GPT's Statistical Analysis of Success Rate Improvement (Example 2)

Statistical Analysis of Success Rate Improvement

Baseline success rate: 95% (0.95)
Number of trials: 47
Observed failures: 1
Observed success rate:

$$\frac {46} {47} = 0.979 \text{ (97.9\%) }$$

Expected number of successes:

$$47 \times 0.95 = 44.65$$

We need to determine if this observed success rate is statistically significant.

The standard error (SE) of a proportion is given by:

$$SE = \sqrt{\frac{p(1 - p)}{n}}$$

where:

$$SE = \sqrt{\frac{0.95 \times 0.05}{47}} = \sqrt{\frac{0.0475}{47}} = \sqrt{0.00101} \approx 0.0318$$

To compare our observed success rate (( $\hat{p} = 0.979$ )) against the expected success rate (( $p = 0.95$ )), we calculate the Z-score:

$$Z = \frac{\hat{p} - p}{SE}$$

$$Z = \frac{0.979 - 0.95}{0.0318} = \frac{0.029}{0.0318} \approx 0.91$$

A Z-score of 0.91 corresponds to a p-value of approximately 0.36 in a one-tailed test (or 0.18 in a two-tailed test).

✅ Interpretation:

This means there is a 36% probability that our observed success rate could happen by chance if the true success rate was still 95%.
A typical significance threshold is 5% (p < 0.05).
Since p = 0.36 is much larger than 0.05, we cannot confidently conclude that the improvement is real—it could still be due to random chance.

A 90% confidence interval (CI) for the observed success rate is:

$$\hat{p} \pm Z_{0.95} \times SE$$

Using ( $Z_{0.95} = 1.645$ ):

$$0.979 \pm 1.645 \times 0.0318$$

$$0.979 \pm 0.0524$$

$$[0.926, 1.031]$$

🔹 Key Takeaway:

The lower bound (0.926) is below 0.95, meaning we cannot rule out that the true success rate could still be at or below 95%.
Since the CI overlaps with 95%, our observed improvement is not statistically significant at the 90% confidence level.

While we observed an increase in success rate (97.9% vs. 95%), it does not pass a statistical significance test.
There is a 36% chance that the observed improvement is just a statistical fluke.
Our confidence interval includes 95%, so we cannot conclude with high certainty that the change led to real improvement.
To increase confidence, we need more trials (larger sample size) to reduce statistical uncertainty.

📌 Next Steps: Would you like help designing a better test to confirm the improvement?