Sampling Distribution of the Sample Proportion Calculator
Calculate probabilities for sampling distributions of sample proportions using normal approximation
Calculate Sampling Distribution Probabilities
True proportion in the population (value between 0 and 1)
Number of observations in the sample
Results
Sampling Distribution Parameters
Probability Result
Formula: μₚ̂ = p, σₚ̂ = √[p(1-p)/n]
Z-score formula: Z = (p̂ - p) / σₚ̂
✓ Normal approximation is valid for these parameters
Example Calculation
Presidential Approval Example
Scenario: 70% of the population approves of the president
Population proportion (p): 0.7
Sample size (n): 500
Question: What's the probability of finding at least 65% approval in a random sample?
Solution
σₚ̂ = √[0.7 × (1-0.7) / 500] = √[0.21 / 500] = 0.0205
Z = (0.65 - 0.7) / 0.0205 = -2.44
P(p̂ > 0.65) = P(Z > -2.44) = 0.9927
Result: 99.27% probability
Normal Approximation Conditions
Success Condition
np ≥ 15
At least 15 expected successes
Failure Condition
n(1-p) ≥ 15
At least 15 expected failures
Random Sampling
Sample must be random
Each observation independent
Key Concepts
Sample proportion (p̂) estimates population proportion (p)
Larger samples give more accurate estimates
Sampling distribution is approximately normal under conditions
Mean of sampling distribution equals population proportion
Understanding Sampling Distribution of Sample Proportions
What is a Sample Proportion?
A sample proportion (p̂) is the fraction of individuals in a sample that have a particular characteristic. It's calculated as the number of successes divided by the sample size: p̂ = x/n.
Sampling Distribution
The sampling distribution of the sample proportion shows all possible values that sample proportions can take and their probabilities. Under certain conditions, this distribution is approximately normal.
Applications
- •Quality control in manufacturing
- •Political polling and surveys
- •Medical research and clinical trials
- •Market research and consumer studies
Key Formulas
Mean: μₚ̂ = p
Standard Deviation: σₚ̂ = √[p(1-p)/n]
Z-Score: Z = (p̂ - p) / σₚ̂
When to Use Normal Approximation
✓ Valid when:
- • np ≥ 15 (at least 15 successes expected)
- • n(1-p) ≥ 15 (at least 15 failures expected)
- • Sample is randomly selected
⚠ Use caution when:
- • Sample size is small (n < 30)
- • Proportion is very close to 0 or 1
- • Conditions above are not met