Final answer:
The total probability under the normal distribution curve is always one. To determine specific probabilities, one calculates z-scores and refers to a z-table. The central limit theorem ensures normal distribution for sample proportions with a large enough size.
Step-by-step explanation:
The total proportion, or probability, under the normal distribution curve is always equal to one (100%). The normal distribution is a bell-shaped curve that represents the distribution of many types of data. It is defined by the mean (μ) and standard deviation (σ), and when we refer to a standard normal distribution, the mean is zero and the standard deviation is one.
To calculate specific probabilities or proportions for given values, you use z-scores, which represent how many standard deviations away a point is from the mean. By finding the z-score and referencing it on a z-table, you can find the proportion of the data within a certain range. For example, if you are looking for the probability of a value being below a certain point, you would find the area to the left of the z-score on the curve, denoted as P(X < x).
When dealing with sample proportions in statistics, such as estimating the percentage of the population that will vote for a certain candidate or has a college education, the central limit theorem can apply. This theorem states that the distribution of sample proportions will be normally distributed given a large enough sample size, with a mean equal to the population proportion (p) and a standard deviation calculated using the formula √[p(1-p)/n].