Chapter 9 One and TwoSample Estimation Problems

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Chapter 9 One- and Two-Sample Estimation Problems

• Wen-Hsiang Lu (???)
• Department of Computer Science and Information
  Engineering,
• National Cheng Kung University
• 2004/05/23

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9.2 Statistical Inference

• Statistical inference
• Classical method inferences are based strictly
  on information obtained from a random sample
  selected from the population.
• Bayesian method utilizes prior subjective
  knowledge about the probability distribution of
  the unknown parameters in conjunction with the
  information provided by the sample data. (Section
  9.13)
• Most of this chapter use classical methods.
• Statistical inference may be divided into two
  major areas
• Estimation
• Tests of hypotheses

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9.3 Classical Methods of Estimation

• A point estimate of some population parameter ?
  is a single value of a statistic
• The value of the statistic , is a point
  estimate of the population parameter ?.
• is a point estimate of the true
  proportion p for a binomial experiment.
• Definition 9.1 A statistic is said to be an
  unbiased estimator of the parameter ? if

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Classical Methods of Estimation

• Example 9.1 Show that S2 is an unbiased
  estimator of the parameter ?2.
• This example illustrates why we divide by n-1
  rather than n when the variance is estimated.

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Classical Methods of Estimation

• Definition 9.2 If we consider all possible
  unbiased estimators of some parameter ?, the one
  with the smallest variance is called the most
  efficient estimator of ?.

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Interval Estimation

• Unlikely to estimate the population parameter
  exactly.
• Accuracy increases with large samples.
• In many situations, preferable to determine an
  interval within which we would expect to find the
  value of the parameter.
• Such an interval is called an interval estimate.
• As the sample size increases, we know
  thatdecreases, and consequently our estimate is
  likely to be closer to the parameter ?, resulting
  in a shorter interval.
• An interval estimate might be more informative.

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Interpretation of Interval Estimation

• We have a probability of 1 - ? of selecting a
  random sample that will produce an interval
  containing ?.
• The interval is called a (1 -
  ?)100 confidence interval.
• 1 - ? is called confidence degree.
• are called the lower and upper
  confidence limits.
• We prefer a short interval with a high degree of
  confidence.

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9.4 Single Sample Estimating the Mean

• According to the central limit theorem, we can
  expect the sampling distribution of to be
  approximately normally distributed with mean
  and standard deviation
• Confidence interval of ?

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Single Sample Estimating the Mean

• Different samples will yield different values of
  and therefore produce different interval
  estimates of the parameter ?.

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Single Sample Estimating the Mean

• Ex9.2 The average zinc concentration recovered
  from a sample of zinc measurements in 36
  different locations is found to be 2.6 grams per
  milliliter. Find the 95 and 99 confidence
  intervals for the mean zinc concentration in the
  river. Assume that the population standard
  deviation is 0.3.
• Solution

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Single Sample Estimating the Mean

• Theorem 9.1 If is used as an estimate of ?,
  we can then be (1 - ?)100 confident that the
  error will not exceed
• Theorem 9.2 If is used as an estimate of ?,
  we can then be (1 - ?)100 confident that the
  error will not exceed a specified amount e when
  the sample size is
• Example 9.3 How large a sample is required in
  Example 9.2 if we want to be 95 confident that
  our estimate of ? is off by less than 0.05?
• Solution

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Estimating the Mean with ? Unknown

• If we have a random sample from a normal
  distribution, then the random variable T has a
  students t-distribution with n 1 degrees of
  freedom.
• Confidence interval of ?

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Estimating the Mean with ? Unknown
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Large-Sample Confidence Interval

• Large-sample confidence interval when normality
  cannot be assumed, ? is unknown, and n ? 30, s
  can replace ? and the confidence interval may
  be used.
• s will be very close to the true ? and thus the
  central limit theorem prevails.

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9.5 Standard Error of a Point Estimate

• Width of confidence intervals become shorter as
  the quality of the corresponding point estimate
  becomes better.

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9.6 Prediction Interval

• Sometimes, some experimenters may also be
  interested in predicting the possible value of a
  future observation.
• Some customers may require a statement regarding
  the uncertainty of one single observation.
• The type of requirement is nicely fulfilled by
  the construction of a prediction interval.
• Assume a natural point estimator of a new
  observation is , and the variance of
  ?2/n
• The development of a prediction interval is
  displayed by beginning with a normal random
  variable x0 -

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Prediction Interval

• For a normal distribution of measurements with
  unknown mean ? and known variance ?2, a (1 -
  ?)100 prediction interval of a future
  observation, x0, iswhere z?/2 is the z-value
  leaving an area of ?/2 to the right.

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Prediction Interval

• Example 9.5 Due to the decreasing of interest
  rates, the First Citizens Bank received a lot of
  mortgage applications. A recent sample of 50
  mortgage loans resulted in an average of
  128,300. Assume a population standard deviation
  of 15,000. If a next customer called in for a
  mortgage loan application, find a 95 prediction
  interval on this customers loan amount.
• Prediction interval provides a good estimate of
  the location of a future observation.
• The estimation of future observation is quite
  different from the estimation of sample mean.

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Prediction Interval

• For a normal distribution of measurements with
  unknown mean ? and unknown variance ?2, a (1 -
  ?)100 prediction interval of a future
  observation, x0, iswhere t?/2 is the t-value
  with v n -1 degree-of-freedom, leaving an area
  of ?/2 to the right.

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Prediction Interval

• Example 9.6 A meat inspector has randomly
  measured 30 packs of acclaimed 95 lean beef. The
  sample resulted in the mean 96.2 with the sample
  standard deviation of 0.8. Find a 99 prediction
  interval for a new pack. Assume normality.
• An observation is an outlier if it falls outside
  the prediction interval computed without
  inclusion of the questionable observation in the
  sample.

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Tolerance Limits

• Tolerance limits For a normal distribution of
  measurements with unknown mean ? and unknown
  variance ?2, tolerance limits are given by
  where k is determined so that one can assert
  with 100(1 - ?) confidence that the given
  limits contain at least the proportion (1 - ?) of
  the measurements.
• Example 9.7 A machine is producing metal pieces
  that are cylindrical in shape. A sample of these
  pieces is taken and the diameters are found to be
  1.01, 0.97, 1.03, 1.04, 0.99, 0.98, 0.99, 1.01,
  and 1.03 centimeters. Find the 99 tolerance
  limits that will contain 95 of the metal pieces
  produced by this machine, assuming an approximate
  normal distribution.
• Solution

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Distinction Among Confidence Intervals,
Prediction Intervals, and Tolerance Intervals

• Confidence intervals population mean
• Tolerance limits a tolerance interval must
  necessarily be longer than a confidence interval
  with the same degree of confidence.
• Prediction limits determine a bound of a future
  observation value.

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9.8 Two Samples Estimating the Difference
Between Two Means

• Two populations (?1, ?1), (?2, ?2)
• Confidence interval

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Two Samples Estimating the Difference Between
Two Means

• Example 9.8 An experiment was conducted in which
  two types of engines, A and B, were compared. Gas
  mileage in miles per gallon was measured.
• Fifty experiments were conducted using engine
  type A and 75 experiments were done for engine
  type B.
• The average gas mileage for engine A was 36 miles
  per gallon and the average for machine B was 42
  miles per gallon.
• Find a 96 confidence interval on ?B - ?A , where
  ?B and ?A are population mean gas mileage for
  machines B and A, respectively.
• Assume that the population standard deviations
  are 6 and 8 for machines A and B, respectively.
• Solution

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Two Samples Estimating the Difference Between
Two Means

• Example 9.8 An experiment was conducted in which
  two types of engines, A and B, were compared. Gas
  mileage in miles per gallon was measured.
• Fifty experiments were conducted using engine
  type A and 75 experiments were done for engine
  type B.
• The average gas mileage for engine A was 36 miles
  per gallon and the average for machine B was 42
  miles per gallon.
• Find a 96 confidence interval on ?B - ?A , where
  ?B and ?A are population mean gas mileage for
  machines B and A, respectively.
• Assume that the population standard deviations
  are 6 and 8 for machines A and B, respectively.
• Solution

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Two Samples Estimating the Difference Between
Two Means with Unknown ?

• Variance unknown If ?12 and ?22 are unknown, but
  ?12 ?22 ?2, we obtain

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Two Samples Estimating the Difference Between
Two Means with Unknown ?

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Two Samples Estimating the Difference Between
Two Means with Unknown ?

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Two Samples Estimating the Difference Between
Two Means

• Example 9.9 Two independent sampling stations
  were chosen for the study of acid mine pollution.
• For 12 monthly samples collected at the
  downstream station the species diversity index
  had a mean value 3.11 and a standard
  deviation s1 0.771, while 10 monthly samples
  collected at the upstream station the species
  diversity index had a mean value 2.04 and
  a standard deviation s2 0.448.
• Find a 90 confidence interval for the difference
  between the population means for the two
  locations, assuming that the population are
  approximately normally distributed with equal
  variances.
• Solution

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Two Samples Estimating the Difference Between
Two Means with Unequal Variances

• Unequal Variances

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Two Samples Estimating the Difference Between
Two Means with Unequal Variances

• Example 9.10 Zinc is measured in milligrams per
  liter. 15 samples were collected from station 1
  had an average zinc content of 3.84 milligrams
  per liter and a standard deviation of 3.07
  milligrams per liter, while the 12 samples from
  station 2 had an average zinc content of 1.49
  milligrams per liter and a standard deviation of
  0.80 milligrams per liter. Find a 95 confidence
  interval for the difference in the true average
  zinc contents at theses two stations, assuming
  that the observations came from normal population
  with different variance.
• Solution

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9.9 Paired Observations

• Consider that the samples are not independent and
  the variances of the two populations are not
  necessarily equal.
• If and sd are the mean and standard deviation
  of the normally distributed differences of n
  random pairs of measurements, a (1-a)100
  confidence interval for ?D ?1 - ?2 iswhere
  t?/2 is the t-value with v n -1 degrees of
  freedom, leaving an area of ?/2 to the right.
• Example 9.11 For a study of dioxin, find a 95
  confidence interval for ?1 - ?2, where ?1 and ?2
  represent the true mean TCDD in plasma and in fat
  tissue, respectively. Assume the distribution of
  the differences to be approximately normal.

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Paired Observations

• Solution

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9.10 Single Sample Estimating a Proportion

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Single Sample Estimating a Proportion

• If is the proportion of successes in a random
  sample of size n, and an
  approximate (1-?)100 confidence interval for the
  binomial parameter p is given by iswhere z?/2
  is the z-value with leaving an area of ?/2 to the
  right.

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Single Sample Estimating a Proportion

• If is the proportion of successes in a random
  sample of size n, and an
  approximate (1-?)100 confidence interval for the
  binomial parameter p is given by iswhere z?/2
  is the z-value leaving an area of ?/2 to the
  right.

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Single Sample Estimating a Proportion

• Ex 9.12 In a random of n 500 families owning
  television sets in the city of Hamilton, Canada,
  it is found that x 340 subscribed to HBO. Find
  a 95 confidence interval for the actual
  proportion of families in the city who subscribe
  to HBO.
• Solution

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Single Sample Estimating a Proportion

• Theorem 9.3 If is used as an estimate of p,
  we can be (1 - ?)100 confident that the error
  will not exceed
• Theorem 9.4 If is used as an estimate of p,
  we can be (1 - ?)100 confident that the error
  will be less than a specified amount e when the
  sample size is approximately

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Single Sample Estimating a Proportion

• Example 9.13 How large a sample is required in
  Example 9.12 if we want to be 95 confident that
  our estimate of p is within 0.02?
• Solution

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Single Sample Estimating a Proportion

• Theorem 9.5 If is used as an estimate of p,
  we can be at least (1 - ?)100 confident that the
  error will not exceed a specified amount e when
  the sample size is approximately
• Example 9.14 How large a sample is required in
  Example 9.12 if we want to be at least 95
  confident that our estimate of p is within 0.02?
• Solution

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9.11 Two Samples Estimating the Difference
Between Two Proportions

• p1 might be the proportion of smokers with lung
  cancer and p2 the proportion of non-smokers with
  lung cancer.

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Two Samples Estimating the Difference Between
Two Proportions

• Large-sample confidence interval for p1 - p2If
  are the proportion of successes in
  a random sample of size n1 and n2,
  , an approximate (1-?)100
  confidence interval for the difference of two
  binomial parameters p1 - p2 is given by where
  z?/2 is the z-value leaving an area of ?/2 to the
  right.

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Two Samples Estimating the Difference Between
Two Proportions

• Example 9.15 A certain change in a process for
  manufacture of component parts is being
  considered.
• Sample are taken using both the existing and new
  procedure so as to determine if the new process
  results in an improvement.
• If 75 of 1500 items from the existing procedure
  were found to be defective and 80 of 2000 items
  from the new procedure were found to be
  defective.
• Find a 90 confidence interval for the true
  difference in the fraction of defectives between
  the existing and the new process.
• Solution