• A confidence interval is a range of values that contains the true population parameter
with a chosen confidence level.
• We have a set of data from a population with both m and s unknown. We use y̅ to
estimate m, and s to estimate s, using a t distribution (df=n − 1).
• Centered around the mean
̅ ± 𝑡 𝑠/ 𝑛 √
Standard Deviation Versus Standard Error
• For a sample of size n, 1
theo n − 1 is the “degrees of freedom.” (y - y) 2
• The value s/√n is called the standard error of å i
the mean, SEM. n-1
• Scientists often present their sample results as the mean ± SEM.
• Example: A medical study examined the effect of a new medication on the seated
systolic blood pressure. The results, presented as mean ± SEM for 25 patients, are 113.5
± 8.9. What is the standard deviation s of the sample data?
o SEM = s/√n <=> s = SEM*√n
s = 8.9*√25 = 44.5
• The t distributions are wider for smaller sample sizes, reflecting the lack of precision in
estimating from s.
• When n is large, s is a good estimatdf n – 1ribution is close to the
standard normal distribution.
o SEE BELOW
1 Stat 1400
t distribution, df 4
t distribution, df 1
t distribution, df 100
t distribution, df 20
2 Stat 1400
Link Between Confidence Level and Margin of Error
• Higher confidence implies a larger margin of error (less precision more accuracy).
• A lower confidence level produces a smaller margin of error (more precision less
• Win/lose situation
• For the same confidence level, narrower confidence intervals can be achieved by using
larger sample sizes:
3 Stat 1400
Sample Size and Experimental Design
• A study may have a limit set on its margin of error (e.g., drug trial, manufacturing specs).
In many cases, the population variability (s) is fixed, but we can choose the number of
• Using simple algebra, you can find what sample size is needed to obtain a desired
margin of error.
s æ t*s ö
m =t* Û n = ç ÷