Proportion of defective integrated chips (IC) /incidence rate of a disease

We refer to BLOGS of March 13 and 14. A company can allow only up to 10% defectives. Then maximum number of defective in a lot of 52 IC is 5. A sampling scheme is designed. A sample of size 10 is drawn from a lot of 52 IC. But 10% of this sample of size 10 is 1 defective IC. But due to variation in sampling the distribution of number of defectives in 95% of the samples will be between 0 to 2.8~3.  Here C(52, 10) = 15820024220  unique samples of size 10 can be drawn from this lot of size 52. This example can also be related to number of patients suffering from a disease. Suppose that the incidence of disease is 10% in the entire population. Then in a sample of 10 people,  0 to 3 people suffer from this disease if the incidence rate is 10%. But if the number of patients suffering is higher than 3 in the sample of size 10, then we conclude that the incidence of disease is higher than 10% in the entire population.

H₀: p = 0.10

H₁: p > 0.10

The mathematics is given in the figure below.