Calculate confidence intervals using either standard deviation or standard error and input confidence level, Z-score, or p-value.

## Understanding Confidence Intervals

A confidence interval (CI) is a range of values used to estimate the true value of a population parameter. For example, a 95% confidence interval for the mean suggests that if we take repeated samples and compute a confidence interval for each sample, approximately 95% of those intervals will contain the population mean.

## Margin of Error, Lower Bound, and Upper Bound

The **Margin of Error** represents the range within which we expect the true population parameter to fall. It is calculated as the product of the Z-score and the standard error or standard deviation. The **Lower Bound** is the lowest value within the confidence interval, calculated by subtracting the Margin of Error from the sample mean. The **Upper Bound** is the highest value within the confidence interval, calculated by adding the Margin of Error to the sample mean. Together, the lower and upper bounds define the confidence interval range.

## Formula for Confidence Interval

The confidence interval for a sample mean \( \bar{x} \) is calculated as:

$ CI = \bar{x} \pm Z \cdot \frac{\sigma}{\sqrt{N}} $

Where:

- \( \bar{x} \): Sample mean
- \( Z \): Z-score corresponding to the confidence level (e.g., 1.96 for 95%)
- \( \sigma \): Population standard deviation (or sample standard deviation)
- \( N \): Sample size

## Real-Life Example: Average Weight of Apples

Imagine you’re a researcher interested in determining the average weight of apples in an orchard. You randomly sample 100 apples and find an average (mean) weight of 150 grams with a standard deviation of 20 grams. If you want to calculate a 95% confidence interval for the mean weight of all apples in the orchard, follow these steps:

- Since the confidence level is 95%, use a Z-score of 1.96 (common for 95% confidence).
- Calculate the standard error: \( SE = \frac{20}{\sqrt{100}} = 2 \) grams.
- Calculate the margin of error: \( ME = 1.96 \times 2 = 3.92 \) grams.
- Determine the confidence interval by adding and subtracting the margin of error from the sample mean:

$$ CI = 150 \pm 3.92 $$

Therefore, the 95% confidence interval is approximately [146.08, 153.92] grams. This means we are 95% confident that the average weight of all apples in the orchard falls between 146.08 and 153.92 grams.

## Z-values for Common Confidence Levels

Confidence Level (%) | Z-value | p-value |
---|---|---|

70% | 1.04 | 0.30 |

75% | 1.15 | 0.25 |

80% | 1.28 | 0.20 |

85% | 1.44 | 0.15 |

90% | 1.645 | 0.10 |

95% | 1.96 | 0.05 |

98% | 2.33 | 0.02 |

99% | 2.576 | 0.01 |

99.5% | 2.807 | 0.005 |

99.9% | 3.291 | 0.001 |

99.99% | 3.891 | 0.0001 |

99.999% | 4.417 | 0.00001 |

Suf is a senior advisor in data science with deep expertise in Natural Language Processing, Complex Networks, and Anomaly Detection. Formerly a postdoctoral research fellow, he applied advanced physics techniques to tackle real-world, data-heavy industry challenges. Before that, he was a particle physicist at the ATLAS Experiment of the Large Hadron Collider. Now, he’s focused on bringing more fun and curiosity to the world of science and research online.