Interval estimation is a range of values used to estimate a population parameter. It provides more information than a point estimate by offering a range where the parameter is likely to be.

A confidence interval is a type of interval estimate of a population parameter. It indicates a range of values that are believed to contain the parameter with a certain degree of confidence.

For example, a 95% confidence interval means that 95% of the time, the interval will contain the population parameter.

For a normally distributed population, the confidence interval for the population mean \(\mu\) when the population standard deviation \(\sigma\) is known is given by:

\[\mu \pm Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}\]

Where: - \(Z_{\alpha/2}\) is the critical value from the standard normal distribution. - \(n\) is the sample size.

When the population standard deviation is unknown, the confidence interval for the mean is given by:

\[\mu \pm t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}}\]

Where: - \(t_{\alpha/2, n-1}\) is the critical value from the t-distribution. - \(s\) is the sample standard deviation.

Consider a sample of 10 observations: \(2.1, 2.5, 3.0, 2.8, 2.9, 3.1, 2.7, 2.6, 2.8, 3.0\). Assume the population standard deviation is 0.5.

The 95% confidence interval for the mean is calculated as:

\[\mu \pm Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} = 2.75 \pm 1.96 \cdot \frac{0.5}{\sqrt{10}}\]

The result is:

\[ 2.75 \pm 0.31 \]

So, the 95% confidence interval is \((2.44, 3.06)\).

For estimating a population proportion \(p\), the confidence interval is:

\[ \hat{p} \pm Z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]

Where \(\hat{p}\) is the sample proportion and \(n\) is the sample size.

For example, if 60 out of 100 surveyed individuals prefer a certain product, the sample proportion \(\hat{p}\) is 0.6. The 95% confidence interval is:

\[ 0.6 \pm 1.96 \cdot \sqrt{\frac{0.6(1 - 0.6)}{100}} \]