Summary of Summary of Confidence Confidence IntervalsIntervals

for for

Estimating Estimating and and pp

Basic Form of a Confidence Basic Form of a Confidence IntervalInterval

((estimateestimate) ) ++ ( (table valuetable value)()(SE of the estimateSE 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 (CI for the Population Mean (

If we assume the population we are If we assume the population we are sampling is approximately normal or sampling is approximately normal or if our sample size is “sufficiently if our sample size is “sufficiently large” the confidence interval for the large” the confidence interval for the population mean is given by:population mean is given by:

nstX

XSEquantiletabletX

)() (

“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 Finding the t-Quantile MultiplierMultiplier

The t-distribution has degrees of freedom The t-distribution has degrees of freedom = n – 1, i.e. the sample size (n) minus 1.= n – 1, i.e. the sample size (n) minus 1.

If If nn is “small” you can use the t-table from is “small” you can use the t-table from the Assignment 4 resource links.the Assignment 4 resource links.

If If nn is such that the table cannot be used is such that the table cannot be used you can use the t-Quantile Calculator in you can use the t-Quantile Calculator in JMP.JMP.

t-Quantile Calculatort-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 Find 95% CI for the mean APACHE score for patients APACHE score for patients with lung cancer in RHC with lung cancer in RHC StudyStudyFrom these From these

datadata

39

06.16

31.34

n

s

X

1st) Find t-quantile for 95% confidence with df = 39 -1 = 38

2nd) Find CI for

)50.39 , 11.29(

19.531.34)57.2(02.231.34

3906.1602.231.34

nstX

InterpretationInterpretation

We estimate that the mean We estimate that the mean APACHE score for critically ill APACHE score for critically ill patients admitted to the ICU patients admitted to the ICU that had lung cancer as their that had lung cancer as their primary disease category is primary disease category is between 29.11 and 39.50 with between 29.11 and 39.50 with 95% confidence.95% confidence.

JMP gives CI’s for JMP gives CI’s for

95% CI for mean APACHE score for lung cancer patients (29.102, 39.514)

CI for the Population CI for the Population Proportion (Proportion (pp))

If our sample size is sufficiently largeIf our sample size is sufficiently large** the confidence interval for the the confidence interval for the population proportion (population proportion (pp) is given by:) is given by:

n

ppquantilenormalp

)ˆ1(ˆ) (ˆ

* 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 = 1.645

For 99% Normal quantile z = 2.576

Find a 95% for 30-day Find a 95% for 30-day Mortality for Right Heart Mortality for Right Heart

Catheter PatientsCatheter PatientsResults:Results: n = 2,184 patients had a n = 2,184 patients had a

Swan-Ganz line put in during their Swan-Ganz line put in during their treatment, of these 830 died within treatment, of these 830 died within 30-days of admission to the ICU.30-days of admission to the ICU.

40%) , (36%or .40) , 36(.

02.38.)0104(.96.138.2184

)62)(.38(.96.138.

)ˆ1(ˆ96.1ˆ

38%or 38.184,2

830ˆ

n

ppp

p

InterpretationInterpretation

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