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Mean

Mean. Variance. Size. n. Sample. N. Population. IME 301. b = is a random value = is probability. means For example:. means. Then from standard normal table: b = 1.96. Also: For example. IME 301. Point estimator and Unbiased estimator

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Mean

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  1. Mean Variance Size n Sample N Population IME 301

  2. b = is a random value = is probability means For example: means Then from standard normal table: b = 1.96 Also: For example IME 301

  3. 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% =<CL=< 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) IME 301

  4. 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: IME 301 (new Oct 06)

  5. Interval length is: • Half-width length is: • Critical Points are: and IME 301

  6. 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 IME 301 new Oct 06

  7. 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

  8. 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” IME 301

  9. FACT H0 is trueH0 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 IME 301

  10. 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)

  11. Test of hypothesis on mean, two-sided No information on population distribution Test statistic: Reject H0 if or P-value = IME 301

  12. Test of hypothesis on mean, one-sided No information on population distribution IME 301

  13. Test of hypothesis on mean, two-sided, variance known population is normal or conditions for central limit theoremholds Test statistic: Reject H0 if or, p-value = IME 301

  14. Test of hypothesis on mean, one-sided, variance known population is normal or conditions for central limit theorem holds IME 301 and 312

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