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Sampling and Estimation Theories
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Sampling Distributions

In statistics, we often work with samples to make inferences about a larger population. A sampling distribution is the probability distribution of a statistic (like the sample mean, $\bar{x}$, or sample proportion, $\hat{p}$) that is obtained through a large number of samples of a specific size drawn from a specific population.

Key Ideas:

• If we were to take many random samples of the same size from a population, calculate a statistic for each sample (e.g., the mean), the distribution of these sample statistics would form the sampling distribution.
• The sampling distribution of the sample mean ($\bar{x}$) has important properties, especially as described by the Central Limit Theorem (CLT).

Central Limit Theorem (CLT):

The CLT states that if you have a population with mean $\mu$ and standard deviation $\sigma$, and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This holds true regardless of the shape of the population distribution, provided the sample size ($n$) is large enough (typically $n \ge 30$).

• The mean of the sampling distribution of sample means ($\mu_{\bar{x}}$) is equal to the population mean ($\mu$).
  $$ \mu_{\bar{x}} = \mu $$
• The standard deviation of the sampling distribution of sample means, also known as the standard error of the mean (SEM), is $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$.
  $$ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} $$

Understanding sampling distributions is fundamental for hypothesis testing and constructing confidence intervals.

Point and Interval Estimates

When we want to estimate an unknown population parameter (like the population mean $\mu$ or population proportion $p$), we can use information from a sample.

Point Estimate:

A point estimate is a single value (a statistic) calculated from sample data that is used to estimate the unknown population parameter.

• The sample mean ($\bar{x}$) is a point estimate for the population mean ($\mu$).
• The sample proportion ($\hat{p}$) is a point estimate for the population proportion ($p$).

While point estimates provide a single best guess, they are unlikely to be exactly equal to the true population parameter due to sampling variability.

Interval Estimate (Confidence Interval):

An interval estimate provides a range of plausible values for the population parameter, along with a level of confidence that the true parameter lies within that range. This range is called a confidence interval (CI).

A confidence interval is typically expressed as:

The confidence level (e.g., 90%, 95%, 99%) indicates the proportion of such intervals, constructed from repeated sampling, that would contain the true population parameter in the long run. For example, a 95% confidence interval means that if we were to take many samples and construct a CI from each, about 95% of those intervals would capture the true population parameter.