Final answer:
The 'sampling distribution' of the sample proportion is the distribution of all possible sample proportions. As the sample size increases, the distribution becomes more normally distributed, centered on the population proportion with a standard deviation shrinking in size. Ensuring np and nq are both greater than 5 is crucial for the distribution to approximate normality for hypothesis testing.
Step-by-step explanation:
The term sampling distribution of the sample proportion refers to the probability distribution of all possible values of the sample proportion (p') based on samples of a certain size (n) from a population. When focused on a categorical data set that can be divided into two categories, such as 'Success or Failure' or 'Yes or No', we consider the central limit theorem for proportions to understand the behavior of the sampling distribution.
As the sample size (n) increases, the shape, center, and spread of the sampling distribution change accordingly. The central limit theorem tells us that the sampling distribution of p' will follow a normal distribution with a mean (μ) equal to the population proportion (p), and a standard deviation (σ) equal to √((p * q) / n), where q = 1 - p. When p = 0.1 and we increase the sample size, the distribution's shape becomes more normal, centering around 0.1. The spread or standard deviation decreases, indicating a more precise estimate of the population proportion.
It is important when dealing with a sample proportion problem, that both np and nq must be larger than 5 for the normal approximation to be considered valid. This is critical for conducting a hypothesis test of a single population proportion or when comparing two sample proportions