Confidence Interval

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Example: a sample of 36 light bulbs lasted 1,200 hours on average. The production's standard deviation is known to be 120 hours. Between which values does the mean life of all bulbs lie, at 95% confidence? Change the values to work out your own case.

Sample mean x̄ Standard deviation σ Sample size n Confidence (%)

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What each term means

Confidence Interval (CI): A range of values that, at a given confidence level (usually 95% or 99%), contains the true population parameter. It expresses the uncertainty in the estimate.
Sample Mean (x̄): The mean computed from the sample. It is the center of the confidence interval and the best estimate of the population parameter.
Plus or Minus (±): Shows that the interval extends both above and below the sample mean, making it symmetric.
Critical Z Value (z_α/2): The value of the standard normal distribution that matches the chosen confidence level. For 95% confidence, z_α/2 ≈ 1.96; for 99%, z_α/2 ≈ 2.58.
Standard Deviation (σ): The population standard deviation, which the formula assumes is known. When it is estimated from the sample itself, z should be replaced by the critical value of the t distribution.
Square Root of n (√n): The square root of the sample size. In the denominator, it shrinks the margin of error as the sample grows, making the estimate more precise.