# [R] Discrepency between confidence intervals from t.test and computed manually -- why?

Prof Brian Ripley ripley at stats.ox.ac.uk
Wed Jan 4 10:37:38 CET 2006

Some user error it appears.  I googled, got the data file from

http://www3.interscience.wiley.com:8100/legacy/college/davis/0471172758/datafiles/data_index.html

and did

> Cs137 <- temp$X137Cs > t.test(Cs137)$conf.int
[1] 5.115488 7.239712
attr(,"conf.level")
[1] 0.95

> mu <- mean(Cs137); n <- length(Cs137)
> se.mean <- sqrt(var(Cs137)/n)
> # two-tail alphas
> alpha <- c(1, 5, 10, 20)/100
> # t-values for each tail
> t.vals <- qt(1-(alpha/2), n-1)
> # name them for the respective alpha
> names(t.vals) <- alpha
> # low and high ends of the confidence interval
> round(ci.low <- mu - se.mean * t.vals, 2)
0.01 0.05  0.1  0.2
4.77 5.12 5.29 5.49
> round(ci.hi <- mu + se.mean * t.vals, 2)
0.01 0.05  0.1  0.2
7.58 7.24 7.07 6.87
> c(ci.low[2], ci.hi[2])
0.05     0.05
5.115488 7.239712

which agrees with t.test and not what you reported you got.

On Wed, 4 Jan 2006, David Rossiter wrote:

> I am sure there is something simple here I am missing, so please bear
> with  me.
>
> It concerns the computation of the confidence interval for a population
> mean.
>
> The data are 125 measurements of Cs137 radation, a sample data set from
> Davis "Statistics and Data Analysis in Geology" 3rd ed. (CROATRAD.TXT)
> ------------------
> method 1: using textbook definitions: mean \pm se_mean * t-value
>
> mu <- mean(Cs137); n <- length(Cs137)
> se.mean <- sqrt(var(Cs137)/n)
> # two-tail alphas
> alpha <- c(1, 5, 10, 20)/100
> # t-values for each tail
> t.vals <- qt(1-(alpha/2), n-1)
> # name them for the respective alpha
> names(t.vals) <- alpha
> # low and high ends of the confidence interval
> round(ci.low <- mu - se.mean * t.vals, 2)
> round(ci.hi <- mu + se.mean * t.vals, 2)
>
> Output:
> 0.01 0.05  0.1  0.2
> 5.66 5.81 5.90 5.99
>
> 0.01 0.05  0.1  0.2
> 6.69 6.54 6.46 6.36
>
> -----------------
>
> So for the 95% confidence level I seem to get a CI of 5.81 .. 6.54
>
> ------------------
> method 2: using t.test.  I am not really testing for any specific mean,
> I just want the confidence interval of the mean, which t.test seems to
> give to me:
>
> Input:
> t.test(Cs137)
>
> Output:
>
>        One Sample t-test
>
> data:  Cs137
> t = 11.5122, df = 124, p-value < 2.2e-16              <-- not relevant
> alternative hypothesis: true mean is not equal to 0   <-- not relevant
> 95 percent confidence interval:
> 5.115488 7.239712
> sample estimates:
> mean of x
>   6.1776
> ------------------------------
>
> So with t.test I seem to get a CI of 5.12 .. 7.24 which is considerably
> wider than the directly computed interval 5.81 .. 6.54.  Perhaps I am
> mis-understanding the CI which t.test is reporting?
>
> Any help would be appreciated.
>
> Thank you.
>
> D G Rossiter
> Senior University Lecturer
> Department of Earth Systems Analysis (DESA)
> International Institute for Geo-Information Science and Earth
> Observation (ITC)
> Hengelosestraat 99
> PO Box 6, 7500 AA Enschede, The Netherlands
> mailto:rossiter at itc.nl,  Internet: http://www.itc.nl/personal/rossiter

--
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595