# Statistical test for rates that are very low with low count

I want to know if dataset suitable for test of significance, and if 'no' (which I am suspect) why exactly not as I must report it why not.

I have three apples species (Gala, Pink Lady and 'new') and apples have been inspected for damage. I want to know if 'new' have less damages. All apples sampled in same way, but only 50 apples for each variety have been tested (150 total).

Out of 150 apples test, Gala had 5 apples with defects, Pink Lady had 1 and 'new' had 0 defect. I was expect a defect rate of ~15%, so now I have dataset with less defect I expect. What can I do?

Thank you any help would be really appreciate!

• You can certainly perform a significance test with those data, but you might be surprised by the low power that such a test will give you. There is just not much information concerning the mean rate of defects in 1 or 0 defects out of 150 trials. Confidence intervals for the proportions might be a better expression of the uncertainties in your data. Oct 16, 2023 at 20:35
• how do I calculate confidence interval for groups Mr Lew? Oct 17, 2023 at 0:00
• Your system is equivalent to what statisticians often call 'Bernoulli sampling'. See here for intervals: stats.stackexchange.com/questions/4756/… and here: stats.stackexchange.com/questions/411699/… and note that the normal approximation method ('Wald') has bad properties when the number of successes is close to zero, as in your situation. I would suggest a Wilson scores interval as a well behaved option. Oct 17, 2023 at 4:53

One simple approach here is to use the Fisher test, which is valid for counts of zero.

x = c('Gala'=5, 'Pink Lady'=1, 'New'=0)
d = rbind('Damage'=x, 'No Damage'=150-x)
d
#> Damage       5         1   0
#> No Damage  145       149 150

mosaicplot(t(d), color=T, main='Proportions', dir='h')

fisher.test(d)
#>
#>  Fisher's Exact Test for Count Data
#>
#> data:  d
#> p-value = 0.05158
#> alternative hypothesis: two.sided