IS: Confidence Intervals

Confidence intervals are a statistical concept that provides a range of values within which a population parameter is estimated to lie with a certain level of confidence. They are used to quantify the uncertainty associated with estimating population parameters based on sample data.

Confidence Level of the Interval

Denoted by:

\[ Confidence \: Level = 1 – \alpha \]

Where:

\[ 0 \le \alpha \le 1 \]

Equation for all Confidence Intervals:

\[ [Point \: Estimate – Reliability \: Factor * Standard \: Error, Point \: Estimate + Reliability \: Factor * Standard \: Error] \]
\[ [\bar\chi – Reliability \: Factor * \frac{\sigma}{\sqrt{n}}, \bar\chi + Reliability \: Factor * \frac{\sigma}{\sqrt{n}}] \]

The Confidence Intervals for a Population with a known Variance is:

\[ [\bar\chi – Z_{\alpha/2} * \frac{\sigma}{\sqrt{n}}, \bar\chi + Z_{\alpha/2} * \frac{\sigma}{\sqrt{n}}] \]
  • Z comes from the standard normal distribution table.
  • Note that the Point Estimate is the same as the sample mean χ hat.
3.9.The-z-tableDownload

Student’s T Distribution uses a similar formula, but has a T-statistic with this formula:

\[t_{n-1,\alpha} = \frac{\bar\chi – \mu}{s/\sqrt{n}} \]

T-Statistic = Sample Mean – Population Mean / Standard error of the sample

The Confidence Intervals for a Population with an Unknown Variance is:

\[ \bar\chi \pm t_{n-1,\alpha/2} * \frac{S}{\sqrt{n}} \]

Where t is retrieved from the T-Table.

3.11.The-t-tableDownload

Margin of Error (ME) refers to the last part of the Two Confidence Interval Equations above. They are:

\[ Z_{\alpha/2} * \frac{\sigma}{\sqrt{n}} \]
\[ t_{n-1,alpha/2} * \frac{S}{\sqrt{n}} \]

Confidence Intervals with Two Populations

Confidence Intervals with two or more populations can have samples that are Dependent one one another, or Independent of one another.

Dependent Samples

  • before and after situations
  • cause and effect situations

The formula for Confidence Interval for the difference of two means (with Dependent samples):

\[ \bar{d} \pm t_{n-1,\alpha/2} * \frac{S_d}{\sqrt{n}} \]
  • d hat is the mean of the differences.
  • Sd is the standard deviation of the differences.

Independent Samples

  • population variance known
  • population variance unknown but assumed to be equal
  • population variance unknown but assumed to be different

Find Confidence Intervals of two Means when Population Variance is Known and Samples are Independent

The formula for the Variance of the difference of two means with Independent samples and known variance.

\[ \sigma^{2}_{diff} = \frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2} \]

The formula for the Confidence Interval of the difference of two means with Independent samples and known variance.

\[ (\bar{x} – \bar{y}) \pm z_{\alpha/2} \sqrt{\frac{\sigma^2_x}{n_x} + \frac{\sigma^2_y}{n_y}} \]

Find Confidence Intervals of two Means when Population Variance is Unknown but Assumed to be Equal and Samples are Independent

The unbiased estimator in this case is called the Pooled Sample Variance. Here is the formula for it:

\[ s^2_p = \frac{(n_x – 1)s^2_x + (n_y – 1)s^2_y}{n_x + n_y – 2} \]

Note that this is based solely on the dataset samples sizes and their Standard Deviations.

The formula for the Confidence Interval of the difference of two means with Independent samples and Unknown Variance that is assumed to be equal.

\[ (\bar{x} – \bar{y}) \pm t_{n_x + n_y – 2, \alpha/2} \sqrt{\frac{s^2_p}{n_x} + \frac{s^2_p}{n_y}} \]

Find Confidence Intervals of two Means when Population Variance is Unknown but Assumed to be Different and Samples are Independent

The formula for the Confidence Interval of the difference of two means with Independent samples and Unknown Variance that is assumed to be different.

\[ (\bar{x} – \bar{y}) \pm t_{v, \alpha/2} \sqrt{\frac{s^2_x}{n_x} + \frac{s^2_y}{n_y}} \]

Where:

\[ v = \frac{(\frac{s^2_x}{n_x} + \frac{s^2_y}{n_y})^2} {\frac{(\frac{s^2_x}{n_x})^2}{n_x – 1} + \frac{(\frac{s^2_y}{n_y})^2}{n_y – 1}} \]