STA 291 Summer 2008 Lecture 14 Dustin Lueker

Confidence Intervals 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

Interpreting Confidence Intervals 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

Confidence Interval 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

Facts about Confidence Intervals The width of a confidence interval Increases as the confidence level increases Increases as the error probability decreases Increases as the standard error increases Increases as the sample size n decreases Why are each of these true? STA 291 Summer 2008 Lecture 14

Choice of Sample Size Start with the confidence interval formula assuming that the population standard deviation is known Mathematically we need to solve the above equation for n STA 291 Summer 2008 Lecture 14

Example 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 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

Confidence Interval for Unknown σ To account for the extra variability of using a sample size of less than 30 the student’s t- distribution is used instead of the normal distribution STA 291 Summer 2008 Lecture 14

t-distribution 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

Finding tα/2 Need to know α and degrees of freedom (df) α=.05, n=23 STA 291 Summer 2008 Lecture 14

Example 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