STA 291 Spring 2008 Lecture 14 Dustin Lueker

Confidence Interval To calculate the confidence interval, we use the Central Limit Theorem Use this formula if σ is known Also, we need a that is determined by the confidence level Formula for 100(1-α)% confidence interval for μ STA 291 Spring 2008 Lecture 14

Confidence Interval To find a confidence interval for μ with σ unknown and n≥30 use the following formula If the sample size is smaller than 30 we will use a different distribution from the normal STA 291 Spring 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 Spring 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 Spring 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 Decreases as the sample size n increases STA 291 Spring 2008 Lecture 14

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

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

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

Example A sample of 12 individuals yields a mean of 5.4 and a variance of 16. Construct a 98% confidence interval for the population mean. STA 291 Spring 2008 Lecture 14