As a statistical expert with a focus on probability and statistical inference, I'm well-versed in calculating various statistical measures, including the estimation of population proportions. When it comes to calculating the sample proportion, often denoted as \( \hat{p} \), it is a fundamental concept in statistics that is used to estimate the true proportion of a characteristic in the entire population based on a sample.

The calculation of \( \hat{p} \), or the sample proportion, is straightforward. It involves taking the number of successes (X) in your sample and dividing it by the total number of observations (n) in that sample. The formula is as follows:

\[ \hat{p} = \frac{X}{n} \]

Here's how you can calculate it step by step:


1. Identify the Sample Size (n): Determine the total number of observations in your sample. This is the count of all individuals or items in the sample from which you are drawing your conclusions.


2. Count the Number of Successes (X): Identify the number of times the characteristic of interest, often referred to as a "success," occurs within your sample. A success is any outcome that meets the criteria you are interested in studying.


3. Apply the Formula: Divide the number of successes (X) by the sample size (n) to find the sample proportion (\( \hat{p} \)).


4. Interpret the Result: The resulting value of \( \hat{p} \) is an estimate of the true population proportion (p). It gives you an idea of the likelihood that a randomly selected member from the population would exhibit the characteristic of interest.

It's important to note that the sample proportion is just an estimate. The larger the sample size, the more reliable this estimate is likely to be, assuming the sample is representative of the population. However, even with a large sample, there is always some degree of uncertainty, which can be quantified using confidence intervals or by conducting hypothesis tests.

Now, let's move on to the translation of the answer into Chinese.

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If the sample size is n, the proportion can be obtained from the count by division by n: p-hat = X/n (where X is the number of people who approve of Clinton, and p-hat is defined below).read more >>