A confidence interval can help you interpret data.
A confidence interval in statistics helps analyze the reliability of a population mean estimate. The interval results in two values, low and high, and describes how confident an estimate for a value can be given the available information used to create the estimate. Calculate a confidence interval mathematically in a few different ways, each dependent on the type of data you are analyzing.
Finding a Confidence Interval for Data with a Known Population Standard Deviation
To find the confidence interval when population standard deviation is known, use these equations:
Xbar + z(alpha/2) x (sigma / sqrt(n))
Xbar - z(alpha/2) x (sigma / sqrt(n))
In the equations:
Xbar = sample mean
alpha = level of confidence
z = z-value based on alpha and the normal distribution
sigma = population standard deviation
sqrt = square root function
n = population size
As an example, a company produces wood beams used in the construction of houses. Using a sample of 256 beams, the company determined the average weight the beam can hold is 792 pounds before showing signs of stress. The beams produced have a standard deviation of 25 pounds. Find the 95-percent confidence interval.
Xbar = 792 pounds
alpha = 0.95 or 95%
z(0.95/2) = 1.96 (found using a normal distribution)
sigma = 25 pounds
n = 256
The equation: 792 + 3.0625 and 792 - 3.0625; or 795.0625 and 788.9375.
Finding z-values requires looking up the value on a normal distribution table using the level of confidence divided by 2. To simplify this process, it is best to use some commonly used levels of confidence: 90% yields z = 1.28, 95% yields z = 1.96 and 99% yields z = 2.58. The value of 1.96 takes the place of the entire value z(alpha/2) in the equation above.
Finding a Confidence Interval for Data with an Unknown Population Standard Deviation
To find the confidence interval when the population standard deviation is unknown, use:
Xbar + t(alpha/2) x (s / sqrt(n))
Xbar - t(alpha/2) x (s / sqrt(n))
In the equations:
Xbar = sample mean
alpha = level of confidence
t = t-value based on alpha and the t-distribution
s = sample standard deviation
sqrt = square root function
n = population size
The process is the same as when a population standard deviation is known, instead using a t-distribution and the sample standard deviation. A t-distribution uses both the level of confidence and the degrees of freedom to find the t-value. Degrees of freedom is defined as n-1, and in the previous example, that would be 24. T-values, much like z-values, must be looked up in a table using the level of confidence.
Finding a Confidence Interval for Proportion Data
At times, data can be given as a proportion rather than a set of values. If this is the case, then you will want to use a confidence interval for a proportion. The equations:
p + Z(1-alpha/2) x Sqrt(p x (1 - p) / n)
p - Z(1-alpha/2) x Sqrt(p x (1 - p) / n)
In the equations:
p = proportion
z = z-value found in the normal distribution table
alpha = the 1 - the level of confidence
n = the number of data in the set
As an example, a candidate running for mayor of a small town wishes to know the confidence interval for an upcoming election. Approximately 53 percent of voters in a sample of 300 people favor him for the election. He would like to use the 95-percent level of confidence.
p = 53% or 0.53
n = 300
alpha = 100% - 95% or 1 - 0.95 or 0.05
z(0.05/2) = 1.96, we get this value by looking up the value in a normal distribution table
The confidence interval equation will result in: 0.53 + 0.0692 and 0.53 - 0.0692; or 0.599 and 0.461. This indicates you are 95 percent sure the results of the election will fall between 46.1-69.9 percent.
How a Confidence Interval can Change
When finding a confidence interval, it is important to remember the interval you find is not the only confidence interval available for the data you are analyzing. It is the interval at the level of confidence you choose. If you decide to use a 95-percent level of confidence, the interval is telling you that you are 95-percent confident your estimated data falls between the bounds you found. However, using the z or t distribution, you can find confidence intervals for any level of confidence from 0-100 percent.
Tags: level confidence, confidence interval, confidence interval, normal distribution, standard deviation, population standard deviation, Xbar alpha