Confidence Interval

Basic steps for calculating Confidence Interval (CI): (Wikipedia)

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

  1. Identify the sample mean, \bar{x} . While \bar{x} differs from \mu, population mean, they are still calculated the same way: \sum {x_i \over n} .
  2. Identify whether the standard deviation is known, \sigma, or unknown, s.
    • If standard deviation is known then z* is used as the critical value. This value is only dependent on the confidence level for the test. Typical confidence levels are:[24]
99% 2.58
95% 1.96
90% 1.64
    • If the standard deviation is unknown then t* is used as the critical value. This value is dependent on the confidence level (C) for the test and degrees of freedom. The degrees of freedom is found by subtracting one from the number of observations, n-1. The critical value is found from the t-distribution table. In this table the critical value is written as tα(r), where r is the degrees of freedom and \alpha = {1-C \over 2}.
  1. Plug the found values into the appropriate equations:
    • For a known standard deviation: (\bar{x} - z^* {\sigma \over \sqrt{n}}, \bar{x} + z^* {\sigma \over \sqrt{n}})
    • For an unknown standard deviation: (\bar{x} - t^* {\sigma \over \sqrt{n}}, \bar{x} + t^* {\sigma \over \sqrt{n}})
  2. The final step is to interpret the answer. Since the found answer is an interval with an upper and lower bound it is appropriate to state that based on the given data we are __ % (dependent on the confidence level) confident that the true mean of the population is between __ (lower bound) and __ (upper bound).[25]
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