Confidence Interval

CONFIDENCE INTERVAL

        In statistics, a confidence interval (CI) is a type of interval estimate of a population parameter. It is an observed interval (i.e., it is calculated from the observations), in principle different from sample to sample, that potentially includes the unobservable true parameter of interest. How frequently the observed interval contains the true parameter if the experiment is repeated is called the confidence level. In other words, if confidence intervals are constructed in separate experiments on the same population following the same process, the proportion of such intervals that contain the true value of the parameter will match the given confidence level.

         Where as two-sided confidence limits form a confidence interval, and one-sided limits are referred to as lower/upper confidence bounds (or limits).

        Confidence intervals consist of a range of values (interval) that act as good estimates of the unknown population parameter. However, the interval COMPUTED from a particular sample does not necessarily include the true value of the parameter. When we say, "we are 99% confident that the true value of the parameter is in our confidence interval", we express that 99% of the hypothetically observed confidence intervals will hold the true value of the parameter. After any particular sample is taken, the population parameter is either in the interval, or not. Since the observed data are random samples from the true population, the confidence interval obtained from the data is also random. The 99% confidence level means that 99% of the intervals obtained from such samples will contain the true parameter. The desired level of confidence is set by the researcher (not determined by data). If a corresponding hypothesis test is performed, the confidence level is the complement of the level of significance, i.e. a 95% confidence interval reflects a significance level of 0.05. 

 The confidence interval contains the parameter values that, when tested, should not be rejected with the same sample. Confidence intervals of difference parameters not containing 0 imply that there is a statistically significant difference between the populations.

In applied practice, confidence intervals are typically stated at the 95% confidence level.  However, when presented graphically, confidence intervals can be shown at several confidence levels, for example 90%, 95% and 99%.

        Certain factors may affect the confidence interval size including size of sample, level of confidence, and population variability. A larger sample size normally will lead to a better estimate of the population parameter.

        Confidence intervals were introduced to statistics by Jerzy Neyman in a paper published in 1937.

        For example, a confidence interval can be used to describe how reliable survey results. In a poll of election–voting intentions, the result might be that 40% of respondents intend to vote for a certain party. A 99% confidence interval for the proportion in the whole population having the same intention on the survey might be 30% to 50%. From the same data one may calculate a 90% confidence interval, which in this case might be 37% to 43%. A major factor determining the length of a confidence interval is the size of the sample used in the estimation procedure, for example the number of people taking part in a survey.

        Confidence interval - limits statistic that with a given confidence level γ   will be in the interval when sampling a larger volume. Denoted as

P(θ - ε < x < θ + ε) = γ. A measure of confidence θ assessment considered the probability of γ, that the estimation error | θ - x | will not exceed the specified accuracy ε.

        In practice, choosing a confidence probability of γ sufficiently close to unity values γ = 0.9, γ = 0.95, γ = 0.99.

• confidence interval for the general average, the confidence interval for the variance;

• confidence interval for the standard deviation, the confidence interval for the proportion of the general;

Classification of confidence intervals by type of estimated parameter:

1. The confidence interval for the general average (mathematical expectation);

[pic 1]

2. The confidence interval for the variance:

[pic 2] 

 where s2 - sample variance; Χ2 - quantile of the distribution of Pearson.

3. The confidence interval for the standard deviation;

[pic 3]

4. The confidence interval for the proportion of the general;

[pic 4]

According to the sample type:

1. The confidence interval for an infinite sample;

2. The confidence interval for the final sample;

The formulas for calculating the number of samples at random self-sampling method

The basic breakdown of how to calculate a confidence interval for a population mean is as follows: