Confidence Intervals
Understand sample proportions, calculate margins of error, and construct 95% confidence intervals to estimate population parameters.
Sample Proportion
When we take a sample and measure a characteristic, the sample proportion (p̂) estimates the true population proportion (p). Since each sample gives a different p̂, we need a way to express our uncertainty.
p̂ = x/n
where x = number of successes in the sample, n = sample size
For example, if 120 out of 400 surveyed students prefer maths, then p̂ = 120/400 = 0.30 (30%).
The 95% Confidence Interval
A 95% confidence interval gives a range of plausible values for the population proportion. It is calculated using the sample proportion and the margin of error:
95% CI: p̂ ± z* x √p̂(1-p̂)/n
For a 95% confidence level, z* = 1.96 (approximately 2).
The margin of error (E) = 1.96 x √p̂(1-p̂)/n
Interpretation
We are 95% confident that the true population proportion lies within the interval. This means if we repeated the sampling process many times, about 95% of the resulting intervals would contain the true proportion.
Effect of Sample Size on Margin of Error
The margin of error depends on the sample size n. Larger samples produce smaller margins of error and hence narrower (more precise) confidence intervals.
Visual: Margin of Error vs Sample Size (p̂ = 0.5)
To halve the margin of error, you need to quadruple the sample size (since E is proportional to 1/√n).
Key Vocabulary
Confidence Interval
A range of values, calculated from sample data, that is likely to contain the true population parameter.
Margin of Error (E)
The maximum expected difference between the sample proportion and the true population proportion.
Sample Proportion (p̂)
The proportion of successes observed in a sample, used to estimate the population proportion.
Confidence Level
The probability (e.g., 95%) that the interval contains the true parameter if sampling is repeated many times.
Worked Examples
A survey of 500 voters finds 280 support a policy. Calculate the 95% confidence interval for the true proportion.
Step 1: p̂ = 280/500 = 0.56.
Step 2: E = 1.96 x √(0.56 x 0.44 / 500) = 1.96 x √(0.000493) = 1.96 x 0.02220 = 0.0435.
Step 3: CI = (0.56 - 0.0435, 0.56 + 0.0435) = (0.517, 0.604).
Answer: The 95% CI is (0.517, 0.604), or approximately 51.7% to 60.4%.
A sample of 200 finds p̂ = 0.35. What is the margin of error at the 95% confidence level?
Step 1: E = 1.96 x √(0.35 x 0.65 / 200).
Step 2: E = 1.96 x √(0.001138) = 1.96 x 0.03373 = 0.0661.
Answer: The margin of error is approximately 0.066 or 6.6%.
A 95% confidence interval is (0.42, 0.58). Find the sample proportion and margin of error.
Step 1: p̂ = centre of interval = (0.42 + 0.58) / 2 = 0.50.
Step 2: E = half the width = (0.58 - 0.42) / 2 = 0.08.
Answer: p̂ = 0.50 and E = 0.08 (8%).
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
In a sample of 400 people, 160 said yes. What is the sample proportion p̂?
Question 2
What is the z* value used for a 95% confidence interval?
Question 3
If you increase the sample size from 100 to 400, what happens to the margin of error?
Question 4
A 95% confidence interval for a proportion is (0.32, 0.48). What is the sample proportion?
Question 5
Which statement best describes what "95% confident" means?
Key Concepts Summary
- ●The sample proportion p̂ = x/n estimates the population proportion p.
- ●A 95% CI = p̂ ± 1.96 x √(p̂(1-p̂)/n).
- ●Larger samples produce smaller margins of error and narrower confidence intervals.
- ●To halve the margin of error, quadruple the sample size.
- ●"95% confident" means that 95% of similarly constructed intervals would contain the true proportion.