JOURNAL ARTICLE Extra Credit

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JOURNAL ARTICLE Extra Credit

• Talk to a professor in a field that interests
  you.
• Choose a subject area, like business, nursing,
  geography, education, environment, psychology.
• Choose a scientific journal in that field.
• Choose an article in that journal which compares
  two populations or treatments using a t-test.
• Do not choose an article which simply discusses
  the theory of a t-test or how to use it.
• Choose an article which has sections Abstract,
  Introduction, Method, Results, Conclusions.
• Xerox the Title Page and also a page (or two)
  which show t-scores, degrees of freedom, and
  corresponding p-values based on data from a study
  or an experiment.
• Give me the Xeroxed pages by the class after Test
  3.

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Anything z can do, t can do better!
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3.2 Testing a Mean of One Normal Population
  EXAMPLE D A large group of bolts has just
been received, but before allowing them to be
used in manufacturing, the Quality Assurance
Department wants to see whether they appear to
meet the specifications of the order. These bolts
are to be an important part of a large aircraft,
so one specification is that they be no heavier
than 1.5 grams on the average. In this case, we
happen to know from experience that such
populations have nearly normal histograms. Suppose
a random sample of 16 bolts from the shipment
has an average weight of 1.55 grams and a
standard deviation of 0.12 gram. Certainly the
sample mean exceeds the specification, but at
what level is the evidence that the shipment
(population) mean is too heavy? Can we support
this hypothesis ("too heavy") at the .05 level?
Can we support it at the .10 level?
M155 L31 Testing a Mean of One Normal Population
-- Slide 1
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SOLUTION D The problem implies two hypotheses
about ?. RIGHT-SIDED H1 ? gt 1.5 H0 ? ?
1.5   Here is a summary of what we
know.   POPULATION SAMPLE !! normally
distributed n 16 (small) ? unknown
1.55 (data ) ? unknown s .12 The
sample event is , so the
supporting event is . Notice
that the given mean and standard deviation belong
to the sample!! And since the sample is small
(nlt30) from a normal population, we must treat
as being t-distributed.
M155 L31 Testing a Mean of One Normal Population
-- Slide 2
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3.2.1 Testing a Mean of One Population, Using the
t-table   Instead of we compute

.
Suppose H0 is true and ?1.5 . Then we can
place the 1.5 on the histogram and calculate the
probability of the supporting event .
We then compute the data t-score
M155 L31 Testing a Mean of One Normal Population
-- Slide 3
6
Figure 2
.058
_
X, n16
1.55
1.5
.05/.030
t, df15
1.67
0
7
M155 L31 Testing a Mean of One Normal Population
-- Slide 5
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To compute the tail probability for t1.67, we
enter the t-table T-6 at the 15
degree-of-freedom row and notice that 1.67 is
between 1.65 and 1.70 . Therefore, we must
interpolate between .060 (for t1.65) and .055
(for t1.7). Since 1.67 is 2/5 of the way (up)
from 1.65 to 1.70, we suppose the unknown
probability (italicized) is 2/5 of the way (down)
from .060 to .055
M155 L31 Testing a Mean of One Normal Population
-- Slide 5
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Since 1.67 is 2/5 of the way (up) from 1.65 to
1.70, we suppose the unknown probability
(italicized) is 2/5 of the way (down) from .060
to .055   .060 - (2/5)(.060-.055) .060 -
(2/5).005 .060 - .002 .058   Figure 8 shows
that   if ? 1.5, then (and thus if ? ?
1.5, then   CONCLUSION At the .058 level,
there is evidence that the average weight of all
the bolts in the shipment is more than 1.5 gram.
 
M155 L31 Testing a Mean of One Normal Population
-- Slide 6
10
SUMMARY OF HYPOTHESIS TEST PROCEDURES Write both
hypotheses first! Gather the values of n,
, s . Since n lt 30, verify that the
population is normal. Write the supporting
event Compute Draw a histogram of
with ?0 at the middle and to one side.
(DON'T mix these up!) Find the probability to
the side of , using the t-table T-6. This
is the observed level of significance ( ols
). CONCLUSION At the .. level, there is
evidence that .. REMEMBER If ols ? ?, the data
also support H at the ? level. NOTE The
larger t is, the smaller ols is, and the more
the evidence supports H1 .
M155 L31 Testing a Mean of One Normal Population
-- Slide 7
11
EXAMPLE E Suppose subjects in the students data
set were chosen randomly from all freshmen and
sophomores at the university, and we know that
Verbal SAT is normally distributed in the
population. At the 10 level is there evidence
that the mean Verbal SAT score for all freshmen
and sophomores is above 500?   SOLUTION E The
hypotheses are   RIGHT-SIDED H1 ? gt
500 H0 ? ? 500.   The population is normal, so
we can use the t statistic, and MINITAB will
compute the sample standard deviation s. Open
the students worksheet, choose Stat gt Basic
Statistics gt 1-Sample t, double-click on the
variable Verbal, and enter 500 for the Test mean.
Choose Options and select the alternative
hypothesis greater than.
M155 L31 Testing a Mean of One Normal Population
-- Slide 8
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M155 L31 Testing a Mean of One Normal Population
-- Slide 8
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Test of mu 500 vs mu gt 500   Variable
N Mean StDev SE Mean Verbal
10 547.0 93.6 29.6   Variable
95.0 Lower Bound T P Verbal
492.8 1.59 0.073   CONCLUSION
E At the .073 level, there is evidence that the
mean Verbal SAT of all freshmen and sophomores is
above 500. There is also evidence at the 10
level.
M155 L31 Testing a Mean of One Normal Population
-- Slide 8
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EXAMPLE F Suppose subjects in the students data
set were chosen randomly from all freshmen and
sophomores at the university, and we know that
Verbal SAT is normally distributed in the
population. At the 10 level is there evidence
that the mean Verbal SAT score for all freshmen
and sophomores is different from 500?
  SOLUTION F The hypotheses are   TWO-SIDED H
1 ? ? 500 H0 ? ? 500.   The population is
normal, so we can use the t statistic, and
MINITAB will compute the sample standard
deviation s. Open the students worksheet, choose
Stat gt Basic Statistics gt 1-Sample t,
double-click on the variable Verbal, and enter
500 for the Test mean. Choose Options and select
the alternative hypothesis not equal.
M155 L31 Testing a Mean of One Normal Population
-- Slide 8
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M155 L31 Testing a Mean of One Normal Population
-- Slide 8
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Test of mu 500 vs mu not 500   Variable
N Mean StDev SE Mean Verbal
10 547.0 93.6 29.6   Variable
95.0 CI T P Verbal
( 480.1, 613.9) 1.59 0.147   CONCLUSION
F At the .147 level, there is evidence that the
mean Verbal SAT of all freshmen and sophomores is
different from 500. There is not evidence at the
10 level. In exercises 7-9, assume each
variable is normally distributed in the
population, and write that on your paper. Write
the hypotheses and then answer the question with
MINITAB, giving an appropriate conclusion.
M155 L31 Testing a Mean of One Normal Population
-- Slide 8