Statistics 270 - Lecture 22

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Statistics 270 - Lecture 22
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• Last Daycompleted 5.1
• Today Parts of Section 5.3 and 5.4

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Example

• Government regulations indicate that the total
  weight of cargo in a certain kind of airplane
  cannot exceed 330 kg. On a particular day a plane
  is loaded with 81 boxes of a particular item
  only. Historically, the weight distribution for
  the individual boxes of this variety has a mean
  3.2 kg and standard deviation 1.0 kg.
• What is the distribution of the sample mean
  weight for the boxes?
• What is the probability that the observed sample
  mean is larger than 3.33 kg?

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• Statistical Inference deals with drawing
  conclusions about population parameters from
  sample data
• Estimation of parameters
• Estimate a single value for the parameter (point
  estimate)
• Estimate a plausible range of values for the
  parameter (confidence intervals)
• Testing hypothesis
• Procedure for testing whether or not the data
  support a theory or hypothesis

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Point Estimation

• Objective to estimate a population parameter
  based on the sample data
• Point estimator is a statistic which estimates
  the population parameter

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• Suppose have a random sample of size n from a
  normal population
• What is the distribution of the sample mean?
• If the sampling procedure is repeated many times,
  what proportion of sample means lie in the
  interval
• If the sampling procedure is repeated many times,
  what proportion of sample means lie in the
  interval

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• In general, 100(1-a) of sample means fall in the
  interval
• Therefore, before sampling the probability of
  getting a sample mean in this interval is

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• Could write this as
• Or, re-writingwe get

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• The interval below is called a
  confidence interval for
• Key features
• Population distribution is assumed to be normal
• Population standard deviation, s, is known

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Example

• To assess the accuracy of a laboratory scale, a
  standard weight known to be 10 grams is weighed 5
  times
• The reading are normally distributed with unknown
  mean and a standard deviation of 0.0002 grams
• Mean result is 10.0023 grams
• Find a 90 confidence interval for the mean

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Interpretation

• What exactly is the confidence interval telling
  us?
• Consider the interval in the previous example.
  What is the probability that the population mean
  is in that particular interval?
• Consider the interval in the previous example.
  What is the probability that the sample mean is
  in that particular interval?

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Large Sample Confidence Interval for m

• Situation
• Have a random sample of size n (large)
• Suppose value of the standard deviation is known
• Value of population mean is unknown

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• If n is large, distribution of sample mean is
• Can use this result to get an approximate
  confidence interval for the population mean
• When n is large, an approximate
  confidence interval for the mean is

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Example

• Amount of fat was measured for a random sample of
  35 hamburgers of a particular restaurant chain
• It is known from previous studies that the
  standard deviation of the fat content is 3.8
  grams
• Sample mean was found to be 30.2
• Find a 95 confidence interval for the mean fat
  content of hamburgers for this chain

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Changing the Length of a Confidence Interval

• Can shorten the length of a confidence interval
  by
• Using a difference confidence level
• Increasing the sample size
• Reducing population standard deviation

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Sample Size for a Desired Width

• Frequent question is how large a sample should I
  take?
• Well, it depends
• One to answer this is to construct a confidence
  interval for a desired width

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Sample Size for a Desired Width

• Width (need to specify confidence level)
• Sample size for the desired width

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Example

• Limnologists wishes to estimate the mean
  phosphate content per unit volume of a lake water
• It is known from previous studies that the
  standard deviation is fairly stable at around 4
  ppm and that the observations are normally
  distributed
• How many samples must be sampled to be 95
  confidence of being within .8 ppm of the true
  value?

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Example

• A plant scientist wishes to know the average
  nitrogen uptake of a vegetable crop
• A pilot study showed that the standard deviation
  of the update is about 120 ppm
• She wishes to be 90 confident of knowing the
  true mean within 20 ppm
• What is the required sample size?