Summary of Confidence Intervals for Estimating  and p

Basic Form of a Confidence Interval (estimate) + (table value)(SE of the estimate) MARGIN OF ERROR (E) Important facts to know: The margin of error, and hence the width of the interval, gets smaller the as the sample size increases. The margin of error, and hence the width of the interval, increases and decreases with the confidence.

CI for the Population Mean (  If we assume the population we are sampling is approximately normal or if our sample size is “sufficiently large” the confidence interval for the population mean is given by: “95% Confidence” means that 95% of all possible samples that could be drawn from the population will produce an interval that covers the true population mean (  ).

Finding the t-Quantile Multiplier The t-distribution has degrees of freedom = n – 1, i.e. the sample size (n) minus 1. If n is “small” you can use the t-table from the Assignment 4 resource links. If n is such that the table cannot be used you can use the t-Quantile Calculator in JMP.

t-Quantile Calculator Enter the df = n -1 in this column. You don’t need to do anything else. Here I have found the t-quantile multiplier for a 95% CI for  based on a sample of size n = 10, for a 99% CI based on sample of size n = 20, and for a 90% CI based on a sample of size n = 25. Check these using t- table!

Find 95% CI for the mean APACHE score for patients with lung cancer in RHC Study From these data 1 st ) Find t-quantile for 95% confidence with df = = 38 2 nd ) Find CI for 

Interpretation We estimate that the mean APACHE score for critically ill patients admitted to the ICU that had lung cancer as their primary disease category is between and with 95% confidence. We estimate that the mean APACHE score for critically ill patients admitted to the ICU that had lung cancer as their primary disease category is between and with 95% confidence.

JMP gives CI’s for  95% CI for mean APACHE score for lung cancer patients (29.102, )

CI for the Population Proportion (p) If our sample size is sufficiently large * the confidence interval for the population proportion ( p ) is given by: * Sufficiently large conditions require knowledge of true proportion (p) nonetheless we generally require: np > 10 and n(1-p) > 10 For 95% Normal quantile z = 1.96 For 90% Normal quantile z = For 99% Normal quantile z = 2.576

Find a 95% for 30-day Mortality for Right Heart Catheter Patients Results: n = 2,184 patients had a Swan-Ganz line put in during their treatment, of these 830 died within 30-days of admission to the ICU.

Interpretation We estimate that the 30-day mortality rate for patients that had a Swan-Ganz line used during their treatment is between 36% and 40 % with 95% confidence. We estimate that the 30-day mortality rate for patients that had a Swan-Ganz line used during their treatment is between 36% and 40 % with 95% confidence.