A-Level Further Maths: Inference for the Mean of a Normal Distribution with Unknown Variance | A-Level 进阶数学：方差未知时正态分布的均值推断

📚 A-Level Further Maths: Inference for the Mean of a Normal Distribution with Unknown Variance | A-Level 进阶数学：方差未知时正态分布的均值推断

When we carry out statistical inference about a population mean, the population standard deviation σ is often unknown. In A-Level Further Maths, when the parent population is normal and the variance is unknown, we use the sample standard deviation s to estimate σ, and we base our inference on the Student’s t-distribution rather than the standard normal distribution.

当我们对总体均值进行统计推断时，总体标准差 σ 往往是未知的。在 A-Level 进阶数学中，当总体服从正态分布且方差未知时，我们用样本标准差 s 来估计 σ，并基于学生 t 分布而不是标准正态分布进行推断。

1. Why Unknown Variance Changes the Problem | 为什么方差未知会改变问题的性质

If the population variance σ² is known, the sample mean x̄ has a normal distribution, and the statistic z = (x̄ − μ)/(σ/√n) follows N(0, 1). We can then use z-tables to construct confidence intervals or run hypothesis tests.

如果总体方差 σ² 已知，样本均值 x̄ 服从正态分布，统计量 z = (x̄ − μ)/(σ/√n) 服从 N(0, 1)。此时我们可以使用 z 表来构造置信区间或进行假设检验。

When σ² is unknown, it is natural to replace σ with the sample standard deviation s. However, s itself is a random variable that varies from sample to sample, so the new statistic no longer follows a standard normal distribution.

当 σ² 未知时，自然想到用样本标准差 s 代替 σ。但是 s 本身是随样本变化而变化的随机变量，因此新的统计量不再服从标准正态分布。

The result is that the statistic (x̄ − μ)/(s/√n) follows a t-distribution with n − 1 degrees of freedom. The t-distribution is wider and has heavier tails than the normal distribution, reflecting the extra uncertainty caused by estimating σ.

事实上，统计量 (x̄ − μ)/(s/√n) 服从自由度为 n − 1 的 t 分布。t 分布比正态分布更宽、尾部更厚，这反映了估计 σ 所带来的额外不确定性。

2. The Student’s t-Distribution | 学生 t 分布

The t-distribution was developed by William Sealy Gosset, who published under the pseudonym “Student”. It is defined by a single parameter, the degrees of freedom ν.

t 分布由威廉·西利·戈塞特提出，他以笔名 “Student” 发表文章。t 分布由一个参数决定，即自由度 ν。

If Z is a standard normal random variable and V is an independent chi-squared random variable with ν degrees of freedom, then the random variable T = Z / √(V/ν) follows a t-distribution with ν degrees of freedom.

如果 Z 是标准正态随机变量，V 是服从自由度为 ν 的卡方分布的独立随机变量，那么随机变量 T = Z / √(V/ν) 服从自由度为 ν 的 t 分布。

The t-distribution is symmetric about zero, but its tails are thicker than those of the standard normal distribution. As ν increases, the t-distribution approaches the standard normal distribution.

t 分布关于零对称，但其尾部比标准正态分布更肥厚。随着 ν 增大，t 分布趋近于标准正态分布。

For the one-sample mean problem, ν = n − 1 because one parameter, the population mean μ, is replaced by the sample mean x̄ and the sample standard deviation s is computed from the data.

在单样本均值问题中，ν = n − 1，因为总体均值 μ 被样本均值 x̄ 替代，而样本标准差 s 由数据计算得到，损失了一个自由度。

3. Degrees of Freedom and the Sample Variance | 自由度与样本方差

The sample variance is usually calculated as s² = (1/(n−1)) Σ(xᵢ − x̄)². The use of n − 1 in the denominator is important because it makes s² an unbiased estimator of the population variance σ².

样本方差通常计算为 s² = (1/(n−1)) Σ(xᵢ − x̄)²。分母使用 n − 1 非常重要，因为这使得 s² 成为总体方差 σ² 的无偏估计量。

In A-Level Further Maths, you should be familiar with both the population variance formula and the unbiased sample variance formula. The term “unbiased estimate” means that, on average, the sample estimate is equal to the true parameter.

在 A-Level 进阶数学中，你需要熟悉总体方差公式和无偏样本方差公式。”无偏估计” 的意思是，长期平均而言，样本估计值等于真实参数值。

The degrees of freedom measure the amount of independent information in the sample. For a fixed sample size, a larger number of degrees of freedom corresponds to a more reliable estimate of σ.

自由度衡量样本中独立信息的数量。在固定样本量下，自由度越大，说明对 σ 的估计越可靠。

4. The One-Sample t Statistic | 单样本 t 统计量

When the population is normal and σ² is unknown, the key statistic for inference about μ is:

当总体服从正态分布且 σ² 未知时，关于 μ 进行推断的关键统计量为：

t = (x̄ − μ) / (s/√n)

Here x̄ is the sample mean, μ is the proposed population mean, s is the sample standard deviation, and n is the sample size. This statistic measures how many standard errors x̄ is away from μ.

其中 x̄ 是样本均值，μ 是所提出的总体均值，s 是样本标准差，n 是样本容量。该统计量衡量 x̄ 与 μ 相差多少个标准误。

The denominator s/√n is called the standard error of the sample mean. It is an estimate of the variability of x̄ across repeated samples.

分母 s/√n 称为样本均值的标准误。它是对重复抽样中 x̄ 变异程度的一个估计。

Under the null hypothesis, T follows a t-distribution with n − 1 degrees of freedom. We use critical values from the t-table to decide whether the observed value of t is extreme.

在原假设成立时，T 服从自由度为 n − 1 的 t 分布。我们使用 t 表中的临界值来判断所观测到的 t 值是否极端。

5. Hypothesis Tests for μ with Unknown Variance | 方差未知时对 μ 的假设检验

A hypothesis test for the population mean using the t-distribution is called a one-sample t-test. The procedure is similar to a z-test, but the critical values come from a t-table.

使用 t 分布对总体均值进行的假设检验称为单样本 t 检验。其步骤与 z 检验类似，但临界值来自 t 表。

Step 1: State the hypotheses. For a two-tailed test, H₀: μ = μ₀ and H₁: μ ≠ μ₀. For a one-tailed test, H₁ would be μ < μ₀ or μ > μ₀.

第一步：写出假设。对于双尾检验，H₀: μ = μ₀，H₁: μ ≠ μ₀。对于单尾检验，H₁ 为 μ < μ₀ 或 μ > μ₀。

Step 2: Choose the significance level α, usually 5% or 1%.

第二步：选择显著性水平 α，通常为 5% 或 1%。

Step 3: Calculate the test statistic t = (x̄ − μ₀)/(s/√n).

第三步：计算检验统计量 t = (x̄ − μ₀)/(s/√n)。

Step 4: Find the critical value from the t-distribution with n − 1 degrees of freedom, or compute the p-value.

第四步：查自由度为 n − 1 的 t 分布临界值，或计算 p 值。

Step 5: Compare the test statistic with the critical value. If the absolute value of t exceeds the critical value, reject H₀.

第五步：将检验统计量与临界值比较。如果 |t| 超过临界值，则拒绝 H₀。

6. One-Tailed and Two-Tailed Tests | 单尾检验与双尾检验

For a two-tailed test at significance level α, the critical region is split equally between both tails: each tail has probability α/2. The critical values are ±t_{n−1, α/2}.

在显著性水平 α 下进行双尾检验时，拒绝域对称分配到两个尾部，每个尾部概率为 α/2，临界值为 ±t_{n−1, α/2}。

For a one-tailed test, all of α is placed in one tail. If H₁ is μ > μ₀, the critical region is t > t_{n−1, α}. If H₁ is μ < μ₀, the critical region is t < −t_{n−1, α}.

对于单尾检验，所有 α 集中于一个尾部。若 H₁ 为 μ > μ₀，拒绝域为 t > t_{n−1, α}。若 H₁ 为 μ < μ₀，拒绝域为 t < −t_{n−1, α}。

You must decide whether a test is one-tailed or two-tailed before collecting the data. This decision should be based on the research question, not on what the data appear to show.

你必须先确定检验是单尾还是双尾，再进行数据收集。这个决定应基于研究问题，而不是数据看似显示的结果。

7. Worked Example: One-Sample t-Test | 例题：单样本 t 检验

A machine is supposed to fill bottles with 22.0 ml of liquid. A random sample of 10 bottles is taken, and the volumes in ml are:

一台机器应灌装 22.0 ml 液体。随机抽取 10 个瓶子，容量（单位 ml）为：

22.3

21.8

22.1

22.4

21.9

22.2

21.7

22.0

22.5

22.1

Test at the 5% significance level whether the true mean fill volume differs from 22.0 ml.

在 5% 显著性水平下检验真实平均灌装量是否与 22.0 ml 不同。

First calculate the sample mean: x̄ = 22.10 ml. The sample standard deviation is s ≈ 0.2582 ml, with n = 10.

首先计算样本均值：x̄ = 22.10 ml。样本标准差 s ≈ 0.2582 ml，n = 10。

The null and alternative hypotheses are H₀: μ = 22.0 and H₁: μ ≠ 22.0.

原假设和备择假设为 H₀: μ = 22.0，H₁: μ ≠ 22.0。

The test statistic is:

检验统计量为：

t = (22.10 − 22.00)/(0.2582/√10) ≈ 1.225

The degrees of freedom are n − 1 = 9. The 5% two-tailed critical value is t₉(0.025) = 2.262.

自由度为 n − 1 = 9。5% 双尾检验的临界值为 t₉(0.025) = 2.262。

Since |t| ≈ 1.225 < 2.262, we do not reject H₀. There is insufficient evidence to say that the true mean fill volume differs from 22.0 ml.

因为 |t| ≈ 1.225 < 2.262，所以不能拒绝 H₀。没有充分证据表明真实平均灌装量与 22.0 ml 不同。

8. Confidence Interval for μ with Unknown Variance | 方差未知时 μ 的置信区间

A confidence interval gives a range of plausible values for the population mean. When the variance is unknown, the interval is based on the t-distribution.

置信区间给出总体均值的合理取值范围。当方差未知时，该区间基于 t 分布。

The general formula is:

一般公式为：

x̄ ± t_{n−1, α/2} × s/√n

where t_{n−1, α/2} is the critical value from a t-distribution with n − 1 degrees of freedom, chosen so that the total tail probability is α.

其中 t_{n−1, α/2} 是自由度为 n − 1 的 t 分布临界值，选择该值使得两侧尾部总概率为 α。

Using the previous sample, a 95% confidence interval for μ is:

利用前面样本，μ 的 95% 置信区间为：

22.10 ± 2.262 × (0.2582/√10) = 22.10 ± 0.185

This gives a confidence interval from 21.915 to 22.285 ml.

因此置信区间为 21.915 到 22.285 ml。

We interpret this by saying that, over many repeated samples, 95% of intervals constructed in this way would contain the true population mean.

我们这样解释：在大量重复抽样中，以这种方式构造的区间有 95% 会包含真实的总体均值。

9. Relationship Between Hypothesis Tests and Confidence Intervals | 假设检验与置信区间的关系

A two-tailed hypothesis test at significance level α is equivalent to checking whether the proposed value μ₀ lies inside a 100(1−α)% confidence interval.

显著性水平为 α 的双尾假设检验，等价于检查所提出的值 μ₀ 是否位于 100(1−α)% 置信区间内。

In the worked example above, the 95% confidence interval was (21.915, 22.285). Since μ₀ = 22.0 lies inside this interval, we do not reject H₀.

在上述例题中，95% 置信区间为 (21.915, 22.285)。由于 μ₀ = 22.0 位于该区间内，因此我们不拒绝 H₀。

If μ₀ had been outside the interval, the test statistic would have been significant at the 5% level. Confidence intervals therefore provide more information than a binary test result.

如果 μ₀ 落在区间之外，检验统计量在 5% 水平上就会显著。因此置信区间比 “拒绝或不拒绝” 的二元检验结果提供更多信息。

10. When Does the t-Distribution Become the Normal Distribution? | t 分布何时趋近正态分布

For large sample sizes, s becomes a very accurate estimate of σ, and the t-distribution approaches the standard normal distribution.

当样本量很大时，s 成为 σ 的非常精确的估计，t 分布趋近于标准正态分布。

In practice, when n is large, the difference between t and z critical values is very small. Many textbooks suggest that for n > 30 the normal approximation is acceptable, but the t-distribution is still more accurate when the population is normal.

实际上，当 n 较大时，t 临界值与 z 临界值差异很小。许多教材建议当 n > 30 时可以使用正态近似，但在总体为正态分布时，t 分布仍然更加精确。

For A-Level Further Maths, you should use the t-distribution whenever σ² is unknown, regardless of sample size, unless the question specifically allows a normal approximation.

对于 A-Level 进阶数学，只要 σ² 未知，就应使用 t 分布，无论样本量大小，除非题目明确允许使用正态近似。

A common shortcut is: when n is large, t_∞ critical values are the same as z critical values. Many t-tables include a row for infinite degrees of freedom corresponding to the normal distribution.

一个常用技巧是：当 n 很大时，t_∞ 的临界值就是 z 临界值。许多 t 表包含一行 “无穷大” 自由度，对应正态分布。

11. Assumptions and Common Pitfalls | 假设条件与常见误区

For the one-sample t-test to be valid, the data must be a random sample, the observations must be independent, and the population distribution must be approximately normal.

要使单样本 t 检验有效，数据必须是随机样本，观测值必须相互独立，且总体分布必须近似服从正态分布。

For small samples, the normality assumption is especially important. If the population is heavily skewed, the t-test may not be reliable.

对于小样本，正态性假设尤其重要。如果总体严重偏斜，t 检验可能不可靠。

A common mistake is to use s instead of s/√n in the denominator of the test statistic. Always divide s by √n.

一个常见错误是在检验统计量的分母中使用 s，而不是 s/√n。始终要用 s 除以 √n。

Another common mistake is confusing the sample standard deviation s with the standard error s/√n. The standard error is the standard deviation of the sampling distribution of x̄.

另一个常见错误是混淆样本标准差 s 与标准误 s/√n。标准误是 x̄ 抽样分布的标准差。

Also remember that the degrees of freedom are n − 1, not n. Losing one degree of freedom is the price we pay for using the sample to estimate σ.

还要记住自由度是 n − 1，而不是 n。失去一个自由度是我们用样本估计 σ 所付出的代价。

Finally, do not choose a one-tailed test after looking at the data. This inflates the probability of a false positive and is bad statistical practice.

最后，不要在看到数据后才选择单尾检验。这样会增大误报概率，是糟糕的统计实践。

12. Summary and Exam Tips | 总结与考试要点

When the population variance is unknown and the data come from a normal distribution, use the one-sample t statistic rather than the z statistic.

当总体方差未知且数据来自正态分布时，应使用单样本 t 统计量，而不是 z 统计量。

The key formula to remember is t = (x̄ − μ₀)/(s/√n), with n − 1 degrees of freedom.

需要记住的关键公式是 t = (x̄ − μ₀)/(s/√n)，自由度为 n − 1。

Confidence intervals have the form x̄ ± t_{n−1, α/2} × s/√n.

置信区间形式为 x̄ ± t_{n−1, α/2} × s/√n。

Make sure you can read a t-table, identify the correct degrees of freedom, and distinguish between one-tailed and two-tailed critical values.

确保你能读懂 t 表，确定正确的自由度，并区分单尾与双尾临界值。

In exam questions, show all substitution steps clearly. State your hypotheses, write the formula for t, calculate x̄ and s correctly, and compare with the critical value.

在考试题目中，请清楚地写出所有代入步骤。写出假设，写出 t 的公式，正确计算 x̄ 和 s，并与临界值进行比较。

With practice, inference for a normal mean with unknown variance becomes one of the most predictable topics in A-Level Further Maths. Master the t-distribution, and you will be well prepared for questions on confidence intervals and hypothesis testing.

通过练习，方差未知时正态总体均值的推断会成为 A-Level 进阶数学中最容易把握的题型之一。掌握 t 分布，你就能为置信区间和假设检验类问题做好充分准备。

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A-Level Further Maths: Inference for the Mean of a Normal Distribution with Unknown Variance | A-Level 进阶数学：方差未知时正态分布的均值推断