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Confidence Level — Definition, Formula & Examples

Confidence level is the percentage of the time your method for building a confidence interval will capture the true population value, if you repeated the sampling many times. Common choices are 90%, 95%, and 99%.

The confidence level CC of an interval procedure is the probability, computed before the sample is drawn, that the random interval it produces contains the true parameter. Equivalently, C=1−αC = 1 - \alpha, where α\alpha is the long-run proportion of intervals that miss the parameter. For a z-interval, CC fixes the critical value z∗z^* through P(−z∗≤Z≤z∗)=CP(-z^* \le Z \le z^*) = C, where ZZ is standard normal.

Key Formula

C=1−α,P(−z∗≤Z≤z∗)=CC = 1 - \alpha, \qquad P(-z^* \le Z \le z^*) = C
Where:
  • CC = Confidence level, written as a decimal (0.95 means 95%)
  • α\alpha = Proportion of intervals expected to miss the parameter in the long run
  • z∗z^* = Critical value that cuts off area α/2\alpha/2 in each tail of the standard normal curve
  • ZZ = A standard normal random variable, mean 0 and standard deviation 1

Worked Example

Problem: A random sample of n=100n = 100 students has mean sleep time xˉ=50\bar{x} = 50 hours per week. The population standard deviation is known to be σ=10\sigma = 10 hours. Build a 95% confidence interval for the true mean μ\mu.
Step 1: Turn the confidence level into a critical value. With C=0.95C = 0.95, you get α=0.05\alpha = 0.05, so each tail holds 0.0250.025. The standard normal value with area 0.9750.975 to its left is about 1.961.96.
α=1−0.95=0.05,z∗≈1.96\alpha = 1 - 0.95 = 0.05, \qquad z^* \approx 1.96
Step 2: Compute the standard error, then the margin of error.
SE=σn=10100=1,ME=z∗⋅SE=1.96⋅1=1.96SE = \frac{\sigma}{\sqrt{n}} = \frac{10}{\sqrt{100}} = 1, \qquad ME = z^* \cdot SE = 1.96 \cdot 1 = 1.96
Step 3: Add and subtract the margin of error from the sample mean.
50±1.96  ⇒  (48.04, 51.96)50 \pm 1.96 \;\Rightarrow\; (48.04,\ 51.96)
Answer: The 95% confidence interval is (48.04,51.96)(48.04, 51.96) hours. About 95% of intervals built this way from repeated samples would contain μ\mu.

Why It Matters

On the AP Statistics exam, free-response questions often ask you to interpret a confidence level in context, and the scoring rewards the repeated-sampling wording. The confidence level also controls a tradeoff. With the sample size fixed, raising it from 95% to 99% raises z∗z^* from about 1.96 to about 2.576, which makes the interval wider and less precise.

Common Mistakes

Mistake: Saying there is a 95% probability that μ\mu lies in the specific interval (48.04,51.96)(48.04, 51.96).
Correction: Once the interval is computed, it either contains μ\mu or it does not. The 95% describes the method: in repeated sampling, about 95% of the intervals it produces capture μ\mu.

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