Answer :
The z-test for "no difference" may be potentially inaccurate if the sample has small counts of successes and failures, but it is reasonably accurate if the sample sizes are large and other conditions for inference are met, such as known population variance or a large enough sample for the normal approximation to hold. We never accept the null hypothesis; we only reject or fail to reject it based on the evidence.
The accuracy of a z-test for "no difference" mainly depends on whether the conditions for inference are met, including sample size and the variance in the population. If the counts of successes and failures in the sample are too small, which can occur in samples with a small number of observations, the z-test may be less accurate because the normal approximation may not be valid. Conversely, the z-test's accuracy is reasonable if the sample sizes are sufficiently large and the population variance is known or the sample is large enough for the sample variance to approximate it well.
The use of a t-test is recommended over a z-test when dealing with small sample sizes and unknown population variances. However, the z-test can be used when there is a large sample size or when the actual population variance is known. It is important to note that a z-test should not be discredited merely because the pooled sample proportion is large; problems only arise when the sample size is not adequate to satisfy the conditions for a normal approximation.
Furthermore, it is crucial to understand that we never accept the null hypothesis; we either reject it or fail to reject it based on the evidence at hand. In this sense, a small z statistic (e.g., z = 0.13) indicates a failure to reject the null hypothesis, suggesting there is not enough statistical evidence to conclude a difference exists between the sample statistic and the population parameter.
