# Expressing Confidence Intervals as a percentage for simplicity

I am running some stats on a feeding experiment where 6 rodents were given the choice between control seeds and treated seeds for 12 hours. seed count consumed was the response variable. I've analyzed the data using a paired t-test and got a p-value of 0.0007. I'm assuming this is the right analysis to do but let me know if you disagree. Here is the raw data.

Mean control seed consumed 1082 Mean treated seed consumed 32.5

For the sake of making the data easy for the layperson to interpret, I would like to present the results as a percentage which I got using the following equation which I applied to the means

control seeds consumed / total seeds consumed

1082/(1082+32.5)= 96.8%

I noticed that if I apply this same equation to the 6 replicates individually I get

100% 100% 92.9% 99.4% 99.8% 100%

which if averaged yields 98.1%

So which is correct 98.1% or 96.8% Should I be taking the average of the proportions or the proportion of the averages

In the end I would like to be able to say "rodents showed a (##)% +/- (#)% preference for untreated seeds"

I'm also not sure how to do the +/-. should this plus or minus be the standard error? Should I calculate it from the equation...

σp = sqrt [ P(1 - P) / n ]

using the proportion of the averages 96.8%

or use the equation

σx = σ / sqrt( n )

using the list of proportions to find the standard deviation and applying the standard error to the average of the proportion 98.1%

• Please don’t take this the wrong way, but do you really need statistics to say the vermin loved the control seeds and hated the treated seeds? Like the famed statistician Bob Dylan said: “you don’t need a weatherman to know which way the wind blows.” – Ed V Aug 20 at 23:34
• I know right, The trend is obvious so the statistics aren't really necessary. But with as few reps as I have, some stats might help convince someone that the trend is unlikely the result of chance – Justin Taylor Aug 22 at 4:08
• Well, your reps will increase and best of success with your work! – Ed V Aug 22 at 11:51

You have seen the behavior of only six mice. Four avoided eating treated seeds altogether and two showed a substantially reduced appetite for treated seeds. There are many more than six mice of various kinds in the universe, so it is a stretch to say the treatment deters most of them. Nevertheless, the data you have give an early indication that the treatment may be promising.

The difference ctrl - trt passes a Shapiro-Wilk test for normality. It seems OK to look at results of a paired test, which (not surprisingly) strongly rejects the null hypothesis that that there is no significant difference in preferences of half a dozen mice. They prefer eating untreated seeds. Because one can't 'interview' mice, asking what they think of treated seeds, you can take their eating behavior as a proxy for "Treated seeds are yucky."

Your measure of 'proportion of control seeds consumed' might be OK if half of several thousand randomly sampled seeds were treated and half not, and then seeds were deployed to thousands of independent places. Then there would be a basis for statements about the fortunes of treated vs. untreated seeds.

In your experiment, however, mice were the experimental units. You have limited information on the behavior of mice. In summary, you might say something like what I did at the start:

"Four of six mice avoided eating treated seeds altogether and remaining two showed a substantially reduced appetite for treated seeds."

First, before you start generalizing (drawing inferences) from six mice, you should establish what population of mice these six are a random sample of. If you can't do that, then I'd go with what @BruceET said.

Second, these results pass the IOTT - the interocular trauma test - it hits you between the eyes. It seems like overkill to do all the work you are doing.

Third, your two numbers are answers to different questions. The first question is "Of all seeds eaten, what proportion were control and what proportion were treated?" and the second is "What is the average proportion of control seeds eaten?" They get different answers because different rats ate different numbers of seeds.

Finally, for the standard error, the usual formula is $$SE = \sqrt{\frac{p(1-p)}{n}}$$. This isn't perfect and a bunch of alternatives have been proposed, but, given how extremely clear the results are, this estimate is probably fine for your purposes.

• Sorry, I oversimplified my experiment a bit and it raised some concerns. I actually tested 10 different treated seeds on 60 rodents, so I have more replicates than I originally let on. The effect is the same for all 10 kinds of treated seeds. Rodents don't like any of them and usually avoid them entirely. – Justin Taylor Aug 22 at 4:23
• The rodents were trapped from the wild at a number of locations and although there could be some trap bias between individuals it is usually assumed that trapped individuals are a semi-random sample. So I understand that any inference I make is to the populations at our trap sites – Justin Taylor Aug 22 at 4:38