Inferential statistics comprises the use of statistics to make inferences concerning some unknown as

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Inferential statistics comprises the use of
statistics to make inferences concerning some
unknown aspect of a population. A statistical
hypothesis test is a method of making statistical
decisions using data. These decisions are almost
always made using null-hypothesis tests that is,
ones that answer the question Assuming that the
null hypothesis is true, what is the probability
of observing a value for the test statistic that
is at least as extreme as the value that was
actually observed? One use of hypothesis testing
is deciding whether observed results contain
enough information to cast doubt on conventional
wisdom.
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• H0 is the null hypothesis (conventional wisdom)
• The type I error rate is specified by the
  experimenter (always known)
• The type II error rate is determined by which
  alternative hypothesis is true, the process
  variability, sample size, and confidence level
  (never known unless one assumes they know the
  truth)

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An analogy between hypothesis testing and the
American legal system
At the start of the trial, the jury must assume
that the accused is innocent (H0) The prosecutor
tries to prove the accused is guilty
(H1) Evidence is presented (the data) The jury
deliberates (compute the test statistic) The
accused is acquitted (fail to reject H0) or
convicted (reject H1) Guilty person goes to jail
(power) Innocent person is acquitted
(confidence) Guilty person goes free (type II
error) Innocent person goes to jail (type I error)
Note An analogy is not an equivalence
relationship. Statistical inference is not based
on the American legal system. Visit your local
library if you dont understand what an analogy
is.
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• Two approaches
• Critical value approach (typically done by hand
  with tables)
• P-value approach (typically done using computer
  software)

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• Critical value approach
• State null and alternative hypothesis
• Determine distribution of test statistic if null
  hypothesis is true
• Determine critical value(s)
• State rejection rule
• Compute value of test statistic from data
• Make decision
• State conclusion

• P-value approach
• State null and alternative hypothesis
• Determine distribution of test statistic if null
  hypothesis is true
• State rejection rule
• Compute value of test statistic from data
• Compute p-value
• Make decision
• State conclusion

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Null hypothesis H0 Parameter some
number Alternative hypothesis H1 Parameter
the number listed above
Examples
Test if the mean is greater than 75
Test if two means are different from one another
Test if a proportion is less than 0.3
Test if proportion 1 is bigger than proportion 2
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Distribution (depends on assumptions made and
parameters being tested) Zobs N(0,1) if H0 is
true tobs tv if H0 is true X2obs c2v if
H0 is true (chi-square distribution later in
course) Fobs Fa,b if H0 is true
(F-distribution later in course)
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Suppose the confidence level is selected by the
researcher to be 95 and that the test statistic
is normally distributed.
Right tailed alternative
Critical values z0.95 1.64 z0.05
-1.64 z0.025 -1.96 and z0.975 1.96
Left tailed alternative
Two tailed alternative
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Suppose the confidence level is selected by the
researcher to be 95 and that the test statistic
is normally distributed.
Critical value approach
Reject H0 if Zobs lt -1.64
Reject H0 if Zobs gt 1.64
Reject H0 if Zobs lt -1.96 or Zobs gt 1.96
P-value approach Reject H0 if the p-value lt 0.05
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Some of the test statistics well compute.
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• Decision
• Reject H0
• Fail to reject H0

• Conclusion
• State level of confidence (or type I error rate
  or level of significance)
• State the alternative hypothesis using the
  context of the problem
• State decision sufficient evidence (reject H0)
  or insufficient evidence (fail to reject H0)

Example With 98 confidence there is sufficient
evidence to conclude that the ammonium sulfate
level exceeds 150ppm. 0.02 Type I error rate H1
m gt 150 Reject H0