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Browse grades and teaching videos01 · Read and understand
What you will learn
- Calculate the standard error of a mean.
- Construct a symmetric z-interval.
- Give a correct frequentist interpretation.
Before you start
Means, square roots, normal standardization, and the distinction between sample and population.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Thirty-six independent waiting times from a normal population have sample mean 82 seconds. The population standard deviation is known to be 12 seconds. Construct a 95% confidence interval for the population mean.
Why this math matters
A sample mean changes when a new sample is collected. Confidence intervals express that sampling uncertainty using a procedure with a known coverage rate under stated assumptions. The population mean is treated as fixed. The intervals vary from sample to sample, and the confidence level describes how often the procedure captures that fixed mean.
Set up the model
A useful answer starts with clear assumptions:
- The thirty-six observations are independent and selected by the stated random sampling process.
- The population is normal and its standard deviation 12 seconds is known, not estimated from these data.
- The 95% central normal critical value is approximately 1.96.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
What does a 95% confidence interval actually promise?
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Thirty-six independent waiting times from a normal population have sample mean 82 seconds. The population standard deviation is known to be 12 seconds. Construct a 95% confidence interval for the population mean.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
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Measure the mean's sampling variability
SE = σ/√n = 12/√36 = 2 seconds
The sample mean varies less than individual waits. Dividing by √n measures the variability of the statistic rather than the spread of raw observations.
Find the margin of error
margin = 1.96 × 2 = 3.92 seconds
The critical multiplier encloses the central 95% of the standardized sample-mean distribution under this model.
Place the margin around the estimate
82 ± 3.92 ⇒ [78.08, 85.92] seconds
Both endpoints are computed from the observed sample. A new sample would generally produce a different pair of endpoints.
The result
The 95% confidence interval is [78.08, 85.92] seconds under the stated assumptions.
In repeated sampling, approximately 95% of intervals constructed this way would contain the fixed population mean. After observing this interval, frequentist coverage does not assign a 95% posterior probability to that fixed parameter being inside it.
Common mistakes to catch
- The interval is for the population mean, not for 95% of individual waiting times.
- Replacing a known population SD with a sample SD requires reconsidering the method, commonly using a t-interval under suitable conditions.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
If n increases to 144 with the same known SD, what are the SE and 95% margin?
Show a hint
√144 = 12.
Reveal answer and explanation
SE 1 second; margin 1.96 seconds
12/12 = 1. Quadrupling sample size halves this standard error.
Practice 2
Can this margin of error correct a sample chosen only during quiet hours?
Show a hint
Separate sampling variability from selection bias.
Reveal answer and explanation
No
The formula handles variability under the sampling model. A systematically unrepresentative sample violates the basis for the stated population inference.
Take the idea with you
Report the population, sampling procedure, interval method, and assumptions alongside any confidence interval.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: What evidence does a small p-value provide?
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