1. MeanVariance Sample Population Size n N IME 301

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

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 H 0 Alternative hypothesis : not H 0, H 1, H A Two-sided alternative hypothesis, uses One-sided alternative hypothesis, uses > or < IME 301

8. Hypothesis Testing Process: 1.Read statement of the problem carefully ( * ) 2.Decide on “hypothesis statement”, that is H 0 and H A ( ** ) 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: Z 0, t 0, …. 6.Calculate “critical limits” such as: 7.Compare “test statistics” with “critical limit” 8. Conclude “accept or reject H0” IME 301

9. FACT H 0 is true H 0 is false Accept no errorType II H 0 error Decision Reject Type I no error H 0 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

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 H 0 P-value > 0.1 generally leads to acceptance of H 0 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 H 0 if or P-value = IME 301

12. Test of hypothesis on mean, one-sided No information on population distribution IME 301 Test statistic: Reject Ho if P-value = OR Reject H0 if

13. Test of hypothesis on mean, two-sided, variance known population is normal or conditions for central limit theorem holds Test statistic: Reject H 0 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 Test statistic: Reject Ho if P-value = Or, Reject H0 if