Sampling Distribution and Confidence Intervals

Sampling Distribution: The distribution of all possible values taken by a statistic from all possible samples of a fixed size $n$ taken from a population.
It's a theoretical construct and represents the probability distribution of that statistic.

Multiple random samples of size $n$ are taken from a population with mean $µ$ and standard deviation $σ$ .
Some sample means will be above $µ$ and some below, forming the sampling distribution.
The mean of the sampling distribution of $\overline{x}$ is equal to the population mean $µ$ .
The standard deviation of the sampling distribution is $σ/\sqrt{n}$ , where $n$ is the sample size.
There's no systematic tendency for a sample mean to fall above or below $µ$ , even if the raw data is skewed.
The mean of the sampling distribution is an unbiased estimate of the population mean $µ$ , meaning it will be "correct on average" in many samples.
The standard deviation of the sampling distribution measures how much the sample statistic varies from sample to sample.
It is smaller than the population's standard deviation by a factor of $\sqrt{n}$ . Averages are less variable than individual observations.

Central Limit Theorem: When sampling randomly from any population with mean $µ$ and standard deviation $σ$ , if $n$ is large enough, the sampling distribution of $\overline{x}$ is approximately normal.

Confidence Level: The overall capture rate if the method is used many times.
The sample mean varies from sample to sample, but the method estimate $\pm$ margin of error is used to get an interval based on each sample.
C% of these intervals capture the unknown population mean $µ$ . The actual mean will be located within the interval C% of the time.
Confidence interval = sample mean $\pm$ margin of error
The population mean for a certain variable is estimated by computing a confidence interval for that mean.
If several random samples were collected, the mean for that variable would be slightly different from one sample to another. When researchers estimate population means, instead of providing only one value, they specify a range of values (or an interval) within which this mean is likely to be located.
To obtain this confidence interval, add and subtract the margin of error from the sample mean. This result is the upper limit and the lower limit of the confidence interval.
The confidence interval may be wider or narrower depending on the degree of certainty, or estimation precision, that is required.

The formulas for the confidence interval and margin of error can be combined into one formula.

Quantitative Data: Data that can be counted or measured and averaged across individuals in the population (e.g., height, age, IQ score).
Categorical Data: Data that falls into one of several categories. What can be counted is the proportion of individuals in each category (e.g., gender, hair color, blood type—A, B, AB).