed, yet the graph of the sample mean
  data was bell-shaped.
• In both cases the graphs of the sample mean data
  is centered at the population mean.
• In both cases the standard error is the
  population standard deviation divided by the
  square root of the sample size.
• Hmmmmm..

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Coincidence?

• No, we couldnt be that lucky! In fact, this is
  the Central Limit Theorem in action.

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Central Limit Theorem

• Suppose that a random sample of size n is taken
  from a large population in which the variable you
  are measuring has mean and standard deviation
  .
• Then, provided n is at least 30, the sampling
  distribution of the sample means is roughly
  bell-shaped, centered at the population mean,
  , with standard error equal to .

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Since the graph of sample means will always be
bell-shaped.

• 68 of the sample means should come within one
  standard error of the center (population mean).
• 95 of the sample means should come within two
  standard errors of the center (population mean).
• 99.7 of the sample means should come within
  three standard errors of the center (population
  mean).

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Confidence Intervals to Estimate

• What is the average number of hours Ship students
  sleep per night during final exam week?
• Who is the population?
• What is the parameter we are interested in
  estimating?

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Confidence Intervals, Contd

• Estimate the population mean by taking a random
  sample of 50 college students and finding a
  sample mean.
• Suppose you find that the sample mean is 5.8

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Confidence Intervals, contd.

• Can we say that is 5.8?
• If the sample was indeed random is it reasonable
  to believe is close to 5.8?
• Reason? Central Limit Theorem.

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• So the 95 confidence interval to estimate
  is
• What is the formula for a 68 confidence
  interval?
• How about a 99.7 confidence interval?