S2 · How to Read Confidence Intervals

Alberta Grade 11 Mathematics

Statistics

Interpreting estimates, confidence levels, and margins of error

What you will learn

• Explain what a confidence interval tells us about an unknown population value.
• Interpret a confidence level without claiming that it is the probability the value lies in one particular interval.
• Identify the estimate, margin of error, and endpoints in a supplied confidence interval.
• Use the context and units to judge what a reported interval means.

1. Review: estimates and populations

• Population means the whole group of interest; sample means the group that provided information.
• An estimate is based on the sample and may differ from the population value.
• Interpret an interval in the context of the question and with the correct units.

2. Interval, estimate, and margin of error

• The margin of error is the distance from the estimate to an endpoint in a centred interval.
• The two endpoints are the estimate decreased and increased by the margin of error.
• A margin of error is not a guarantee that the interval contains the population value.

3. What a confidence level means

• A confidence level describes the long-run performance of the interval-making method.
• Do not describe a 95% confidence level as a 95% chance that a fixed population value is in this particular interval.
• With other conditions held the same, greater confidence generally means a wider interval.

4. A practical interpretation routine

• Name the group, the quantity, and the units before interpreting the numbers.
• Separate what the interval says from what the confidence level says.
• Use the reported values; this expectation asks you to interpret them, not calculate them from raw data.

Worked example

A school start-time survey

1. Find the endpoints
   The margin of error is 4 percentage points on either side of the estimate. The reported estimate is 46%, so the interval extends 4 percentage points below and above it.
   $$46\% - 4 = 42\%,\quad 46\% + 4 = 50\%$$
2. Interpret the interval
   The interval is from 42% to 50%. In context, the report's method gives a range of plausible percentages of all students at this school who prefer a later start time.
   $$42\% \leq p \leq 50\%$$
3. Interpret the confidence level
   If the same kind of sampling and interval-making method were repeated many times, about 95% of the intervals made this way would contain the true percentage for all students at the school. This is not a statement that this one interval has a 95% chance of containing that percentage.

Worked example

A community survey

1. Read the range
   The reported endpoints are 57% and 67%. They give a range of plausible percentages of all residents who support the cycling path, according to the survey's method.
   $$57\% \leq p \leq 67\%$$
2. Relate the range to its estimate
   The estimate, 62%, is 5 percentage points above the lower endpoint and 5 percentage points below the upper endpoint. That distance is the margin of error. $62\% - 57\% = 67\% - 62\% = 5$ percentage points
3. State the confidence meaning
   A 90% confidence level means that about 90% of intervals made by this method over many repetitions would contain the true percentage of residents who support the path. It does not assign a 90% chance to the true percentage being in this one reported interval.

Worked example

Comparing two reports carefully

1. Interpret Report A
   Report A's estimate is 48%. Its margin of error gives an interval from 45% to 51%. The report describes the share of all residents who use a public library each month.
   $$48\% \pm 3\% = [45\%, 51\%]$$
2. Interpret Report B
   Report B's estimate is also 48%, but its margin of error is larger. Its interval runs from 42% to 54%.
   $$48\% \pm 6\% = [42\%, 54\%]$$
3. Compare without overclaiming
   Report B has a wider interval and a higher confidence level. The values are consistent with the idea that a method aiming for higher confidence can give a wider interval, when other conditions are the same. The reports alone do not establish that their other conditions are the same, so do not claim that confidence level is the only reason for the difference.

Common mistakes and how to avoid them

Lesson summary

• A confidence interval is a range of plausible values for a population quantity based on a sample and a method.
• For a centred interval, the estimate is in the middle and the margin of error is the distance to either endpoint.
• The confidence level describes how often the method would produce intervals containing the true value over repeated samples.
• Interpret the interval in context, state its units, and avoid treating the confidence level as a probability for one particular interval.

Check your understanding

Key terms

Population

The whole group a question or survey is about.

Sample

The smaller group that provides information about the population.

Estimate

A value based on sample information that is used to describe a population quantity.

Confidence interval

A range of values made by a method to describe plausible values for a population quantity.

Margin of error

For a centred interval, the distance from the estimate to either endpoint.

Confidence level

The long-run proportion of intervals made by a method that would contain the true population value when the sampling process is repeated.

Percentage point

The difference between two percentages, such as the difference between 46% and 42%.

Continue through Math 20-2

View the complete Math 20-2 Alberta Grade 11 Mathematics curriculum and lessons

• S1 · Interpret normal distributions, standard deviation, and z-scores
• RF1 · Describe quadratic vertices, intercepts, domain, range, and symmetry
• M1 · Apply and compare rates and unit rates in context
• M3 · Relate scale factors to area, surface area, and volume
• G1 · Derive direct proofs involving angles and triangles
• G2 · Solve problems using properties of angles and triangles

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Published by DoAssignment. This reviewed lesson follows Alberta Grade 11 Mathematics (Math 20-2), expectation S2. It is a study resource, not an official curriculum publication.

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