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MTHE 224

Week 5: Confidence Intervals and Hypothesis Testing Goals: • Study confidence intervals based on a normal population distribution • Study large-sample confidence intervals • Study Hypothesis Testing Suggested Textbook Readings: Chapter 7: 7.1 - 7.2, Chapter 8: 8.1, 8.2 Practice Problems: Section 7.1: 1, 3, 5, 7 Section 7.2: 13, 15, 17 Section 8.1: 7, 9, 11 Section 8.2: 1 Week 5: Confidence Intervals and Hypothesis Testing 2 What is a Confidence Interval? It is important not to think of a confidence interval as the probability the true mean lies inside X ± 1.96 √n is 0.95. The correct interpretation is more elusive. Confidence Interval Meaning: the long run proportion of random samples of size n from a normal (µ,σ) population that have true mean µ inside the calculated confidence interval is 0.95. Definition: The expression √σ in the CI expression is called the standard error (syn- onomous with the standard deviation of the sample mean of a random sample of size n) We can consider other level of confidence. Similar to 95% CI, other level of confidence, for example 100(1 − α) CI, can be achieved by replacing 1.96 with the appropriate standard normal critical value α/2, that is ▯ ▯ σ σ ¯ − zα/2 ·√ n,¯ + zα/2 ·√ n Example 1: Calculate a 90% confidence interval for the true mean of a normal popu- lation with parameters σ =6 .0, n = 34, and ¯ = 80. Choice of Sample Size Example 2: (Example 7.4, page 259) Extensive monitoring of a computer time-sharing system has suggested that response time to a particular editing command is normally dis- tributed with standard deviation 25 millisec. A new operating system has been installed, and we wish to estimate the true average response time µ for the new environment. As- suming that response times are still normally distributed with σ = 25, what sample size is nessary to ensure that the resulting 95% confidence interval has a width of (at most) 10? In general, the sample size required to get an interval width w satisfies the condition that σ w =2 · z α/2√ n MTHE 224 Fall 2012 Week 5: Confidence Intervals and Hypothesis Testing 3 Large Sample Confidence Intervals To generalize this theory: 1. Because of the Central Limit theorem, if the sample size n is large enough, we know that the sample mean X (when calculated from a random sample) has an approximate normal distribution. 2 2. When n is large, we also expect S to approximate V [X] very well. (Where X has the distribution of the random sample.) Proposition: If n is sufficiently large (n> 40), the rv ¯ Z = X −√µ S/ n has approximately a standard normal distribution, which implies that s ¯ ± zα/2 ·√ n is a 100(1 − α) confidence interval for µ. Example 3: (Example 7.6 and 7.7, page 264 and 265) The sample observations on the alternating current (AC) breakdown voltage of an insulating liquid are given. Summary quantities include n = 48, ▯ x = 2626, and ▯ x = 144,950. i i (a) Find the sample mean and the sample standard deviation. (b) Find the 95% confidence interval for µ. (c) Suppose the investigator believes that virtually all values in the population are between 40 and 70. Estimate the sample size for a 95% CI with width 2. MTHE 224 Fall 2012 Week 5: Confidence Intervals and Hypothesis Testing 4 We may creat a normal plot of the data to check normality. X=xlsread(’example7-6.xlsx’); normplot(X); One-Sided Confidence Intervals (Confidence Bounds) A one-sided confidence bound results from replacing z α/2by z α In all cases the confi- dence level is approximately 100(1 − α)%. A large-sample upper confidence bound for µ is µ< x¯ + zα· √s n and a large-sample lower confidence bound for µ is µ> x¯ − z · √s α n MTHE 2
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