[R] how to interpret t.test output

[(Ted Harding)]

On 09-Aug-08 20:31:33, Felipe Carrillo wrote:
># Hi all:
>      #I got a vector with fish lengths(mm)
>      # Can someone help me interpret the output of
>      # a t.test in plain english?
>      #  Based on the t.test below I can say that
>      # I reject the null hypothesis because
>      # the p-value is smaller than the the significance
>      # level(alpha=0.05). What else can I conclude here?
>     Ho = 36 mm
>     Ha <> 36 mm
>     fishlength <- c(35,32,37,39,42,45,37,36,35,34,40,42,41,50)
>     t.test(fishlength,mu=36)
>        One Sample t-test
> 
> data:  fishlength
> t = 2.27, df = 13, p-value = 0.04087
> alternative hypothesis: true mean is not equal to 36
> 95 percent confidence interval:
>  36.14141 41.71573
> sample estimates:
> mean of x
>  38.92857

The standard interpretation of the above is that, on the hypothesis
that the fish were sampled from a population in which the mean length
was 36 (and the distribution of length is sufficiently close to
Normal for the t-test to be adequately applicable), then the value
of t differs from 0 by an amount which has probability less than
0.05 of being attained if that hypothesis were true. Therefore the
result is evidence of some strength that the hypotesis is not true,
and the mean length of fish in the population is different from 36.

However, it is on the borderline of the mildest criterion of
"significant difference" usually adopted, namely 0.05. A more
stringent criterion could be, for instance, 0.01; and then you
would not be "rejecting this null hypothesis".

As for the following:

>     # I also would like to know how to interpret the output
>     # when mu=0? Notice that the p-value from the t.test above
>     #is different from this t.test. Are we trying to reject the null 
>     # hypothesis here too?
> 
>     t.test(fishlength)
>          One Sample t-test
> 
> data:  fishlength
> t = 30.1741, df = 13, p-value = 2.017e-13
> alternative hypothesis: true mean is not equal to 0
> 95 percent confidence interval:
>  36.14141 41.71573
> sample estimates:
> mean of x
>  38.92857
> 
> Thanks in advance for your help.

In terms of interpreting a statistical test, using your data,
of the hypothesis that the mean length in the population is 0,
the P-value of 0.0000000000002017 is very strong evidence indeed
that the mean is not 0.

However, I do not know why you are asking the question. No test
is needed. The length of any living fish, even while it is still
in the egg, is greater than 0; and whatever population you have
taken your sample from will have a mean length which is greater
than 0.

That is not to say that the result of a t-test on any sample
will necessarily give a significant result. You could have a
small catch with lengths, say,

   fishlengths <- c(2,4,9,20,50)
   t.test(fishlengths,mu=0)

#         One Sample t-test
# data:  fishlengths 
# t = 1.9273, df = 4, p-value = 0.1262
# alternative hypothesis: true mean is not equal to 0 
# 95 percent confidence interval:
#  -7.489442 41.489442 
# sample estimates:
# mean of x 
#        17 

And all you can conlude from that is that the sample, *in itself*,
does not carry sufficient information to confirm what you know
is true (i.e. mu > 0). Even the one-sided test of mu=0 with alternative
alt="greater" does not give a result significant at 5%:

 t.test(fishlengths,mu=0,alt="greater")

#         One Sample t-test
# data:  fishlengths 
# t = 1.9273, df = 4, p-value = 0.0631
# alternative hypothesis: true mean is greater than 0 
# 95 percent confidence interval:
#  -1.803807       Inf 
# sample estimates:
# mean of x 
#        17 

Hoping this helps!
Ted.

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Date: 09-Aug-08                                       Time: 22:28:34
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