1. Having Confidence in our Means: Confidence Intervals
Scientific Practice
Having Confidence in our Means:
Confidence Intervals

2. Samples are Estimates
When we sample a population, we end up with a sample mean, x
it’s our ‘best guess’ of the real population mean, µ
the ‘real’ mean of the population is ‘hidden’
Our sample also has a measure of the variability of the data comprising it
the sample Standard Deviation, s
which is also an estimate of the population SD, σ
s can be also be used to indicate the variability of the mean itself
SEM = s / √ N
can then use SEM to determine confidence limits

3. Confidence Limits and the SEM
The SEM reflects the ‘fit’ of a sample mean, x , to the underlying population mean, µ
if we calculate two sample means and they are the same, but for one the SEM is high,
we are less ‘confident’ about how well that one estimates the population mean
Just like the ‘raw’ data used to calculate a sample mean follows a distribution, so will repeat estimates of the population mean itself
this is the t Distribution

4. The t Distribution Yet another distribution! 
but distributions are important because they define how we expect our data to behave
if we know that, then we gain insight into our expts!
Generally ‘flatter’ than the Normal Distribution
any particular area is more ‘spread out’ (less clear)
the more ‘pointed’ a curve, the clearer the peak

5. t Distribution Pointedness Varies!
Logic…
the number of samples influences the ‘accuracy’ of our estimate of the population mean from the sample mean
as N increases, the ‘peak’ becomes sharper
a given area of the curve is less ‘spread out’
At high N, t Distribution = Normal Distribution

6. Using the t Distribution
When we calculate a sample mean and call it our estimate of the population mean…
it’s nice to know how ‘confident’ we are in that estimate
One measure of confidence is the 95% Confidence Interval (95%CI)
the range over which we are 95% confident the true population mean lies
derived from our sample mean (we calculate)
and our SEM (we calculate)
and the N (though it’s the ‘degrees of freedom’, N-1)
we use this to look up a ‘critical t vlaue’

7. From t Distribution to 95%CI
The t Distribution is centred around our mean and its shape is influenced by N-1
95%CI involves chopping off the two 2.5% tails
Need a t table to look up how many SEMs along the x-axis this point will be
Value varies with N-1
And level of confidence sought

8. Step 1: The t Table t value varies with… For N = 10, α = 0.05 Row…
DoF is N-1
Column…
level of ‘confidence’
95%CI involves chopping off the two 2.5% tails
α = 0.05 (5%)
For N = 10, α = 0.05
t(N-1),0.05 = 2.262
when N large, t=1.96

9. Step 2: Using the t value
The t value is the number of SEMs along the x-axis (in each direction) that encompasses that % of the t distribution centred on our mean
2.262 in the case of t(N-1),0.05
Eg we measure the FVC (litres) of 10 people…
mean = 3.83, SD = 1.05, N = 10
SEM = 1.05/√10 = litres
t(N-1),0.05 = standard errors to cover 95% curve
So, litres either side of the mean = * 0.332
= litres either side of mean covers 95% of dist
So, 95%CI is 3.83 ± =  litres
(3.079, 4.581)

10. Effect of Bigger N
A larger sample size gives us greater confidence in any population mean we estimate
so 95%CI should be smaller
In previous example…
mean = 3.83, SD = 1.05, N =10, SEM = 0.332
95%CI is (3.079, 4.581)
But say we measured 90 more people…
mean = 3.55, SD = 0.915, N = 100
mean and SD similar to before, but
SEM now a lot smaller, at 0.915/√100 =
so too is t(N-1),0.05 = 1.96 (rather than 2.262)
95%CI = 3.55 ± (1.96 * ) =  3.729

11. Effect of Bigger N
A bigger N ‘sharpens’ the t distribution so that the 95% boundaries are less far apart
ie our confidence interval will become smaller
95%CI also shrinks because SEM = SD/√N

12. Summary Sample means are estimates of population means
Bigger samples give more confident estimates
SEM reflects the distribution of mean estimates
SEM = s / √ N
Estimates of means follow the t Distribution
t Distribution becomes ‘sharper’ with higher N
‘Width’ of t dist covering 95% is called 95%CI
range in which 95/100 mean estimates would fall
95%CI = mean ± (t(N-1),0.05 * SEM)
t is the number of SEMs along dist covering that %