STATISTICAL INFERENCE:

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STATISTICAL INFERENCE Making decisions or
drawing conclusions about a population.
Suppose we have collected a representative sample
that gives some information concerning a mean or
other statistical quantity.
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One question on the basis of this information
would be are the sample quantity and a
corresponding population quantity close enough
together so that it is reasonable to say that the
sample might have come from the population? Or
are they far enough apart so that they likely
represent different populations?
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Statistical inference may be divided into two
major areas parameter estimation hypothesis
testing
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Test of Hypothesis We are testing the hypothesis
that a sample is similar enough to a particular
population so that it might have come from that
population. If the hypothesis is true, we say
that the sample is consistent with the population
and all disagreement between the sample and the
population is due to random variation.
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Suppose that we are interested in the burning
rate of a solid fuel. Burning rate is a random
variable that can be described by a probability
distribution. We are interested in the mean
burning rate. To be specific, we are interested
in deciding whether or not the mean burning rate
is 70 cm/s. We may express this formally as
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IMPORTANT
Hypotheses are always statements about the
population or distribution under study, not
statements about the sample.
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The value of the population parameter specified
in the null hypothesis (mean of 70 cm/s in our
example) is usually determined in one of the
three ways.

• Past experience or knowledge of the process

2. Theory or model regarding the process

• Engineering specifications or contractual
  obligations

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Objective of Hypothesis Testing
(Parameter from Past experience or knowledge of
the process) To determine whether the parameter
value has changed
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Objective of Hypothesis Testing
(Parameter from Theory or model regarding the
process) To verify the theory or model
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Objective of Hypothesis Testing
(Parameter from Engineering specifications or
contractual obligations) To test conformance or
compliance
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A procedure leading to a decision about a
particular hypothesis is called a test of a
hypothesis. Hypothesis testing procedures rely
on using the information in a random sample from
the population of interest. If this information
is consistent with the hypothesis, then we will
conclude that the hypothesis is true. However,
if this information is inconsistent with the
hypothesis, we will conclude that the hypothesis
is false.
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The truth or falsity of a particular hypothesis
can never be known with certainty unless we can
examine the entire population. This is usually
impossible in most practical situations.
Therefore, a hypothesis testing procedure should
be developed with the probability of reaching a
wrong conclusion in mind.
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Test of Hypothesis
Consider the burning rate of the solid fuel. The
null hypothesis is that the mean burning rate is
70 cm/s, and the alternative hypothesis is that
it is not equal to 70 cm/s. We wish to test the
following
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The sample mean is an estimate of the true
population mean m.
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On the other hand, a sample mean that is
considerably different from 70 cm/s is evidence
in support of the alternative hypothesis H1.
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The sample mean can take on many different
values. Suppose that if
we will not reject the null hypothesis
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But if
or
we will reject the null hypothesis in favour of
the alternative hypothesis
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Decision criteria
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The boundaries between the critical regions and
acceptance region are called the critical values.
In our example the critical values are 68.5 and
71.5.
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The decision procedure can lead to either of two
wrong conclusions.
The true mean burning rate of the solid fuel
could be equal to 70 cm/s. However, the value of
the sample mean observed from the randomly
selected specimens may fall into the critical
region. We would then reject the null hypothesis
H0 in favour of the alternative H1 when in fact
H0 is really true. This type of wrong conclusion
is called a type I error.
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The true mean burning rate of the solid fuel
could be different from 70 cm/s. However, the
value of the sample mean observed from the
randomly selected specimens may fall into the
acceptance region. We would then fail to reject
the null hypothesis H0 when in fact H0 is false.
This type of wrong conclusion is called a type II
error.
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Definitions
Rejecting the null hypothesis H0 when it is true
is defined as a type I error.
Failing to reject the null hypothesis H0 when it
is false is defined as a type II error.
The probability of making a type I error is
called the significance level or size of the test.
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Example Consider the burning rate of the solid
fuel. The true mean burning rate is 70 cm/s and
the standard deviation of the burning rate is s
2.42 cm/s.
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What percentage of all random samples would lead
to the rejection of the null hypothesis when the
true mean burning rate is really 70 cm/s?
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Critical Region
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Standard deviation of the sample mean
a P(type I error)
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Critical Region
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The z-values that correspond to the critical
values 68.5 and 71.5 are
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Therefore,
5 of all samples would lead to the rejection of
the null hypothesis.
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CONFIDENCE INTERVAL
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If the population mean is m and the normal
distribution applies, the probability that a
random sample mean will fall by chance in the
region between z1 -1.96 and z2 1.96 is 95.
This is called 95 confidence interval.
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When a manufacturing process is operating
properly, the mean length of a certain part is
known to be 6.175 inches, and lengths are
normally distributed. The standard deviation of
this length is 0.008 inches. If a sample
consisting of 6 items taken from current
population has a mean length of 6.168 inches, is
there evidence at the 5 level of significance
that some adjustment of the process is required?
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At the 5 level of significance, the critical
regions are
and
Standard deviation of the sample mean
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At the 5 level of significance z -2.12 falls
in the critical (rejection) region. Therefore,
some adjustment of the process is required.
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Carbon composition resistors with a mean
resistance of 560 W and a standard deviation of
51 W are produced by a factory. They are sampled
each hour in the quality control laboratory.
What sample size would be required so that there
is 95 probability (confidence) that the mean
resistance of the sample lies within 10 W of 560
W if the population mean has not changed.
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For a 95 confidence interval, a random sample
mean will fall in the region between z1 -1.96
and z2 1.96 if the manufacturing process has
not changed.
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A sample size of 100 would be required.