AP Stats Confidence Interval for a Proportion Calculator
Use this calculator to determine a confidence interval for a population proportion based on sample data. This is a fundamental concept in AP Statistics for estimating population parameters.
Understanding Confidence Intervals for a Population Proportion in AP Statistics
In AP Statistics, a confidence interval for a population proportion (p) is a range of plausible values for the true proportion of a characteristic in a population, based on data from a sample. It's a crucial inferential technique used when we want to estimate an unknown population proportion (e.g., the proportion of all voters who support a certain candidate, or the proportion of defective items in a production batch).
What Does a Confidence Interval Tell Us?
A 95% confidence interval, for example, means that if we were to take many samples and construct a confidence interval from each, about 95% of those intervals would contain the true population proportion. It does NOT mean there's a 95% chance the true proportion is within *this specific* interval.
The Formula and Conditions
The formula for a one-sample z-interval for a population proportion is:
Sample Proportion (p̂) ± Critical Value (z*) × Standard Error (SE)
Where:
- p̂ (p-hat) is the sample proportion, calculated as
x / n(number of successes / sample size). - z* is the critical z-value corresponding to the chosen confidence level (e.g., 1.645 for 90%, 1.960 for 95%, 2.576 for 99%).
- SE (Standard Error) is calculated as
sqrt(p̂ * (1 - p̂) / n).
Before constructing a confidence interval, several conditions must be met:
- Random Condition: The data must come from a well-designed random sample or randomized experiment.
- 10% Condition: When sampling without replacement, the sample size (n) should be no more than 10% of the population size (N). This ensures independence.
- Large Counts Condition (Normal Condition): Both
n * p̂ ≥ 10andn * (1 - p̂) ≥ 10. This ensures that the sampling distribution of p̂ is approximately Normal, allowing us to use z-scores.
Interpreting the Results
Once you calculate the interval, you would state it in context. For example, "We are 95% confident that the true proportion of [population characteristic] is between [lower bound]% and [upper bound]%."
Example Calculation:
Let's say a survey of 100 randomly selected students found that 60 of them prefer online learning. We want to construct a 95% confidence interval for the true proportion of all students who prefer online learning.
- Number of Successes (x) = 60
- Sample Size (n) = 100
- Confidence Level = 95% (z* = 1.960)
1. Calculate Sample Proportion (p̂):
p̂ = 60 / 100 = 0.60
2. Check Large Counts Condition:
n * p̂ = 100 * 0.60 = 60 (≥ 10)
n * (1 – p̂) = 100 * (1 – 0.60) = 100 * 0.40 = 40 (≥ 10)
Conditions met.
3. Calculate Standard Error (SE):
SE = sqrt(0.60 * (1 – 0.60) / 100) = sqrt(0.60 * 0.40 / 100) = sqrt(0.24 / 100) = sqrt(0.0024) ≈ 0.04899
4. Calculate Margin of Error (ME):
ME = z* × SE = 1.960 × 0.04899 ≈ 0.0960
5. Construct the Confidence Interval:
Interval = p̂ ± ME = 0.60 ± 0.0960
Lower Bound = 0.60 – 0.0960 = 0.5040
Upper Bound = 0.60 + 0.0960 = 0.6960
The 95% confidence interval is (0.5040, 0.6960). We are 95% confident that the true proportion of students who prefer online learning is between 50.40% and 69.60%.