Inferences Concerning Means

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Inferences Concerning Means
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Hypothesis Testing

• Hypothesis tests also address the uncertainty of
  the sample estimate. A hypothesis test attempts
  to refute a specific claim about a population
  parameter based on the sample data.
• For example, the hypothesis might be one of the
  following
• the population mean is equal to 10
• the population standard deviation is equal to 5
• the means from two populations are equal
• the standard deviations from 5 populations are
  equal

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• To reject a hypothesis is to conclude that it is
  false. However, to accept a hypothesis does not
  mean that it is true, only that we do not have
  evidence to believe otherwise.
• Thus hypothesis tests are usually stated in terms
  of both a condition that is doubted (null
  hypothesis) and a condition that is believed
  (alternative hypothesis).

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Terminology

• There are two kinds of errors that can be made in
  significance testing (1) a true null hypothesis
  can be incorrectly rejected and (2) a false null
  hypothesis can fail to be rejected.
• The former error is called a Type I error and the
  latter error is called a Type II error. The
  probability of a Type I error is designated by
  the Greek letter alpha (a).

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Confidence

• a establishes the risk you are willing to take in
  rejecting the null hypothesis when it is true.
  This value tell where your computed value must
  fall on the distribution.

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• It is important to distinguish between
  statistical significance and practical
  significance. Statistical significance simply
  means that we reject the null hypothesis.
• For example, for a particularly large sample,
  the test may reject the null hypothesis that two
  process means are equivalent. However, in
  practice the difference between the two means may
  be relatively small to the point of having no
  real engineering significance.
• Similarly, if the sample size is small, a
  difference that is large in engineering terms may
  not lead to rejection of the null hypothesis. The
  analyst should not just blindly apply the tests,
  but should combine engineering judgment with
  statistical analysis.

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Testing the Population Mean

• Suppose we wish to ascertain whether the value
  for the population mean is correct.
• We will take a sample from the population and
  compute the sample mean.
• We know the population variance.

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Problem 1

• The CEO submitted a white paper indicating a few
  changes in the software development process are
  in order. His statements include a claim that
  the average effort devoted to unit testing on
  projects is 7.8 person-months. You collect a
  random sample of 75 effort-logs from projects
  and determine the average effort for unit testing
  was 7.5 person-months with a standard deviation
  of 1.75 person-months. Does the data you
  collected support or refute the CEO?

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Solution - Problem 1
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Critical Values
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Problem 2

• A server vendor provides ratings from their high
  performance series. It claims the mean time
  between failure(MTBF) is at least 325 days. A
  recent independent study of 50 of these units
  resulted in an average MTBF of 304 days with a
  standard deviation of 50.3 days. You suspect the
  MTBF is less than the vendors claim.

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Solution - Problem 2
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Problem 3

• Recently many companies have been experimenting
  with telecommuting, allowing employees to work at
  home on their computers. Telecommuting is
  supposed to reduce the number of sick days taken.
  One firm, where the average is 5.4 sick days,
  introduces telecommuting. Management choose a
  random sample of 80 employees to follow. At the
  end of the year this group has an average of 4.5
  sick days with a standard deviation of 2.7 days.
  What conclusion can be drawn?

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Problem 4

• A consumer watchdog group suspects that a paper
  produced for laser printers that is advertised as
  98 acid free has higher acid content. The group
  can take action against the company if it can
  substantiate its suspicion with data. The group
  takes a sample of 25 reams of papers and
  determines that the mean acid content was 2.12
  with a standard deviation of .98. Is there
  sufficient evidence for the group to act?

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Inferences with Two Means

• Test the difference of means to determine whether
  alternative A is better than alternative B.
• x1 and x2 are sample means from large sample
  sizes n1 and n2 respectively taken from
  populations having means m1 and m2, and standard
  deviations s1 and s2.

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We will use s1 and s2 as estimators of s1 and s2,
if the standard deviations are unknown and
samples sizes are large.
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Problem 5

• A large national corporation is looking to
  outsource the hosting of its web services. There
  are two national ISPs that offer the levels of
  service they require. The corporations CIO
  conducts a study of ISPs pricing structures
  around the country so local ISPs would host
  services near corporate sites.

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Solution - Problem 5
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Small Sample Means
Assuming both populations are normal and they
have the same variance.
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Problem 6

• As part of employee training, some trainees are
  instructed in company processes using a
  computer-based training program (Method A), while
  others are instructed using mentors with weekly
  small group discussion sessions. At the end of
  the training session new employees are given an
  achievement test to assess their mastery of the
  material. Using the data from the next slide,
  can you determine if one method is better than
  the other?

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Scores

• Method A
• 71 75 65 69 73 66 68 71 74 68
• Method B
• 72 77 84 78 69 70 77 73 65 75

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Matched Pairs

• For some studies we will have a two treatment
  conditions which are applied to each experimental
  object. The research question is whether the two
  treatments are different.

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If the sample is large the difference need not be
normally Distributed. In which case you can you
a z-test.