27 October 2003

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27 October 2003

• 6.1 Estimating with Confidence

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Sampling

• We have a known population.
• We ask what would happen if I drew lots and lots
  of random samples from this population?

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Inference

• We have a known sample.
• We ask what kind of population might this sample
  have been drawn from?

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Looking ahead

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The Central Limit Theorem

• If you draw simple random samples of size n
• from a population with mean m and variance s2
• then
• the expected mean of x-bar is m
• the expected variance of x-bar is s2 / n
• the expected histogram of x-bar is approximately
  normal

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Estimating mu from sample data

• estimated mu sample mean
• Why?
• Because the Central Limit Theorem tells us that,
  if we drew lots and lots of sample, the sample
  means would average out to mu.
• (The sample mean is an unbiased estimator of mu.)

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Estimating mu from sample data

• Is this true?
• mu sample mean
• Why not?
• Because the Central Limit Theorem tells us that,
  if we drew lots and lots of sample, the sample
  means vary. Some are bigger than mu and others
  are smaller than mu.

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Estimating mu from sample data

• What abou this?
• mu somewhere in the neighborhood
• of the sample mean
• But how do we define neighborhood?

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Example 6.1

• We have a sample of 500 high-school seniors,
  selected at random from the population of all
  high-school seniors in California. For the 500
  kids in the sample, their average score on the
  math section of the SAT is 461.
• Known sample mean is 461
• Unknown population mean
• Assumed population sigma is 100

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The Central Limit Theorem

• If you draw simple random samples of size 500
  from a population with mean m and standard
  deviation of 100, then
• the expected mean of x-bar is m
• the expected st dev of x-bar is about 4.5
• the expected histogram of x-bar is approximately
  normal

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Table A tells us...

• ...about 68 of sample means should fall within
  4.5 points of mu
• ...about 95 of sample means should fall within 9
  points of mu
• ...about 99.75 of sample means should fall
  within 13.5 points of mu

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About 95 of sample means should fall within 9
points of mu
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The 95 Confidence Interval

• If mu is any number less than 452, then our
  sample mean would be surprisingly large.
• If mu is any number greater than 470, then our
  sample mean would be surprisingly small.
• Therefore, the 95 confidence interval for mu is
  the range from 452 to 470.
• If mu is inside this range, then our sample is
  not unusual (according to the 95 rule).

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Other confidence intervals

• If we suppose that the sample mean is within
  1.645 standard deviations of mu, then we get a
  90 confidence interval.
• If we suppose that the sample mean is within
  2.576 standard deviations of mu, then we get a
  99 confidence interval.

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Effect of sample size on the confidence interval

• As n gets larger, the expected variability of the
  sample means gets smaller.
• Larger sample sizes produce narrower confidence
  intervals (other things equal).
• Smaller sample sizes produce wider confidence
  intervals (other things equal).

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Some cautions

• The data must be a simple random sample from the
  population
• The sample mean, and therefore the confidence
  interval, may be too heavily influenced by one or
  more outliers
• If the sample size is small and population is not
  approximately normal, then the CLT doesnt
  promise the approximately normal distribution for
  the sample means

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One more caution

• There is a 95 chance that mu lies in the
  confidence interval.
• In Example 6.1
• P(452 lt mu lt 470) .95

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One more caution

• There is a 95 chance that mu lies in the
  confidence interval.
• In Example 6.1
• P(452 lt mu lt 470) .95