02 // Visual Math & Finance SolversCI = x̄ ± z* · σ, Central Area = 2·Φ(|z*|) − 1
Visual Confidence Interval ($90\%, 95\%, 99\%$) Critical Z Explorer
A two-sided $(1-\alpha)$ confidence interval captures central probability $1 - \alpha = \Phi(z^*) - \Phi(-z^*) = 2\Phi(z^*) - 1$, with critical values $z^* = 1.645$ ($90\%$), $1.960$ ($95\%$), and $2.576$ ($99\%$).
CI = x̄ ± z* · σ, Central Area = 2·Φ(|z*|) − 1Inputs & Parameters (3)
0±1200
1±0.540
σ
0.1σ±0.013.5σ
Left-Tail CDF Φ(Z)
97.5%
P(X ≤ 119.6) at Z = +1.96σ
Symmetric Central Area
95%
P(80.4 ≤ X ≤ 119.6)
Right-Tail P(X > x)
2.5%
Two-tailed p = 5%
Observed Score (x)
119.6
PDF f(x) = 0.00584
20-Row Comparison Table
Compare across:
| # | Sample Mean Estimate (x̄) | Left-Tail Percentile Φ(Z) | Raw Value x | Central ±|Z| Area |
|---|---|---|---|---|
| 01 | 0 | 97.5 % | 19.6 | 95% |
| 02 | 11 | 97.5 % | 30.6 | 95% |
| 03 | 21 | 97.5 % | 40.6 | 95% |
| 04 | 32 | 97.5 % | 51.6 | 95% |
| 05 | 42 | 97.5 % | 61.6 | 95% |
| 06 | 53 | 97.5 % | 72.6 | 95% |
| 07 | 63 | 97.5 % | 82.6 | 95% |
| 08 | 74 | 97.5 % | 93.6 | 95% |
| 09 | 84 | 97.5 % | 103.6 | 95% |
| 10 | 95 | 97.5 % | 114.6 | 95% |
| 11 | 105 | 97.5 % | 124.6 | 95% |
| 12 | 116 | 97.5 % | 135.6 | 95% |
| 13 | 126 | 97.5 % | 145.6 | 95% |
| 14 | 137 | 97.5 % | 156.6 | 95% |
| 15 | 147 | 97.5 % | 166.6 | 95% |
| 16 | 158 | 97.5 % | 177.6 | 95% |
| 17 | 168 | 97.5 % | 187.6 | 95% |
| 18 | 179 | 97.5 % | 198.6 | 95% |
| 19 | 189 | 97.5 % | 208.6 | 95% |
| 20 | 200 | 97.5 % | 219.6 | 95% |
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Key Analytical Takeaways
- 01Setting $z^* = 1.96$ shades exactly $95.0\%$ of the central bell curve, leaving $\alpha/2 = 2.5\%$ in each rejection tail.
- 02Setting $z^* = 2.58$ widens the interval to capture $99.0\%$ confidence, illustrating the trade-off between precision and certainty.
- 03To halve the margin of error $ME = z^* \cdot \frac{\sigma}{\sqrt{n}}$, you must quadruple the sample size $n$.
Frequently Asked Questions
Why is 1.96 used for a 95% confidence interval instead of 2.0?+
Under the exact standard normal distribution, ±1.9600 standard deviations encloses 95.00% of the area, whereas ±2.0000 encloses 95.45%.
What is the p-value in terms of the shaded tails?+
For a two-tailed Z-test, the p-value is the combined area in both outer tails beyond ±|Z_obs|: p = 2 × (1 − Φ(|Z_obs|)).
How much sample size is needed to cut the margin of error in half?+
Because standard error scales inversely with the square root of sample size (SE = σ / √n), halving your confidence interval width requires quadrupling (4×) the sample size n.