Final answer:
The statement regarding the binomial probability distribution and large sample sizes approximating a normal distribution is true due to the central limit theorem. As the sample size increases, the binomial distribution increasingly resembles a normal distribution, simplifying the calculation of probabilities.
Step-by-step explanation:
True. The statement that the larger the sample size, the more closely a binomial probability distribution will approximate a normal distribution is correct. This assertion is based on the central limit theorem, which posits that as the sample size increases, the distribution of sample means becomes more normally distributed, regardless of the shape of the population distribution. This is subject to the sample size being sufficiently large (commonly suggested as greater than or equal to 30) and also assumes that the binomial distribution has a reasonably large number of trials and is not heavily skewed.
One historical note regarding the normal approximation to binomial distribution highlights this point. When computing binomial probabilities with a large number of trials, using the normal approximation simplifies the calculations significantly, allowing us to estimate probabilities that would otherwise be cumbersome to compute directly from the binomial formula.
In application, when sample sizes increase, statistical measures such as the confidence level and standard deviation of the sampling distribution adjust to more closely align with the parameters of the underlying population. The large sample sizes lead to decreased standard deviation of the sampling distribution, reflecting reduced variability around the mean.