Sample size and type of hypothesis test for QC of products (frozen cattle embryos) I have a question regarding sample size and type of hypothesis test to be used.
I am currently working on buying beef cattle embryos from overseas. The livestock department in my country is going to purchase say 400-500 frozen embyos of good genetic cattle per breed (we look for 4 breeds). They are quite expensive (~$1000 per embryo), thus assurance for quality is essential.
We are discussing how to set the quality control of these embryos when the company delivered to us. Few year ago, our process was that the company delivered us all ordered embryos. Then we took only 1 sample for QC. Now, our colleagues are concerned about such a small sample size and we need to review this QC process again.
Generally, the QC process uses Microscope to check if the embryo after thawed passes QC or not. (Pass = embryo is alive and is in a good condition needed to stay alive in recipient cow and has high chance to implant)
We know that in general 10% of the frozen embryos will be of bad quality (dead or not being able to reproduce) after thawing.
To do QC, I am deciding how could we do sampling and testing if the embyos will be acceptable before purchasing them.
I am thinking to use one sample proportion z test:
H0: p = 0.10
H1: p > 0.10
where p is proportion of bad embryo (failed QC)
As stated, the embryos are quite expensive and we may need just to check few samples from the whole lot (e.g. 400-500 embryos per breed).
Hence, my questions are:

*

*Is the hypothesis test that I proposed suitable for this kind of problem?


*following question 1, how many (minimum) samples do we need to take for QC? (taking into consideration high cost per embryo, so we might not be able to take many sample)
We initially discussed that maybe we might take, e.g., 1% of the whole embryos (400-500) as samples. However, we are not sure this number is statistically sound.
Best regards,
Pattarapol

As @BruceET suggested, I have provided more information as follows:
Preliminary questions:

*

*Q1: If an embryo is used for your test, it is destroyed? Or are we monitoring use of embryos is actual use?


*A: Most of the time, the embryo will be destroyed during QC when the company delivers to us. (but if we can prepare recipient cows, we can transfer tested embryos to the recipient shortly after QC. However it is unlikely)


*Q2: Is each tested embryo to be judged either Good or Bad? In your H0 and H1, does p denote proportion Bad?


*A: p is the proportion of bad embryo (not acceptable for transfer)


*Q3: Are you testing at the 5% significance level?


*A: 5% significance level


*Q4: Specific alternative value of p: Considering p, how far from p =0.1 would important to detect? 0.10 vs  0.05?


*A: I am not sure with this alternative p.
How can we determine alternative p?
However as I have changed my alternative hypothesis H1: p > 0.1, I maybe want to set alternative p = 0.2 or 0.3.


*Q5: Power of the test: Suppose that specific alternative value of pp is true. How sure do you want/need to be to detect such a difference? 80%, 90%?


*A: 80%


*Note: I have changed my hypotheses:


*H0: p = 0.10


*H1: p > 0.10
Where p is a proportion of bad embryo
Thank you,
 A: Preliminary questions:

*

*If an embryo is used for your test, it is destroyed? Or are we monitoring use of embryos is actual use?


*Is each tested embryo to be judged either Good or Bad? In your $H_0$ and $H_1,$ does $p$ denote proportion Bad?


*Are you testing at the 5% significance level?


*Specific alternative value of $p:$ Considering $p,$ how far from $p=0.1$ would important to detect?  $.10$ vs $.05?$


*Power of the test: Suppose that specific alternative value of $p$ is true. How sure do you want/need to be to detect such a difference? 80%, 90%?
You need to give at least approximate answers to these questions before anyone can say what $n$ needs to be.
Output from Minitab:
To get started, here are sample size results from a recent release of Minitab statistical software for a scenario where embryos are monitored during use (normal approximation to binomial), assuming you want power .8 or .9 to detect a difference between $p = .5$ in a one-sided
test at significance level 5%. I am not suggesting these values, just showing
sample results so you will appreciate the necessity of more specific information than you have supplied.
[Some other statistical software programs are available for such
computations, including R. Also, there are 'power and sample size' calculators online--of varying accuracy and ease of use.]
Power and Sample Size 

Test for One Proportion

Testing p = 0.1 (versus < 0.1)
α = 0.05

              Sample  Target
Comparison p    Size   Power  Actual Power
        0.05     184     0.8      0.801729
        0.05     239     0.9      0.900175

The power curves below suggest a variety of different alternative values of $p.$

