A statement about the average value of a particular population, inferred from sample data.

The distribution of sample means will be approximately normal if the sample size is large enough (n ≥ 30), even if the population isn't normal.

The value that determines the margin of error based on the desired confidence level and degrees of freedom.

A measure of the statistical accuracy of an estimate, equal to the standard deviation of the theoretical distribution of a large population of such estimates.

A range of values, calculated from sample data, that is likely to contain the true population parameter.

As sample size increases, both the critical value (t*) and the standard error decrease, leading to a narrower confidence interval and a more precise estimate of the population mean.

Use a t-distribution when the population standard deviation is unknown and you are estimating it with the sample standard deviation. Use a z-distribution when the population standard deviation is known.

If the claimed population mean falls within the confidence interval, you cannot reject the claim. If it falls outside, you have reason to doubt the claim.

Random sampling ensures that the sample is representative of the population, which is a key assumption for the validity of the confidence interval. It reduces bias and allows for accurate inferences about the population mean.

Degrees of freedom (df) represent the number of independent pieces of information available to estimate a parameter. In the context of a t-distribution, df = n - 1, where n is the sample size.

$SE = \frac{s}{\sqrt{n}}$ where s is the sample standard deviation and n is the sample size.

Point estimate ± (critical value)(standard error)

df = n - 1, where n is the sample size.

Margin of Error = Critical Value (t*) * Standard Error

$t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$