Confidence Intervals Scientific Practice Samples are Estimates When we sample a population we end up with a sample mean its our best guess of the real population mean µ the real mean of the population is hidden ID: 651461
Download Presentation The PPT/PDF document "Having Confidence in our Means:" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
Having Confidence in our Means:Confidence Intervals
Scientific PracticeSlide2
Samples are EstimatesWhen we sample a population, we end up with a sample mean, 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, swhich is also an estimate of the population SD, σ s can be also be used to indicate the variability of the mean itselfSEM = s / √ Ncan then use SEM to determine confidence limits
Slide3
Confidence Limits and the SEMThe SEM reflects the ‘fit’ of a sample mean, , to the underlying population mean, µif 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
t
his is the t Distribution Slide4
The t DistributionYet another distribution! but distributions are important because they define how we expect our data to behaveif we know that, then we gain insight into our expts!Generally ‘flatter’ than the Normal Distributionany particular area is more ‘spread out’ (less clear)the more ‘pointed’ a curve, the clearer the peakSlide5
t Distribution Pointedness Varies!Logic…the number of samples influences the ‘accuracy’ of our estimate of the population mean from the sample meanas N increases, the ‘peak’ becomes sharpera given area of the curve is less ‘spread out’At high N, t Distribution = Normal DistributionSlide6
Using the t DistributionWhen 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 estimateOne measure of confidence is the 95% Confidence Interval (95%CI)the range over which we are 95% confident the true population mean liesderived 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
’Slide7
From t Distribution to 95%CIThe t Distribution is centred around our mean and its shape is influenced by N-195%CI involves chopping off the two 2.5% tailsNeed a t table to look up how many SEMs along the x-axis this point will beValue varies with N-1And level of confidence soughtSlide8
Step 1: The t Tablet value varies with…Row…DoF is N-1Column…level of ‘confidence’95%CI involves chopping off the two 2.5% tailsα = 0.05 (5%)For N = 10, α = 0.05
t
(N-1),0.05
= 2.262when N large, t=1.96Slide9
Step 2: Using the t valueThe t value is the number of SEMs along the x-axis (in each direction) that encompasses that % of the t distribution centred on our mean2.262 in the case of t(N-1),0.05Eg we measure the FVC (litres) of 10 people…mean = 3.83, SD = 1.05, N = 10SEM = 1.05/√10 = 0.332 litres
t
(N-1),
0.05 = 2.262 standard errors to cover 95% curveSo, litres either side of the mean = 2.262 * 0.332= 0.751 litres either side of mean covers 95% of distSo, 95%CI is 3.83 ± 0.751 = 3.079 4.581 litres(3.079, 4.581)Slide10
Effect of Bigger NA larger sample size gives us greater confidence in any population mean we estimateso 95%CI should be smallerIn previous example…mean = 3.83, SD = 1.05, N =10, SEM = 0.33295%CI is (3.079, 4.581
)
But say we measured 90 more people…
mean = 3.55, SD = 0.915, N = 100mean and SD similar to before, butSEM now a lot smaller, at 0.915/100 = 0.0915so too is t(N-1),0.05 = 1.96 (rather than 2.262)95%CI = 3.55 ± (1.96 * 0.0915) = 3.371 3.729
Slide11
Effect of Bigger NA bigger N ‘sharpens’ the t distribution so that the 95% boundaries are less far apartie our confidence interval will become smaller95%CI also shrinks because SEM = SD/√
NSlide12
SummarySample means are estimates of population meansBigger samples give more confident estimatesSEM reflects the distribution of mean estimatesSEM = s / √ NEstimates of means follow the t Distributiont 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 %