If the sampling distribution of S is approximately normal , we can expect to find an actual sample statistic S lying in the intervals \(\mu_s -\sigma_s\) to \(\mu_s + \sigma_s\), \(\mu_s -2\sigma_s\) to \(\mu_s + 2\sigma_s\) , \(\mu_s -3\sigma_s\) to \(\mu_s + 3\sigma_s\) about 68.27%, 95.45% and 99.73% of times respectively.
So the confidence-limits or fiducial-limits are 68.27%, 95.45% and 99.73%.
* 95% of confidence level S +/- \(1.96\sigma\)
* 99% of confidence level S +/- \(2.58\sigma\) The percentage of confidence is called confidence level. The number 1.96 , 2.58 etc. in the confidence limits are called confidence coefficients or critical values and are denoted by \(z_c\).
Confidence level Table
| Confidence Level | 99.73% | 99.% | 98.% | 96% | 95.45% | 95% | 90% | 80% | 68.27% | 50% |
|---|---|---|---|---|---|---|---|---|---|---|
| \(z_c\) | 3.00 | 2.58 | 2.33 | 2.05 | 2.00 | 1.96 | 1.645 | 1.28 | 1.00 | 0.6745 |
Probable Error The 50% confidence limits of population paramters corresponding to a statistic S are given by $ +- 0.6745\(\sigma_s\). The quantity 0.6745\(\sigma_s\) is called probable error.
Confidence Interval for Means
If \(\bar{X}\) is the sample mean, then 95 % and 99% confidence limits for estimating the mean population \(\mu\),is in the range \(\bar{X}\) - \(z_c\sigma_{\bar{X}}\) and \(\bar{X}\) + \(z_c\sigma_{\bar{X}}\), where \(z_c\) is 1.96 and 2.58 respectively. We can use a generalized formula of confidence interval between \[\bar{X} - z_c\frac{\sigma}{\sqrt{N}} , \bar{X} + z_c\frac{\sigma}{\sqrt{N}}\].
If the sampling is either from an finite population \(N_p\) without replacement then range is given by \[\bar{X} - z_c\frac{\sigma}{\sqrt{N}}\sqrt{\frac{N_p- N}{N_p -1}},\bar{X} + z_c\frac{\sigma}{\sqrt{N}}\sqrt{\frac{N_p- N}{N_p -1}}\].
Confidence Interval for Means
If the statistic S is the proportion of successes in a sample of size N drawn from a binomial population in which p is the proportion of successes, then confidence limit for p are given by \(P - z_c\sigma_p\) and \(P + z_c\sigma_p\), where P is the proportion of successes in the sample size N. The generalized range is given by \[P - z_c \sqrt{\frac{p(1-p)}{N}},P + z_c \sqrt{\frac{p(1-p)}{N}}\]
Confidence Interval for Standard Deviations
The confidence limit for the standard deviation \(\sigma\) of a normally distributed population, as estimated from a sample with standard deviation s,are given by
\[s-z_c\sigma_s,s+z_c\sigma_s\] or \[s-z_c\frac{\sigma}{\sqrt{2N}} , s+z_c\frac{\sigma}{\sqrt{2N}} \]
In computing the confidence limit , we use s or \(\hat{s}\) to estimate \(\sigma\).