MARIO F' TRIOLA

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STATISTICS
ELEMENTARY
Section 7-4 Testing a Claim about a Mean
Small Samples
MARIO F. TRIOLA
EIGHTH
EDITION
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Assumptions

• for testing claims about population means
• The sample is a simple random sample.
• The sample is small (n lt 30).
• The value of the population standard deviation ?
  is unknown.
• The sample values come from a population with a
  distribution that is approximately normal.

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Test Statistic for a Student t-distribution
x -µx
t
s
n

• Degrees of freedom (df) n -1

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Important Properties of the Student t
Distribution

• 1. The Student t distribution is different for
  different sample sizes (see Figure 6-5 in Section
  6-3).
• 2. The Student t distribution has the same
  general bell shape as the normal distribution
  its wider shape reflects the greater variability
  that is expected with small samples.
• 3. The Student t distribution has a mean of t 0
  (just as the standard normal distribution has a
  mean of z 0).
• 4. The standard deviation of the Student t
  distribution varies with the sample size and is
  greater than 1 (unlike the standard normal
  distribution, which has a ??? 1).
• 5. As the sample size n gets larger, the Student
  t distribution get closer to the normal
  distribution. For values of n gt 30, the
  differences are so small that we can use the
  critical z values instead of developing a much
  larger table of critical t values.

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Choosing between the Normal and Student
t-Distributionswhen Testing a Claim about a
Population Mean µ
Start
Use NORMAL distribution
Yes
n gt 30?
Use s if ? is unknown.
No
Is ? known ?
Use NORMAL distribution Extremely unusual!
Is ? known?
Population normally distributed?
Yes
Yes
No
No
Use nonparametric methods (not covered in this
course)
Use STUDENT t distribution
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EXAMPLE Chicken FeedUsing regular feed, a
poultry farmers newborn chickens have normally
distributed weights with a mean of 62.5 oz. With
enriched feed, the weights (in ounces) shown
below were observed 61.4 62.2 66.9 63.3 66.2 66.
0 63.1 63.7 66.6 Use a .05 significance level to
test the claim that the mean weight is higher
with the enriched feed.
5.

• Claim µ gt 62.5

P .013

• H0 µ 62.5
• H1 µ gt 62.5

? x
µ 62.5
P .013

• a 0.05

6. P lt .05 so reject H0
7. There is sufficient evidence to support the
claim that the mean weight is higher than 62.5
oz. with the enriched chicken feed.

• t distribution because n 30

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The larger Student t critical value shows that
with a small sample, the sample evidence must be
more extreme before we consider the difference is
significant.
EXAMPLE Given µ0 2, s 1, x 2.3, µ ? µ0
_
Use a Z-test to find the P-value with n 50.
Use a t-test to find the P-value with n 20.
P 0.0339
P 0.1955