confidence interval came to us from the field of statistics.This specific range, which is used to estimate the unknown parameters with a high degree of reliability.The easiest way to explain this is with an example.
Suppose you want to explore any random variable, for example, the speed of the server response to a client request.Each time the user dials a specific address, the server responds to it at different speeds.Thus, the test response time is random.So, the confidence interval to determine the boundaries of the parameter, and then it will be possible to assert that with a probability of 95% of the speed server response will be in the range calculated by us.
Or do you need to know how many people are aware of the brand of the company.When the calculated confidence interval, it will be possible, for example, say that with 95% probability the percentage of consumers who are aware of this brand is in the range of 27% to 34%.
this term is closely related to such a value as a confidence level.It represents the probability that the desired parameter is included in the confidence interval.From this value depends on how big will our desired range.The greater the value it receives, the narrower the confidence interval, and vice versa.Typically, it is set at 90%, 95% or 99%.The value of 95% of the most popular.
This indicator also affects the dispersion of observations and sample size.Its definition is based on the assumption that the analyzed attribute obeys a normal distribution law.This statement is also known as the law of Gauss.According to him, this is called the normal distribution of probabilities of a continuous random variable that can describe the probability density.If the assumption of normal distribution proved to be wrong, the assessment may be incorrect.
first deal with how to calculate the confidence interval for the expectation.There are two possible cases.The dispersion (dispersion degree of the random variable) may be known or not.If it is known, our confidence interval is calculated using the following formula:
HSR - t * σ / (sqrt (n)) & lt; = α & lt; = HSR + t * σ / (sqrt (n)), where
α - a sign,
t - option from the table of Laplace distribution,
sqrt (n) - the square root of the sample size,
σ - the square root of the variance.
If the variance is unknown, it can be calculated if we know all the values of the desired trait.To do this, use the following formula:
σ2 = h2sr - (XCP) 2, where
h2sr - the mean value of the squares of the studied trait,
(XCP) 2 - the square of the mean value of the trait.
formula for which in this case is calculated confidence interval slightly changes:
HSR - t * s / (sqrt (n)) & lt; = α & lt; = HSR + t * s / (sqrt (n)), wherein
XCP - sample mean,
α - a sign,
t - parameter, which is located in a table of the Student distribution t = t (ɣ; n-1),
sqrt (n) - the square root of the sample size,
s - the square root of the variance.
Consider this example.We assume that the results of measurements of 7 was determined the average value of the test attribute is 30 and the sampling variance, which is equal to 36. We need to find a probability of 99% confidence interval that contains the true value of the measured parameter.
first define what is the t: t = t (0,99; 7-1) = 3.71.Using the above formula, we get:
XCP - t * s / (sqrt (n)) & lt; = α & lt; = HSR + t * s / (sqrt (n))
30 - 3.71 * 36 / (sqrt(7)) & lt; = α & lt; = 30 + 3.71 * 36 / (sqrt (7))
21.587 & lt; = α & lt; = 38.413
confidence interval for the variance is calculated as is the case with known secondary andwhen there is no data on the mathematical expectation, and we only know the value of a point unbiased estimate of the variance.We shall not give the formula for its calculation, since they are quite complex and, if desired, they can always be found on the net.
We only note that the confidence interval is conveniently determined using Excel or a network service, which is called.
