Estimation of population parameters is an important aspect of Inferential Statistics. Estimates of parameters like mean and proportion which are derived from data collected from samples are often used in statements describing the true parameters of population.
The parameter estimates are of two types, Point and interval estimates. The reliability of a Point estimate as a good estimate is often difficult to find and hence its accuracy is generally questionable, Hence the interval estimates are generally preferred by statisticians over Point estimates.An interval estimate is a range of values to estimate the parameter in question, The interval may or may not contain the parameter. Confidence Interval is an Interval estimate for the parameter which also gives the probability of the true parameter falling in that interval.

The probability that an interval estimate will contain the parameter is expressed as the confidence level.

A confidence interval is a specific interval estimate of a parameter determined by sample data and a specific confidence level of the estimate.

The central limit theorem for sampling distributions is used in determining the confidence interval of a parameter like mean or proportion.

The various formulas used to calculate the confidence interval are based on the sample size and the confidence level used.

Confidence level is a degree of confidence expressed as percents like 90%, 95% and 99% prior assigned to the data before the confidence interval is calculated. There is a general trade off between the confidence level and the range of the confidence interval. When greater confidence levels are assigned the confidence interval becomes wider.
A symbol $Z_{\frac{\alpha }{2}}$ is used in general formulas for confidence interval. The Greek letter α represents 1 - (confidence level expressed as a decimal). The α value corresponding the confidence level 90% = 1 - 0.9 = 0.1 .

$Z_{\frac{\alpha }{2}}$ is the corresponding z score in a standard normal curve with area in each tail value = $\frac{\alpha }{2}$.

Confidence Interval Chart

The chart below gives these Z cores (also sometimes called the critical value) for the commonly used confidence levels.

  Level of
Confidence

1 - α   Critical Value
     $Z_{\frac{\alpha }{2}}$

    90%
 0.90         1.645
    95%
 0.95         1.96
    98%
 0.98         2.33
    99%
 0.99         2.58

These critical values can also be found using any Z-score tables.
The formula that gives the confidence interval of population mean μ when the population standard deviation σ is known is given by
$\overline{X}-Z_{\frac{\alpha }{2}}(\frac{\sigma }{\sqrt{n}})<\mu <\overline{X}+Z_{\frac{\alpha }{2}}(\frac{\sigma }{\sqrt{n}})$where X is the sample mean and n the sample size, and the assumed confidence level = 1 - α.

The value that is added and subtracted to the sample mean to find the upper and lower limit of the confidence interval is called the maximum error and denoted by E.
E = $Z_{\frac{\alpha }{2}}(\frac{\sigma }{\sqrt{n}})$The formula is modified to calculate the confidence interval of population mean when the population variance is unknown and the sample size
n ≥ 30 replacing σ with S the sample standard deviation.
$\overline{X}-Z_{\frac{\alpha }{2}}(\frac{S}{\sqrt{n}})<\mu <\overline{X}+Z_{\frac{\alpha }{2}}(\frac{S}{\sqrt{n}})$When the sample size is small that is n < 30 and the population standard deviation σ is not known, critical value based on student's - t distribution is used.
$\overline{X}-t_{\frac{\alpha }{2}}(\frac{S}{\sqrt{n}})<\mu <\overline{X}+t_{\frac{\alpha }{2}}(\frac{S}{\sqrt{n}})$
The confidence interval for the true (population) proportion p is given by the formula:
$\hat{p}-Z_{\frac{\alpha }{2}}(\sqrt{\frac{\hat{p}\hat{q}}{n}})<p<\hat{p}+Z_{\frac{\alpha }{2}}(\sqrt{\frac{\hat{p}\hat{q}}{n}})$
Where $\hat{p}$ is the sample proportion and $\hat{q}$ = 1 - $\hat{p}$.
where np and nq is each ≥ 5. ($\hat{p}$ and $\hat{q}$ are correspondingly used as point estimators of p and q)

The maximum error of estimate is given by
E = $Z_{\frac{\alpha }{2}}(\sqrt{\frac{\hat{p}\hat{q}}{n}})$
Let us work out few examples to demonstrate the method of calculating the confidence intervals.

Solved Examples

Question 1: The heights of the hypothetical population of Police personnel is normally distributed with a standard deviation of 2 ft. If a sample of size 16 is randomly selected and the mean is computed to be 6.2 ft, calculate the 90% confidence interval for the mean height of the cops.
Solution:
 
Given  that, 
X = 6.2 ft ,  σ = 2ft and n =16    and  α = 1 - 0.90 = 0.10
We know from the critical value tables $Z_{\frac{\alpha }{2}}$ = Z0.05 = 1.645
The Maximum error is calculated using the formula,
E = $Z_{\frac{\alpha }{2}}(\frac{\sigma }{\sqrt{n}})$

   = 1.645 ($\frac{2}{\sqrt{16}}$) = 1.645 x $\frac{1}{2}$ = 0.8225 ≈ 0.823

The Lower limit for the confidence interval = 6.2 - 0.823 = 5.377
The upper limit for the confidence interval = 6.2 + 0.823 = 7.023
Hence the confidence interval for the true mean is
5.377 < μ < 7.023.
 

Question 2: A hypothetical survey of 200 women University students found 62 of them are married. Based on this, calculate the confidence interval for population proportion of married woman students at 95% confidence interval.
Solution:
 
From the info given $\hat{p}$ = $\frac{62}{200}$ = 0.31,   $\hat{q}$ = 1 - 0.31 = 0.69,  n =200,  α = 1 - 095 = 0.05

We also note that both np and nq > 5.
Hence we can use the formula given to calculate the interval estimate at the confidence level = 95%
The critical value to be used from the table is
$Z_{\frac{\alpha }{2}}$ = Z0.025 = 1.96
The maximum error for the interval estimate is

E = $Z_{\frac{\alpha }{2}}$($\sqrt{\frac{\hat{p}\hat{q}}{n}}$)

   = 1.96($\sqrt{\frac{0.31\times 0.69}{200}}$)

   = 1.96 x 0.0327 ≈ 0.064
The Lower limit for the confidence interval = 0.31 - 0.064 = 0.246
The upper limit for the confidence interval = 0.31+ 0.064 = 0.374
Hence the confidence interval for the proportion of married among woman studying in Colleges at 95% confidence level
0.246 < p < 0.374