# Each possible random sample provides a possible

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Each possible random sample provides a possible sample mean value. ! If the population is normal or the sample size is large (n # 30), then the sample mean will have a distribution that is (approximately) normal with mean μ and standard deviation . ! About 95% of the possible sample mean values with be in the interval ! For each of these sample mean values, the interval will contain the true population mean μ . ! Thus, about 95% of the intervals will contain the true population mean μ , in repeated samples of the same size n from the same population. σ / n μ ± 1 . 96 σ n ¯ x ± 1 . 96 σ n ¯ x ± 1 . 96 σ n 16

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Applet #14 of the CD that comes with the textbook ! If our confidence level is 95%, then in the long run, 95% of our confidence intervals will contain ! and 5% will not. 17 Try this by yourselves.
Note: about confidence intervals ! The population parameter is not a random quantity . It does not vary – once we have “looked” (computed) the actual interval, we cannot talk about probability or chance for the particular interval anymore. ! The 95% confidence level applies to the procedure , not to an individual interval; it applies “before you look” and not “after you look” at your data and compute your interval. 18

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100(1- ! )% confidence interval for μ For normal population with known σ : ¯ x ± z α / 2 ( σ n ) What if the population distribution is not normal but we know σ ? Based on CLT, if the sample size is large ( n 30) we can still construct the confidence interval by ¯ x ± z α / 2 ( σ n ) . What if σ is unknown? If the sample size n is large ( n 30), then s is a good estimate for σ . We will use ¯ x ± z α / 2 ( s n ) . What if σ is unknown, and the sample size is small? If we know the population is normal, we can use t-distribution to help us construct the confidence interval. 19
Interpretation of a CI for μ 20 ! We are 100(1- ! )% confident that the population mean μ lies somewhere between the lower and the upper bounds of the confidence interval. ! In repeated sampling, 100(1- ! )% of the resulting intervals will contain the population mean μ .

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Commonly used Confidence Levels 21 1- " " " /2 Z " /2 0.90 0.10 0.05 1.645 0.95 0.05 0.025 1.96 0.98 0.02 0.01 2.33 0.99 0.01 0.005 2.575
Try It! 22 ! Suppose that the amount of time teenagers spend working at part-time jobs is normally distributed with a standard deviation of 20 minutes. A random sample of 100 observations is drawn and the sample mean computed as 125 minutes. Determine the 95% confidence interval estimate of the population mean.

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• Spring '12
• wen-ya

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