Answer :
Final answer:
The Random 10% Large Counts Condition ensures that the sampling distribution of p-hat is approximately normal, and hypothesis testing with such conditions relies on a binomial test to calculate the probability of error or P-value.
Explanation:
When the Random 10% Large Counts Condition are met for a binomial distribution, we can expect that the sampling distribution of the sample proportion π-hat to be approximately normally distributed due to the Central Limit Theorem. This is particularly true when both np and nq are greater than 5, allowing us to use normal approximation for the binomial distribution.
Our sample proportion, p-hat, would then have a mean ( μ ) equal to the population proportion p and a standard deviation ( σ) calculated by √(pq/n), where q is the probability of failure (1-p). When conducting a hypothesis test regarding the proportion p, we can utilize a binomial test to assess whether the observed proportion in our sample significantly deviates from the expected proportion under the null hypothesis.
The P-value obtained from this test represents the probability of error, which is the sum of the probabilities of obtaining our observed outcome plus any more extreme outcomes if our null hypothesis were true. Larger sample sizes reduce the P-value, thus decreasing the probability of mistakenly rejecting the null hypothesis.
