Confidence Interval Formula

The Formula

\text{estimate} \pm \text{margin of error}

When to use: Instead of saying 'the average is 50,' you say 'I'm 95% confident the average is between 47 and 53.' The interval acknowledges uncertainty from sampling.

Quick Example

Poll: 52% support candidate, margin of error \pm 3\%. 95% CI: 49%-55%. True support is probably in this range.

Notation

CI is the confidence interval. z_{\alpha/2} is the critical z-value (1.96 for 95%). \alpha is the significance level. The margin of error is E = z_{\alpha/2} \cdot SE.

What This Formula Means

A confidence interval is a range of values, calculated from sample data, constructed so that the procedure captures the true population parameter a specified percentage of the time (e.g., 95%). It quantifies the uncertainty inherent in using a sample to estimate a population value.

Instead of saying 'the average is 50,' you say 'I'm 95% confident the average is between 47 and 53.' The interval acknowledges uncertainty from sampling.

Formal View

A (1-\alpha)100\% confidence interval for \mu is \bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}. When \sigma is unknown, use \bar{x} \pm t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}}.

Worked Examples

Example 1

hard
A sample of 100 students has a mean test score of \bar{x} = 72 with population standard deviation \sigma = 10. Construct a 95% confidence interval for the population mean.

Solution

  1. 1
    Step 1: For 95% confidence, z^* = 1.96.
  2. 2
    Step 2: Standard error: \text{SE} = \frac{10}{\sqrt{100}} = 1.
  3. 3
    Step 3: CI = \bar{x} \pm z^* \cdot \text{SE} = 72 \pm 1.96(1) = (70.04, 73.96).

Answer

(70.04, 73.96)
A 95% confidence interval means that if we repeated this sampling process many times, about 95% of the intervals constructed would contain the true population mean.

Example 2

hard
A 95% confidence interval for the mean weight of apples is (150g, 170g). Interpret this interval.

Common Mistakes

  • Thinking 95% CI means 95% of data falls there
  • Interpreting as probability for one interval
  • Confusing confidence with probability

Why This Formula Matters

Confidence intervals quantify uncertainty. They're essential for making decisions based on sample data.

Frequently Asked Questions

What is the Confidence Interval formula?

A confidence interval is a range of values, calculated from sample data, constructed so that the procedure captures the true population parameter a specified percentage of the time (e.g., 95%). It quantifies the uncertainty inherent in using a sample to estimate a population value.

How do you use the Confidence Interval formula?

Instead of saying 'the average is 50,' you say 'I'm 95% confident the average is between 47 and 53.' The interval acknowledges uncertainty from sampling.

What do the symbols mean in the Confidence Interval formula?

CI is the confidence interval. z_{\alpha/2} is the critical z-value (1.96 for 95%). \alpha is the significance level. The margin of error is E = z_{\alpha/2} \cdot SE.

Why is the Confidence Interval formula important in Statistics?

Confidence intervals quantify uncertainty. They're essential for making decisions based on sample data.

What do students get wrong about Confidence Interval?

Students say '95% probability the true mean is in this interval.' That is wrong. The true mean is fixed; it is the interval construction process that is 95% reliable.

What should I learn before the Confidence Interval formula?

Before studying the Confidence Interval formula, you should understand: standard error, sampling distribution.

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