Chapter 9: One- and Two- Sample Estimation

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Chapter 9 One- and Two- Sample Estimation

• Statistical Inference
• Estimation
• Tests of hypotheses
• Even the most efficient unbiased estimator is
  unlikely to estimate the population parameter
  exactly. (Walpole et al, pg. 233)
• Interval estimation (1 a)100 confidence
  interval for the unknown parameter.
• Example if a 0.01, we develop a _______
  confidence interval.

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Single Sample Estimating the Mean

• Given
• s is known and X is the mean of a random sample
  of size n,
• Then,
• the (1 a)100 confidence interval for µ is
  given by

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Example

• A traffic engineer is concerned about the delays
  at an intersection near a local school. The
  intersection is equipped with a fully actuated
  (demand) traffic light and there have been
  complaints that traffic on the main street is
  subject to unacceptable delays.
• To develop a benchmark, the traffic engineer
  randomly samples 25 stop times (in seconds) on a
  weekend day. The average of these times is found
  to be 13.2 seconds, and the variance is know to
  be 4 seconds2.
• Based on this data, what is the 95 confidence
  interval (C.I.) around the mean stop time during
  a weekend day?

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Example (cont.)

• X ______________ s _______________
• a ________________ a/2 _____________
• Z0.025 _____________ Z0.975 ____________
• __________________ lt µ lt ___________________

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Your turn

• What is the 90 C.I.? What does it mean?

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What if s2 is unknown?

• For example, what if the traffic engineer
  doesnt know the variance of this population?
• If n is sufficiently large (gt _______), then the
  large sample confidence interval is
• Otherwise, must use the t-statistic

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Single Sample Estimating the Mean(s unknown, n
not large)

• Given
• s is unknown and X is the mean of a random sample
  of size n (where n is not large),
• Then,
• the (1 a)100 confidence interval for µ is
  given by

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Recall Our Example

• A traffic engineer is concerned about the delays
  at an intersection near a local school. The
  intersection is equipped with a fully actuated
  (demand) traffic light and there have been
  complaints that traffic on the main street is
  subject to unacceptable delays.
• To develop a benchmark, the traffic engineer
  randomly samples 25 stop times (in seconds) on a
  weekend day. The average of these times is found
  to be 13.2 seconds, and the sample variance, s2,
  is found to be 4 seconds2.
• Based on this data, what is the 95 confidence
  interval (C.I.) around the mean stop time during
  a weekend day?

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Example (cont.)

• X ______________ s _______________
• a ________________ a/2 _____________
• t0.025,24 _____________
• __________________ lt µ lt ___________________

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Your turn

• A thermodynamics professor gave a physics
  pretest to a random sample of 15 students who
  enrolled in his course at a large state
  university. The sample mean was found to be 59.81
  and the sample standard deviation was 4.94.
• Find a 99 confidence interval for the mean on
  this pretest.

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Solution

• X ______________ s _______________
• a ________________ a/2 _____________
• (draw the picture)
• T___ , ____ _____________
• __________________ lt µ lt ___________________

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Standard Error of a Point Estimate

• Case 1 s known
• The standard deviation, or standard error of X is
• Case 2 s unknown, sampling from a normal
  distribution
• The standard deviation, or (usually) estimated
  standard error of X is
• ______

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9.6 Prediction Interval

• For a normal distribution of unknown mean µ, and
  standard deviation s, a 100(1-a) prediction
  interval of a future observation, x0 is
• if s is known, and
• if s is unknown

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9.7 Tolerance Limits

• For a normal distribution of unknown mean µ, and
  unknown standard deviation s, tolerance limits
  are given by
• x ks
• where k is determined so that one can assert
  with 100(1-?) confidence that the given limits
  contain at least the proportion 1-a of the
  measurements.
• Table A.7 gives values of k for (1-a) 0.9,
  0.95, 0.99 ? 0.05, 0.01 and for selected
  values of n.

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Summary

• Confidence interval ? population mean µ
• Prediction interval ? a new observation x0
• Tolerance interval ? a (1-a) proportion of
• the measurements can be estimated
  with a 100(1-?) confidence

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Estimating the Difference Between Two Means

• Given two independent random samples, a point
  estimate the difference between µ1 and µ2 is
  given by the statistic
• We can build a confidence interval for µ1 - µ2
  (given s12 and s22 known) as follows

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An example

• Look at example 9.8, pg. 248

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Differences Between Two Means Variances Unknown

• Case 1 s12 and s22 unknown but equal
• Where,

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Differences Between Two Means Variances Unknown

• Case 2 s12 and s22 unknown and not equal
• Where,

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Estimating µ1 µ2

• Example (s12 and s22 known)
• A farm equipment manufacturer wants to compare
  the average daily downtime of two sheet-metal
  stamping machines located in two different
  factories. Investigation of company records for
  100 randomly selected days on each of the two
  machines gave the following results
• x1 12 minutes x2 10 minutes
• s12 12 s22 8
• n1 n2 100
• Construct a 95 C.I. for µ1 µ2

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Solution
Picture

• a/2 _____________
• z_____ ____________
• __________________ lt µ1 µ2 lt
  _________________
• Interpretation

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µ1 µ2 si2 Unknown

• Example (s12 and s22 unknown but equal)
• Suppose the farm equipment manufacturer was
  unable to gather data for 100 days. Using the
  data they were able to gather, they would still
  like to compare the downtime for the two
  machines. The data they gathered is as follows
• x1 12 minutes x2 10 minutes
• s12 12 s22 8
• n1 18 n2 14
• Construct a 95 C.I. for µ1 µ2

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Solution
Picture

• a/2 _____________
• t____ , ________ ____________
• __________________ lt µ1 µ2 lt
  _________________
• Interpretation

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Paired Observations

• Suppose we are evaluating observations that are
  not independent
• For example, suppose a teacher wants to compare
  results of a pretest and posttest administered to
  the same group of students.
• Paired-observation or Paired-sample test
• Example murder rates in two consecutive years
  for several US cities (see attached.) Construct a
  90 confidence interval around the difference in
  consecutive years.

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Solution
Picture

• D ____________
• ta/2, n-1 _____________
• a (1-a)100 CI for µ1 µ2 is
• __________________ lt µ1 µ2 lt
  _________________
• Interpretation

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C.I. for Proportions

• The proportion, P, in a binomial experiment may
  be estimated by
• where X is the number of successes in n trials.
• For a sample, the point estimate of the parameter
  is
• The mean for the sample proportion is
• and the sample variance

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C.I. for Proportions

• An approximate (1-a)100 confidence interval for
  p is
• Large-sample C.I. for p1 p2 is
• Interpretation _______________________________

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Example 9.15

1. C.I. (-0.0017, 0.0217), therefore no reason to
   believe there is a significant decrease in the
   proportion defectives using the new process.
2. What if the interval were (0.0017, 0.0217)?
3. What if the interval were (-0.9, -0.7)?