Confidence Intervals for the Mean with Unknown Variance | 方差未知时均值的置信区间

📚 Confidence Intervals for the Mean with Unknown Variance | 方差未知时均值的置信区间

In this revision article, we will explore how to construct a confidence interval for a population mean when the population variance is unknown. This is a key topic in IB Mathematics and appears in both Analysis and Approaches (AA) and Applications and Interpretation (AI).

在本复习文章中，我们将探讨在总体方差未知时如何构建总体均值的置信区间。这是 IB 数学中的一个重要考点，出现在分析与方法（AA）以及应用与解释（AI）课程中。

1. What Is a Confidence Interval? | 引言：什么是置信区间？

A confidence interval is a range of values, calculated from a sample, that is likely to contain the population parameter (such as the mean μ) with a specified level of confidence, usually 90%, 95% or 99%.

置信区间是根据样本计算出的一个数值范围，它以指定的置信水平（通常为 90%、95% 或 99%）包含总体参数（如均值 μ）。

For example, if a 95% confidence interval for a mean is (68.42, 75.58), we say we are 95% confident that the true population mean lies between these two values.

例如，如果均值的 95% 置信区间为 (68.42, 75.58)，我们就说我们有 95% 的把握认为总体均值位于这两个值之间。

The width of the interval depends on three things: the sample size, the variability in the sample, and the confidence level.

区间宽度取决于三个因素：样本量、样本中的变异性以及置信水平。

2. Variance Known vs Variance Unknown | 方差已知与方差未知

When the population variance σ² is known, we use the standard normal distribution (z-distribution) to build the confidence interval for the mean.

当总体方差 σ² 已知时，我们使用标准正态分布（z 分布）来构建均值的置信区间。

However, in most real-life situations the population variance is unknown. We then use the sample standard deviation s as an estimate, and this introduces extra uncertainty.

然而，在大多数实际情况中，总体方差是未知的。此时我们用样本标准差 s 作为估计值，这引入了额外的误差。

Because s varies from sample to sample, using the normal distribution is no longer fully accurate, especially for small samples. In this case, we use the t-distribution.

因为 s 会随样本不同而变化，使用正态分布不再完全准确，特别是对于小样本。在这种情况下，我们使用 t 分布。

3. The t-Distribution | t 分布

The t-distribution, also called Student’s t-distribution, is a symmetric, bell-shaped distribution similar to the normal distribution but with heavier tails.

t 分布，也称学生 t 分布，是一种对称的钟形分布，与正态分布类似，但尾部更厚。

The heavier tails reflect the greater uncertainty caused by estimating the population standard deviation from the sample.

更厚的尾部反映了用样本估计总体标准差所带来的更大不确定性。

As the sample size n increases, the t-distribution approaches the standard normal distribution. For n larger than about 30, the difference is small.

随着样本量 n 增大，t 分布趋近于标准正态分布。当 n 大约超过 30 时，两者差别很小。

However, for small samples, using the t-distribution is essential for obtaining a valid confidence interval.

然而，对于小样本，使用 t 分布对于获得有效的置信区间至关重要。

4. Degrees of Freedom | 自由度

The shape of the t-distribution is determined by the degrees of freedom (df). For a single-sample mean, the degrees of freedom are:

t 分布的形状由自由度（df）决定。对于单样本均值，自由度为：

df = n − 1

where n is the sample size.

其中 n 是样本量。

For example, if a sample has 10 observations, the t-distribution has 9 degrees of freedom.

例如，如果样本有 10 个观测值，则 t 分布的自由度为 9。

The critical t-value depends on both the confidence level and the degrees of freedom. Larger degrees of freedom give t-values closer to z-values.

临界 t 值取决于置信水平和自由度。自由度越大，t 值越接近于 z 值。

Degrees of freedom (df)	Critical t-value for 95% confidence interval
5	2.571
9	2.262
15	2.131
30	2.042
∞ (z-value)	1.960

Notice how the t-values decrease and approach the z-value of 1.960 as df increases.

注意：随着自由度增大，t 值逐渐减小并接近 z 值 1.960。

5. Formula for a Confidence Interval (Unknown Variance) | 方差未知时置信区间公式

When the population variance is unknown, the confidence interval for the population mean μ is calculated using the sample mean x̄, the sample standard deviation s, and the t critical value t*.

当总体方差未知时，总体均值 μ 的置信区间使用样本均值 x̄、样本标准差 s 以及 t 临界值 t* 计算。

We define the standard error of the mean as:

我们定义均值的标准误为：

Standard error = s/√n

The general formula for the confidence interval is:

置信区间的一般公式为：

x̄ ± t* × s/√n

where t* is the critical t-value found from the t-distribution with df = n − 1, corresponding to the desired confidence level.

其中 t* 是从自由度为 df = n − 1 的 t 分布中，根据所需置信水平查得的临界 t 值。

The quantity t* × s/√n is called the margin of error.

t* × s/√n 这一项称为误差范围（margin of error）。

6. Steps for Constructing the Interval | 构建置信区间的步骤

To construct a confidence interval for the mean when the variance is unknown, follow these steps:

在方差未知时，构建均值置信区间的步骤如下：

Check that the conditions are satisfied: the sample is random, independent, and either the sample size is large (n ≥ 30) or the population is approximately normal.

检查条件是否满足：样本是随机的、独立的，并且要么样本量较大（n ≥ 30），要么总体近似正态分布。
Calculate the sample mean x̄ and the sample standard deviation s.

计算样本均值 x̄ 和样本标准差 s。
Determine the degrees of freedom: df = n − 1.

确定自由度：df = n − 1。
Find the critical t-value t* from a t-table or using a graphical calculator, for the chosen confidence level.

使用 t 分布表或图形计算器，根据选定的置信水平查找临界 t 值 t*。
Calculate the standard error: s/√n.

计算标准误：s/√n。
Multiply t* by the standard error to get the margin of error.

将 t* 乘以标准误得到误差范围。
Write the interval as x̄ ± margin of error, or as (x̄ − margin, x̄ + margin).

将区间写为 x̄ ± 误差范围，或写为 (x̄ − margin, x̄ + margin)。

7. Worked Example | 例题演示

A teacher wants to estimate the mean score of an IB Mathematics test. A random sample of 10 students gives a sample mean x̄ = 72 and a sample standard deviation s = 5.

一位老师想估计 IB 数学测试的平均得分。随机抽取 10 名学生，得到样本均值 x̄ = 72，样本标准差 s = 5。

Construct a 95% confidence interval for the population mean score μ.

构建总体平均得分 μ 的 95% 置信区间。

Step 1: n = 10, so df = 10 − 1 = 9.

第一步：n = 10，因此 df = 10 − 1 = 9。

Step 2: From the t-table, the critical value for a 95% confidence level with 9 degrees of freedom is t* = 2.262.

第二步：查 t 分布表，置信水平 95%、自由度 9 对应的临界值为 t* = 2.262。

Step 3: Calculate the standard error:

第三步：计算标准误：

s/√n = 5/√10 ≈ 1.581

Step 4: Calculate the margin of error:

第四步：计算误差范围：

t* × s/√n = 2.262 × 1.581 ≈ 3.58

Step 5: The 95% confidence interval is:

第五步：95% 置信区间为：

x̄ ± 3.58 = 72 ± 3.58 = (68.42, 75.58)

Thus, we are 95% confident that the true mean score lies between 68.42 and 75.58.

因此，我们有 95% 的把握认为真实平均得分位于 68.42 和 75.58 之间。

8. Interpreting the Confidence Interval | 解读置信区间

A common mistake is to say that “there is a 95% probability that μ lies in this interval.” This is not correct from the classical (frequentist) viewpoint.

一个常见错误是说“μ 落在这个区间内的概率为 95%”。从经典（频率学派）角度来看，这是不正确的。

The population mean μ is fixed, not random. The interval computed from the sample is random. A correct interpretation is:

总体均值 μ 是固定的，不是随机的。由样本计算得到的区间是随机的。正确的解释是：

“If we repeated this sampling process many times, approximately 95% of the confidence intervals constructed in this way would contain μ.”

“如果我们多次重复这一抽样过程，约有 95% 以这种方式构建的置信区间会包含 μ。”

In simple terms, we say: “We are 95% confident that the true mean lies in this interval.”

简单来说，我们说：“我们有 95% 的把握认为真实均值位于该区间内。”

9. Conditions and Assumptions | 适用条件与假设

Before using the t-interval, you must check the following conditions:

在使用 t 区间之前，必须检查以下条件：

Random sampling: The sample must be a random sample from the population.

随机抽样：样本必须是来自总体的随机样本。
Independence: Observations must be independent. This is usually satisfied if the sample size is less than 10% of the population, or if the data come from a designed experiment.

独立性：观测值必须相互独立。如果样本量小于总体的 10%，或者数据来自设计好的实验，这个条件通常可以满足。
Normality: The population should be approximately normal, especially for small samples. For n ≥ 30, the Central Limit Theorem allows us to proceed even if the population is not normal.

正态性：总体应近似正态，特别是对于小样本。当 n ≥ 30 时，中心极限定理允许我们在总体并非正态的情况下继续使用。

If these conditions are severely violated, the confidence interval may be misleading.

如果这些条件被严重违反，置信区间可能会产生误导。

10. Using a GDC (Graphical Display Calculator) | 使用图形计算器

In IB examinations, you are often expected to use a GDC to compute a t-confidence interval quickly.

在 IB 考试中，通常要求使用图形计算器快速计算 t 置信区间。

On a TI-84 calculator, the command is:

在 TI-84 计算器上，操作命令为：

STAT → TESTS → TInterval

Choose “Data” if you have raw data, or “Stats” if you already know x̄, s and n.

如果已有原始数据，选择 “Data”；如果已经知道 x̄、s 和 n，则选择 “Stats”。

Enter the sample mean, sample standard deviation, sample size, and confidence level, then select “Calculate” to obtain the interval.

输入样本均值、样本标准差、样本量和置信水平，然后选择 “Calculate” 即可得到置信区间。

Always double-check the degrees of freedom that the calculator uses, and present your interval with appropriate rounding.

始终检查计算器使用的自由度，并以适当的精度呈现你的区间。

11. Common Mistakes and Exam Tips | 常见错误与考试提示

Many students lose marks on this topic due to avoidable errors. Here are the most common mistakes:

许多学生因为可避免的错误而在这一主题上失分。以下是最常见的错误：

Using the z-value (1.96 for 95%) instead of the t-value. Only use z when σ is known or n is very large.

使用 z 值（95% 对应 1.96）而不是 t 值。只有当 σ 已知或 n 非常大时才使用 z。
Using the wrong degrees of freedom. Remember df = n − 1, not n.

使用错误的自由度。记住 df = n − 1，而不是 n。
Confusing sample standard deviation s and population standard deviation σ. The formula for the t-interval uses s.

混淆样本标准差 s 和总体标准差 σ。t 区间公式使用 s。
Forgetting to take the square root of n. The standard error is s/√n, not s/n.

忘记对 n 开平方。标准误是 s/√n，而不是 s/n。
Misinterpreting the confidence level. Do not say “there is a 95% probability that μ is in this interval.”

错误解读置信水平。不要说“μ 在此区间内的概率为 95%”。

Exam tips: always state the formula, show the t* value, and write the interval in the form (lower, upper). Rounding to two or three significant figures is usually acceptable.

考试提示：始终写出公式，标明 t* 值，并以 (下限, 上限) 的形式写出区间。通常保留两位或三位有效数字即可。

12. Summary | 总结

When the population variance is unknown, we use the t-distribution to construct a confidence interval for the population mean.

当总体方差未知时，我们使用 t 分布构建总体均值的置信区间。

The formula is:

公式为：

x̄ ± t* × s/√n

where t* is the critical t-value with n − 1 degrees of freedom.

其中 t* 是具有 n − 1 个自由度的临界 t 值。

The t-distribution is similar to the normal distribution but has heavier tails, and it becomes closer to normal as the sample size increases.

t 分布与正态分布相似，但尾部更厚，并且随着样本量增大而更接近正态分布。

Always check the conditions, use the correct degrees of freedom, and interpret the interval carefully.

始终检查条件，使用正确的自由度，并仔细解读区间。

With practice, you will master this essential IB Statistics skill.

通过练习，你将掌握这一 IB 统计的基本技能。

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Confidence Intervals for the Mean with Unknown Variance | 方差未知时均值的置信区间