confidence intervals confidence interval for the mean (CI) = an interval within which the true mean of the population is believed to be located confidence intervals • statisticians mostly use 95% CI and 99% CI • 95% covers middle 95% of the distribution - uses Z = +/- 1.96 confidence intervals • 99% confidence interval covers middle 99% of the distribution - uses Z = +/-2.58 calculating CI with a known SD CI = Z( + Z( ) ) = margin of error = E CI = + E 2 steps to get CI 1. get the lower end of CI CI = -Z( ) 2. get the upper end of CI CI = + Z ( ) example A sample of 81 people took an IQ test. The mean of the IQ for this sample was 102. IQ scores in the general population are known to have a standard deviation of 15. 1. Estimate the mean of the population with a 95% confidence interval. 2. Show the CI on a graph. example A sample of 81 people took an IQ test. The mean of the IQ for this sample was 102. IQ scores in the general population are known to have a standard deviation of 15. 1. Estimate the mean of the population with a 99% confidence interval. 2. Show the CI on a graph. confidence, precision and interval width A sample of 100 people took an IQ test. The mean of the IQ for this sample was 102. At a CI of 95%, find: a) standard error of the mean b) the confidence interval confidence, precision and interval width Now calculate the 95% CI for samples of different sizes with different means. sample A: n = 81, mean = 99 sample B: n = 100, mean = 103 What is the width of each CI? (the difference between the upper and lower limits) calculating CI with an unknown SD CI = ± t (SE) SE = estimated standard error t = t value estimating the standard error (SE) SE = s/n e.g. A survey of a sample of 25 college students reveals that they spend, on average, $5.50 on lunch with a standard deviation of s=$1.50. What is the estimated standard error? t values and t distributions • t value = a point along the baseline of a sampling distribution • t value depends on size of the sample and width of the confidence interval • t distributions are theoretical probability distributions - similar to SNC, but shape is determined by degrees of freedom (df) t values and t distributions • degrees of freedom = n-1