Confidence intervals are used in statistics for estimating population parameters and are calculated using specialized software. A confidence interval is a range of values that a parameter is likely to fall within with a certain level of confidence. The equation for calculating a confidence interval is provided for normally distributed populations with known means and variances. Confidence intervals are used to determine how well a parameter fits a given data set, and increasing the confidence coefficient narrows the range. Statistical programs are often used for more complex calculations and large datasets.
In statistics, confidence intervals are used as interval estimates for population parameters. They are often used in science and engineering for hypothesis testing, statistical process control, and data analysis. While it is possible to calculate confidence intervals manually, it is usually easier and much faster to use specialized statistics programs or advanced graphical calculators.
If a probability statement of the form P(L≤θ≤U) = 1 – α can be written such that L and U are exclusively functions of the sample data and θ is a parameter, then the interval between L and U is an interval confidence. This definition can be formulated more intuitively and practically by saying that a statement that the parameter θ is in the confidence interval will be true 100(1 – α)% of the time the statement is made. The term (1 – α) is known as the confidence coefficient.
For the case of a normally distributed population with known mean μ and known variance σ2, the confidence interval 100(1 – α) around the mean can be calculated from the equation x – zα/2σ/√n ≤ μ ≤ x + zα/ 2σ/√n, where zα/2 is the upper 100α/2 percentage point of the standard normal distribution curve. This is a simple case, because the true mean and variance of the entire population are usually not known.
Confidence intervals are most often used to determine how well a given parameter fits a given data set. For example, if the confidence interval for a given data set is 45 to 55 with a confidence coefficient of 0.95, it could be argued that any data point that falls within this region belongs to the population with 95% confidence . Increasing the confidence coefficient narrows the range, which means that a smaller range of variables can be more confidently explained. Decreasing the confidence coefficient widens the range but decreases the confidence.
For some applications, such as normally distributed populations with known means and variances, the equations used to calculate confidence intervals are readily available. Statistical tables can be used to find values for zα/2. Other applications, such as data analysis in engineering, require more sophisticated calculation methods. It is usually more practical to use a statistical program to determine confidence intervals for these cases. Statistics programs can be especially useful when the datasets are extremely large and the results need to be presented graphically.
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