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A proportion is a way to express a part of a total or whole. It is often used in statistics to represent the fraction of a population having a specific characteristic. In this exercise, the proportion we are interested in is the number of adults aged 65 and older in Greenville County relative to the total number of adult residents. Calculating this is straightforward:

\[ p = \frac{\text{Number of adults 65 or older}}{\text{Total number of adult residents}} = \frac{59,969}{461,299} \approx 0.1299 \]

Here, 0.1299 is interpreted as about 13% of the adult residents being aged 65 and older. This is a crucial statistic as it provides insight into the demographic makeup of the county.

Understanding proportions is vital because it allows us to compare different subsets within a population or across different populations. They are also used in determining probabilities, making them fundamental in various statistical methods.

A simple random sample is a method used in statistics where every individual has an equal chance of being selected from the population. This helps minimize biases or influences that might affect the outcome of a study. It ensures that the sample accurately represents the whole population.

In the Greenville County survey, residents were contacted using random digit dialing based on landline numbers. However, this method raises concerns about the sample's randomness. Excluding cell phone numbers might lead to a biased sample since it is possible that different age groups have distinct preferences for landlines over cell phones. For instance, older adults might rely more on landlines, thus not truly representing other age groups adequately.

When conducting a survey, ensuring the sample is as random as possible helps validate the results. This concept underscores the importance of precise sample selection processes in statistical studies.

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In the context of sample proportions, the standard deviation helps us understand how much these proportions would vary if we took multiple samples from the same population.

For the sample proportion in Greenville County, the standard deviation is computed as:

\[ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \approx \sqrt{\frac{0.1299 \times (1-0.1299)}{689}} \approx 0.0133 \]

This value tells us that the sample proportion will typically vary by about 1.33% from the true population proportion if we sample repeatedly.

Having a smaller standard deviation indicates that the sample proportions are more tightly clustered around the average, leading to more reliable results.

The empirical rule is a statistical guideline suggesting how data should be distributed in a normal distribution. It states that:

• 68% of the data falls within one standard deviation of the mean.
• 95% falls within two standard deviations.
• 99.7% falls within three standard deviations.

For the Greenville County survey, the 95% confidence interval for the proportion of residents aged 65 and older can be calculated using this rule. The interval is given by:

\[ \hat{p} \pm 2 \times \sigma_{\hat{p}} \]

Substituting the values gives us:

\[ 0.1299 \pm 2 \times 0.0133 = (0.1033, 0.1565) \]

This interval means we are 95% confident that the true proportion of adults 65 and older falls between 10.33% and 15.65%.

The empirical rule helps in understanding the likely range of variation in statistical estimates, making it an essential tool for interpreting data in research.