MeanVariance Sample Population Size n N IME 301. b = is a random value = is probability means For...
Transcript of MeanVariance Sample Population Size n N IME 301. b = is a random value = is probability means For...
Mean Variance
Sample
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IME 301
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b = is a random value = is probability
means
For example:
IME 301
Also:
For example
means
Then from standard normal table: b = 1.96
• Point estimator and Unbiased estimator
• Confidence Interval (CI) for an unknown parameteris an interval that contains a set of plausible values of the parameter. It is associated with a Confidence Level (usually 90% =<CL=< 99%) , which measuresthe probability that the confidence interval actuallycontains 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)
IME 301
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Confidence Interval for Population MeanTwo-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:
IME 301 (new Oct 06)
X
• Interval length is:
• Half-width length is:
• Critical Points are:
and
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IME 301
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Confidence Interval for Population MeanOne-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
IME 301 new Oct 06
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Hypothesis: Statement about a parameter
Hypothesis 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 > or <
IME 301
Hypothesis Testing Process:
1. Read statement of the problem carefully (*)
2. Decide on “hypothesis statement”, that is H0 and HA (**)
3. Check for situations such as: normal distribution, central limit theorem, variance known/unknown, …
4. Usually significance level is given (or confidence level)5. Calculate “test statistics” such as: Z0, t0 , ….
6. Calculate “critical limits” such as: 7. Compare “test statistics” with “critical limit”8. Conclude “accept or reject H0”
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IME 301
IME 301
FACTH0 is true H0 is false
Accept no error Type II H0 error
DecisionReject 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
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 < 0.01 generally leads to rejection of H0
P-value > 0.1 generally leads to acceptance of H0
0.01 < P-value < 0.1 need to have significance level to make a decision
IME 301 (new Oct 06)
Test of hypothesis on mean, two-sidedNo information on population distribution
Test statistic:
Reject H0 if
or P-value =
00 : H
01 : H
IME 301
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Test of hypothesis on mean, one-sidedNo information on population distribution
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IME 301
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Test statistic:
Reject Ho if P-value =
OR
Reject H0 if
Test of hypothesis on mean, two-sided, variance knownpopulation is normal or conditions for central limit theorem holds
Test statistic:
Reject H0 if or, p-value =
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IME 301
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Test of hypothesis on mean, one-sided, variance knownpopulation is normal or conditions for central limit theorem holds
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IME 301 and 312
Test statistic:
Reject Ho if P-value =
Or,
Reject H0 if
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Sample size, for mean with known variance
Where
Two-sided
One-sided
IME 301, Feb. 99