How the Central Limit Theorem Powers Confidence Intervals and Sample Estimates

1. Central Limit Theorem

From the previous lecture we know the Central Limit Theorem: when a sample of size n is drawn from any population with mean μ and variance σ², the sampling distribution of the sample mean approaches a normal distribution with mean μ and variance σ²/n as n becomes large.

Central Limit Theorem: For a population with mean μ and variance σ², a sample of size n yields a sample‑mean distribution that is approximately normal with mean μ and variance σ²/n when n is sufficiently large.

The term standard error (Standard error) refers to the standard deviation of the sampling distribution of the sample mean. It differs from the standard deviation , which measures the dispersion of individual data points within the sample.

Standard deviation reflects the overall spread of the sample data around the sample mean; a smaller value indicates more concentrated data.

Standard error reflects how much the sample mean varies around the population mean; it decreases as the sample size increases, making the sample mean a more reliable estimator of the population mean.

Empirically, a sample size greater than 30 is often considered a large sample.

2. The 3‑σ Rule

The 3‑σ principle for a normal distribution states that approximately 68.27% of values lie within one standard deviation (μ±σ), 95.45% within two (μ±2σ), and 99.73% within three (μ±3σ) of the mean, leaving less than 0.3% outside this range.

Combining this with the Central Limit Theorem, the sample mean of n observations will fall within roughly ±2 standard errors of the population mean with a probability of about 95.4%.

However, the practical problem is not knowing the population mean but estimating it from sample statistics.

One simple approach is point estimation, using the sample mean as an estimate of the population mean. This works well for large samples but can be unreliable for small samples.

To account for sample size, statisticians use interval estimation.

3. Interval Estimation

If a range of about ±2 standard errors around the sample mean captures the population mean with 95% confidence, then the interval formed by the sample mean ±2 standard errors serves as a 95% confidence interval for the population mean.

The width of the confidence interval depends on the standard error: a smaller standard error (larger sample) yields a narrower, more precise interval.

4. Confidence Level and Significance Level

A 95% confidence level implies a 5% chance of error. The diagram below illustrates the proportion of confidence intervals that contain the true mean (blue) versus those that miss it (red).

The confidence level (level of confidence) indicates the degree of certainty, while the significance level (level of significance) represents the probability of making an error.

A 95% confidence level (α = 0.05) corresponds to the following illustration:

5. Large‑Sample Population Mean Estimation

For large samples (n > 30), the (1‑α)% confidence interval for the population mean can be obtained using the standard normal critical value z α/2 :

\[ \bar{X} \pm z_{\alpha/2} \frac{s}{\sqrt{n}} \]

If the population standard deviation is unknown, the sample standard deviation s serves as an approximation.

The sample standard deviation is calculated as:

\[ s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(X_i-\bar{X})^2} \]

6. Summary

This article links the Central Limit Theorem, the 3‑σ rule, interval estimation, and large‑sample mean estimation, providing formulas for confidence intervals and emphasizing the importance of understanding confidence level, confidence interval, and significance level concepts.

References

https://baike.baidu.com/item/置信区间/7442583?fr=aladdin

https://saylordotorg.github.io/text_introductory-statistics/s11-01-large-sample-estimation-of-a-p.html