Tests about a Population Mean

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Tests about a Population Mean
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Population Mean (s known)

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

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Case I A Normal Population With Known
Null hypothesis
Test statistic value
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Case I A Normal Population With Known
Alternative Hypothesis
Rejection Region for Level Test
or
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Recommended Steps in Hypothesis-Testing Analysis

• Identify the parameter of interest and describe
  it in the context of the problem situation.
• Determine the null value and state the null
  hypothesis.
• State the alternative hypothesis.

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Hypothesis-Testing Analysis

• Give the formula for the computed value of the
  test statistic.
• State the rejection region for the selected
  significance level
• Compute any necessary sample quantities,
  substitute into the formula for the test
  statistic value, and compute that value.

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Hypothesis-Testing Analysis
7. Decide whether H0 should be rejected and
state this conclusion in the problem context.
The formulation of hypotheses (steps 2 and 3)
should be done before examining the data.
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Problem 1

• A car manufacturer advertises that its newest
  model, the Bullet, visits the petrol station
  infrequently. It claims its EPA rating for
  highway driving is at least 32.5 mpg, s5.3 mpg.
  Results of an independent study of 50 identical
  models was an average of 30.4 mpg. What would you
  conclude at a.01?

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Solution - Problem 1
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Type II Probability for a Level Test
Type II Probability
Alt. Hypothesis
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Sample Size
The sample size n for which a level test also
has at the alternative value
is
one-tailed test
two-tailed test
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Case II Large-Sample Tests
When the sample size is large, the z tests for
Case I are modified to yield valid test
procedures without requiring either a normal
population distribution or a known
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Large Sample Tests (n gt 40)
For large n, s is close to
Test Statistic
The use of rejection regions for Case I results
in a test procedure for which the significance
level is approximately
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Problem 2

• 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 2
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Case III A Normal Population Distribution
If X1,,Xn is a random sample from a normal
distribution, the standardized variable
has a t distribution with n 1 degrees of
freedom.
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The One-Sample t Test
Null hypothesis
Test statistic value
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The One-Sample t Test
Alternative Hypothesis
Rejection Region for Level Test
or
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Problem 3

• An auditing firm was hired to determine if a
  particular defense contractor was overstating the
  value of their inventory items. It was decided
  that 15 items would be randomly selected. For
  each item, the recorded amount, the audited
  amount and the difference were recorded. If it
  could demonstrate that the average difference
  exceeds 25 the contractor can be subject to
  penalties. The recorded differences in dollars
  were

17, 35, 31, 22, 50, 42, 56, 23, 27, 38, 20, 25,
43, 45, 21
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Solution - Problem 3
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P - Value
The P-value is the smallest level of significance
at which H0 would be rejected when a specified
test procedure is used on a given data set.
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P - Value
The P-value is the probability, calculated
assuming H0 is true, of obtaining a test
statistic value at least as contradictory to H0
as the value that actually resulted. The smaller
the P-value, the more contradictory is the data
to H0.
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P-Values for a z Test
P-value
upper-tailed test
lower-tailed test
two-tailed test
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P-Value (area)
Upper-Tailed
z
0
Lower-Tailed
-z
0
Two-Tailed
z
-z
0
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PValues for t Tests
The P-value for a t test will be a t curve area.
The number of df for the one-sample t test is n
1.