Final answer:
The sampling distribution of the sample proportion p is normally distributed if the sample size is less than 5% of the population and if both np and n(1-p) are equal to or greater than 10. For a large sample, n, with x successes, the mean of the distribution is the population proportion (p) and the standard deviation is the square root of (p(1-p)/n).
Step-by-step explanation:
To describe the sampling distribution of p, the proportion of people who are satisfied with the way things are going in their life, we must ensure that certain conditions are met to apply the normal approximation model. The sample size must be less than 5% of the population size to ensure the independence of trials, and the product of the sample size (n) and the population proportion (p) multiplied by 1-p (i.e., np(1-p)) must be equal to or greater than 10 to ensure the distribution is approximately normal. When these conditions are met, we say the distribution of p is normally distributed with mean μp = p (the population proportion) and standard deviation σp = √(p(1-p)/n).
For example, if we are testing a sample proportion with x = 172 and n = 420,019, the calculation for np would be 420,019 * 0.00034 = 142.8, and for nq it would be 420,019 * 0.99966 = 419,876.2. Since both values are greater than 10, the normal approximation is appropriate. Thus, the sampling distribution of the sample proportion (p') can be considered normally distributed.