Short Answer 
In hypothesis testing for population proportions, it’s essential to ensure that both the expected number of successes and failures are at least 10 to allow for normal approximation of the binomial distribution. A sufficiently large sample size is crucial as it enables the application of the Central Limit Theorem, facilitating valid statistical inferences.
Understand the Conditions
In hypothesis testing for a population proportion, it’s crucial to meet specific conditions. The main rule is that both np0 (the expected number of successes) and n(1-p0) (the expected number of failures) should be at least 10. This ensures that the sample size is large enough to yield reliable results.
Importance of Normal Approximation
The reason behind these conditions is to allow the binomial distribution to be well approximated by a normal distribution. When these conditions are met, it helps in applying statistical methods associated with the normal distribution effectively. This is essential for simplifying analysis and making inferential statistics easier to conduct.
Role of Sample Size
A large sample size plays a pivotal role in hypothesis testing. When the sample size is large enough, the Central Limit Theorem allows the sampling distribution of the sample proportion to approximate a normal distribution. This approximation is critical for accurately conducting one-sample tests, enabling researchers to draw valid conclusions from their data.