With normal approximation method and technology, is it necessary to use the one sample z test for proportion for large sample sizes instead of the binomial formula?
When dealing with large sample sizes in the context of hypothesis testing for proportions, it is often more convenient and efficient to use the normal approximation method, specifically the one-sample z-test, instead of the binomial formula.
The binomial formula calculates probabilities for each possible outcome of a categorical variable, such as success (i.e., meeting a certain criterion) or failure. This formula requires knowing the sample size and the true population proportion. However, it can be computationally intensive for large sample sizes since it involves calculating multiple probabilities.
On the other hand, the one-sample z-test for proportion relies on the assumption that the sampling distribution of the proportion follows an approximately normal distribution when the sample size is sufficiently large. This allows us to estimate probabilities without having to consider every possible outcome, resulting in a simpler and faster calculation.
To determine whether you should use the one-sample z-test or the binomial formula, consider the rule of thumb that the normal approximation is valid if both np and n(1-p) are greater than or equal to 10, where n represents the sample size and p is the estimated proportion.
If the conditions for the normal approximation are met (i.e., np ≥ 10 and n(1-p) ≥ 10), you can confidently use the one-sample z-test for proportion. However, if the conditions are not met, it is recommended to use the binomial formula to calculate probabilities for each possible outcome.
In conclusion, for large sample sizes where the conditions for the normal approximation are satisfied, using the one-sample z-test for proportion is generally preferred due to its computational efficiency.