Confidence Intervals

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Confidence intervals are the likeliness of a data point being found in a population between a certain range using information derived from a sample space.


Confidence Interval for the Mean

To calculate the interval that holds the mean of the population, we use the information given for the sample population. The following formula uses the sample mean and the actual standard deviation:

CI Known Variance.PNG


X = Sample Mean

μ = Population Mean

S = Sample Deviation

σ = Population Deviation

The confidence level is the percentage of confidence that the parameter being estimated lies within the interval being calculated. This is calculated as 100 * (1 - α)%, where α is given. It is often that the α is very small, corresponding to a confidence level of 90% or above.

To calculate z, solve the subscript to find the confidence level, and use the Standard Normal Distribution table. Using the probability table involves finding the confidence level (in the body of the table) and the corresponding z value. The rows indicate the start of the z value, and the columns indicate the second decimal place of the z value.

Given that the sample mean is 250, the deviation is 75, the sample size is 50 and the alpha value is 0.05, find the confidence interval for the population mean.

CI Example.PNG

Therefore, the range in which observations of the population mean would be in 95% of the time would be 229.21 and 270.79.

Central Limit Theorem

The previous formulas have been for normal distributions. If the distribution is not normal, the formulas can still be applied as the Central Limit Theory states that a sample of a sufficiently large number of independent random variables will be approximately normally distributed. The larger the sample space, n, the better the normal approximation, with the general rule that n needs to be at least 30.

If the variance is unknown, but the sample is large enough that the CLT holds true, the following formula may be used to calculate the interval for the population mean:

CI Unknown Variance.PNG

If the variance is unknown and the sample is not large enough, a t-distribution is used instead, using the following equation:

CI Unknown Variance T.PNG

To find the t value, find the degrees of freedom, n - 1, and the confidence level. Use these values in the t distribution table to find the corresponding t value, with the degrees of freedom, df, on the vertical axis to the left, and the confidence level on the horizontal axis on the bottom.

Confidence Interval for a Proportion

A confidence interval can also be used to find the interval in which a proportion of a population can be found that have a certain set of characteristics. The following formula can be used:

CI Proportion.PNG


π = Proportion of Population that has a characteristic of interest

p = Proportion of Sample Population that has a characteristic of interest

One-Sided Confidence Interval

A one-sided confidence interval can also be required. The following formulas allow them to be found:

One Sided CI.PNG

One Sided CI 2.PNG


The formulas shown are on the formula sheet, but they need to be rearranged to be useful. If you understand that the difference between the population mean and the sample mean provides the error of the sample mean, and that adding and subtracting that error from the sample mean will give an interval in which the population mean would reside, the rearranging should be intuitive.


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