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CI

Confidence Intervals — Mean, Proportion, and Difference

Three CI formulas: 95% CI for mean [123.1, 133.7], proportion [64.7%, 77.3%], difference between means [1.04, 10.96].

📊 95% CI for mean: [123.1, 133.7]
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📏 17 · Confidence Intervals

Three Essential Confidence Interval Formulas
CI
① 95% CI for a Population Mean
Scenario: n=30 blood pressure readings, M=128.4 mmHg, SD=14.2. Construct a 95% CI.
C I = X ¯ ± t ∗ S D n , S E = 14.2 30 = 14.2 5.477 = 2.593 , t ( 29 , 0.025 ) ∗ = 2.045 C I = 128.4 ± 2.045 × 2.593 = 128.4 ± 5.30 = [ 123.1 , 133.7 ]  mmHg
128.4
Sample Mean
±5.3
Margin of Error
[123.1, 133.7]
95% CI
② 95% CI for a Proportion
Scenario: 142 out of 200 patients are satisfied with care (p̂=71%). Construct a 95% CI.
C I = p ^ ± z ∗ p ^ ( 1 − p ^ ) n , S E = 0.71 × 0.29 200 = 0.001030 = 0.0321 , z ∗ = 1.96 C I = 0.71 ± 1.96 × 0.0321 = 0.71 ± 0.063 = [ 0.647 , 0.773 ]
71.0%
Proportion
±6.3%
Margin
[64.7%, 77.3%]
95% CI
③ 95% CI for a Difference in Means
Scenario: Drug group (n=20, M=85.2, SD=7.4) vs. Placebo (n=20, M=79.6, SD=8.1). Construct a 95% CI for the mean difference.
S D p = 19 ( 7.4 ) 2 + 19 ( 8.1 ) 2 38 = 1038 + 1246 38 = 60.1 = 7.75 S E d i f f = S D p 1 n 1 + 1 n 2 = 7.75 2 20 = 7.75 × 0.316 = 2.45 C I = ( 85.2 − 79.6 ) ± t ( 38 ) ∗ × 2.45 = 6.0 − 2.024 × 2.45 ⇒ [ 1.04 , 10.96  points ]
6.0
Mean Diff.
±4.96
Margin
[1.04, 10.96]
95% CI
✅ CI does not include 0 → the difference is statistically significant at α=.05. The drug improves scores by 1 to 11 points.
APA-7
A 95% confidence interval for the mean difference indicated that the drug group scored 6.0 points higher than the placebo group, 95% CI [1.04, 10.96]. Because the interval excludes zero, the difference is statistically significant at α = .05.

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