Besides estimation by a value, an interval or range of values is also used to estimate population parameter. The interval is determined based on the observation, sample size and confidence level.
Confidence level
Consider random variable `X` with probability density function `f(x)` for continuous random variable, or probability mass function `p(x)` for discrete random variable, and cumulative distribution `F(x)`.
When we use interval `I` to estimate `X`, the probability that `X` belong to `I` is known as confidence level, denotes as (`1–alpha`) or `(1 – alpha)xx100%`. `alpha` (or `alphaxx100%`) is known as significance level, `I` is known as confidence interval. Fig. 1 illustrates these concepts.
Fig. 1 Confidence interval, confidence level, significance level for continuous random variable
We can recognize that the higher the confidence level is, the wider the `I` and the lower the significance level are. To obtain a balance between these factors, we choose frequently `alpha=0,05`.
One-sided & two-sided confidence interval
There are three cases for confidence interval (Fig. 2):
- smaller than a definite value (Fig. 2a),
- greater than a definite value (Fig. 2b),
- a definite interval (Fig. 2c).
Fig. 2 One-sided confidence level (2a & 2b) and two-sided confidence level (2c)
Two first cases are known as one-sided confidence interval, the last case is denoted as two-sided confidence interval and:
`alpha_1+alpha_2=alpha`(8)
In general, we choose :
`alpha_1=alpha_2=alpha//2`(9)
