Mean

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Mean
Variance
Size
n
Sample
N
Population
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b is a random value is probability
means For example
means
Then from standard normal table b 1.96
Also For example
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• Point estimator and Unbiased estimator
• Confidence Interval (CI) for an unknown parameter
• is an interval that contains a set of plausible
  values of
• the parameter. It is associated with a
  Confidence
• Level (usually 90 ltCLlt 99) , which measures
• the probability that the confidence interval
  actually
• contains the unknown parameter value.
• CI half width, half width
• An example of half width is
• CI length increases as the CL increases.
• CI length decreases as sample size, n, increases.
• Significance level ( 1 CL)

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Confidence Interval for Population
Mean Two-sided, t-Interval Assume a sample of
size n is collected. Then sample mean, ,and
sample standard deviation, S, is
calculated. The confidence interval is
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• Interval length is
• Half-width length is
• Critical Points are
• and

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Confidence Interval for Population
Mean One-sided, t-Interval Assume a sample of
size n is collected. Then sample mean, ,and
sample standard deviation, S, is
calculated. The confidence interval is
OR
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Hypothesis Statement about a parameter Hypothesi
s testing decision making
procedure about the hypothesis Null hypothesis
the main hypothesis H0 Alternative hypothesis
not H0 , H1 , HA Two-sided alternative
hypothesis, uses One-sided alternative
hypothesis, uses gt or lt
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• Hypothesis Testing Process
• Read statement of the problem carefully ()
• Decide on hypothesis statement, that is H0 and
  HA ()
• Check for situations such as
• normal distribution, central limit theorem,
• variance known/unknown,
• Usually significance level is given (or
  confidence level)
• Calculate test statistics such as Z0, t0 , .
• Calculate critical limits such as
• Compare test statistics with critical limit
• Conclude accept or reject H0

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FACT H0 is true H0 is false Accept
no error Type II H0
error Decision Reject Type I
no error H0 error Prob(Type I
error) significance level P(reject H0
H0 is true) Prob(Type II error)
P(accept H0 H0 is false) (1 -
) power of the test
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• The P-value is the smallest level of significance
  that would lead to rejection of the null
  hypothesis.
• The application of P-values for decision making
• Use test-statistics from hypothesis testing to
  find P-value. Compare level of significance with
  P-value.
• P-value lt 0.01 generally leads to rejection of
  H0
• P-value gt 0.1 generally leads to acceptance of
  H0
• 0.01 lt P-value lt 0.1 need to have significance
  level to make a decision

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Test of hypothesis on mean, two-sided No
information on population distribution Test
statistic Reject H0 if
or P-value
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Test of hypothesis on mean, one-sided No
information on population distribution


Test statistic
Reject Ho if P-value
OR Reject H0 if
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Test of hypothesis on mean, two-sided, variance
known population is normal or conditions for
central limit theorem holds Test statistic
Reject H0 if
or, p-value
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Test of hypothesis on mean, one-sided, variance
known population is normal or conditions for
central limit theorem holds


Test statistic
Reject Ho if P-value
Or, Reject H0 if
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