10'2: Estimating a Population Mean no

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10.2 Estimating a Population Mean (no )
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• You are estimating the standard deviation, but
  there will likely be some error involved because
  you are estimating it from sample data.
• Resulting distribution t distribution
• Specify a particular t distribution by degrees of
  freedom
• Notation t(k)

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T distribution features

• Similar in shape to the standard Normal curve
• The spread is a bit greater than the standard
  Normal curve
• As degrees of freedom (k) increases, the t(k)
  density curve approaches the N(0,1) curve more
  closely

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• Got Milk?
• A milk processor monitors the number of bacteria
  per milliliter in raw milk received for
  processing. A random sample of 10 one-milliliter
  specimens from milk supplied by one producer
  gives the following data
• 4890 5100 4500 5260 5150 4900 4760 4700 4870
• Find a 90 CI for the mean number of bacteria per
  milliliter in all the milk from this supplier.

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When degrees of freedom are hard to find

• When the actual df does not appear in Table C,
  use the greatest df available that is less than
  your desired df.
• Rounding down in Table C will widen the interval
  which will safely contain the exact interval.

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• Recall
• ? Matched pairs are a form of block design in
  which just two treatments are compared.
• ?Subjects are matched in pairs and each
  treatment is given to one subject in each pair
• ?Alternatively, each subject receives both
  treatments in some order.
• Comparative studies are more convincing than
  single-sample investigations

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Ex. 10.10, p. 651

• Our subjects are 11 people diagnosed as being
  dependent on caffeine. Each subject was barred
  from caffeine instead, they took capsules
  containing their normal caffeine intake. During a
  different time period, they took placebo
  capsules. The order in which subjects took
  caffeine and the placebo was randomized.
• Depression is the score on the Beck Depression
  Inventory. Higher scores higher depression. We
  are interested in whether or not caffeine
  deprivation affects these outcomes.
• Construct a 90 confidence interval for the mean
  change in depression score.

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• Matched pairs problem (same individual received
  both treatments)
• Study looking to provide evidence that
  withholding caffeine from caffeine-dependent
  individuals may lead to depression

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Random selection vs. Random assignment

• Experiments are rarely done on randomly selected
  subjects the purpose is often to compare 2
  treatments rather than to generalize to a larger
  population.
• Random selection allows us to generalize
• Random assignment allows us to compare
  treatments

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• National Fuelsaver Corporation manufactures the
  Platinum Gasaver, a device they claim may
  increase gas mileage by 22. Here are the
  percent changes in gas mileage for 15 identical
  vehicles, as presented in one of the companys
  advertisements
• 48.3 46.9 46.8 44.6 40.2 38.5 34.6
  33.7 28.7 28.7 24.8 10.8 10.4 6.9
  12.4
• 1. Construct and interpret a 90 confidence
  interval to estimate the mean fuel savings in the
  population of all such vehicles. Follow the
  Inference Toolbox.
• 2. Explain what 90 confidence means in this
  setting.
• 3. Comment on the manufacturers claim based on
  your work in Question 1.
•