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Lecture 14 Dustin Lueker x Z / 2 s n This interval will contain μ with a 100(1-α)% confidence ◦ If we are estimating µ, then why it is unreasonable for us to know σ? Thus we replace σ by s (sample standard deviation) This formula is used for large sample size (n≥30) If we have a sample size less than 30 a different distribution is used, the t-distribution, we will get to this later STA 291 Summer 2008 Lecture 14 2 Incorrect statement ◦ With 95% probability, the population mean will fall in the interval from 3.5 to 5.2 To avoid the misleading word “probability” we say ◦ We are 95% confident that the true population mean will fall between 3.5 and 5.2 STA 291 Summer 2008 Lecture 14 3 Changing our confidence level will change our confidence interval ◦ Increasing our confidence level will increase the length of the confidence interval A confidence level of 100% would require a confidence interval of infinite length Not informative There is a tradeoff between length and accuracy ◦ Ideally we would like a short interval with high accuracy (high confidence level) STA 291 Summer 2008 Lecture 14 4 The width of a confidence interval ◦ ◦ ◦ ◦ Increases Increases Increases Increases as as as as the the the the confidence level increases error probability decreases standard error increases sample size n decreases Why are each of these true? STA 291 Summer 2008 Lecture 14 5 Start with the confidence interval formula assuming that the population standard deviation is known x Z / 2 n xE Mathematically we need to solve the above equation for n 2 Z / 2 n E 2 STA 291 Summer 2008 Lecture 14 6 About how large a sample would have been adequate if we merely needed to estimate the mean to within 0.5, with 95% confidence? Assume 5 Note: We will always round the sample size up to ensure that we get within the desired error bound. STA 291 Summer 2008 Lecture 14 7 To account for the extra variability of using a sample size of less than 30 the student’s tdistribution is used instead of the normal distribution x t / 2 s n STA 291 Summer 2008 Lecture 14 8 t-distributions are bell-shaped and symmetric around zero The smaller the degrees of freedom the more spread out the distribution is t-distribution look much like normal distributions In face, the limit of the t-distribution is a normal distribution as n gets larger STA 291 Summer 2008 Lecture 14 9 Need to know α and degrees of freedom (df) ◦ df = n-1 α=.05, n=23 ◦ tα/2= α=.01, n=17 ◦ tα/2= α=.1, n=20 ◦ tα/2= STA 291 Summer 2008 Lecture 14 10 A sample of 12 individuals yields a mean of 5.4 and a variance of 16. Estimate the population mean with 98% confidence. STA 291 Summer 2008 Lecture 14 11