Hypothesis Testing for a Population Mean with Known Variance (z-test) | 方差已知时总体均值的假设检验（z检验）

📚 Hypothesis Testing for a Population Mean with Known Variance (z-test) | 方差已知时总体均值的假设检验（z检验）

In IB Mathematics, hypothesis testing for a population mean when the population variance is known is a fundamental application of the normal distribution. This procedure relies on the fact that the sample mean is normally distributed when the population is normal, or approximately normal when the sample size is large by the Central Limit Theorem.

在IB数学中，当总体方差已知时对总体均值进行假设检验，是正态分布的一个基本应用。该过程依赖于如下事实：当总体为正态分布时样本均值服从正态分布，或当样本量足够大时根据中心极限定理近似服从正态分布。

1. Core Concepts: Null and Alternative Hypotheses | 核心概念：原假设与备择假设

A hypothesis is a statement about a population parameter. In this context, we are interested in the population mean μ. We begin by stating two mutually exclusive hypotheses.

假设是关于总体参数的陈述。在此情境下，我们关注总体均值 μ。我们首先提出两个互斥的假设。

The null hypothesis, denoted H₀, is a statement of no effect or no change, typically written as H₀ : μ = μ₀, where μ₀ is a specific claimed value. The alternative hypothesis, denoted H₁ or Hₐ, is what we suspect to be true if the null is rejected. It can take one of three forms, depending on the question.

原假设记作 H₀，表示无效应或无变化，通常写成 H₀ : μ = μ₀，其中 μ₀ 是一个具体的声称值。备择假设记作 H₁ 或 Hₐ，是当原假设被拒绝时我们怀疑为真的陈述。根据问题不同，它有三种形式。

H₁ : μ ≠ μ₀ (two-tailed test / 双尾检验) — testing for any difference from μ₀.

H₁ : μ ≠ μ₀（双尾检验）— 检验与 μ₀ 是否有差异。
H₁ : μ > μ₀ (upper-tailed / right-tailed test / 上尾或右尾检验) — testing for an increase.

H₁ : μ > μ₀（上尾或右尾检验）— 检验是否增大。
H₁ : μ < μ₀ (lower-tailed / left-tailed test / 下尾或左尾检验) — testing for a decrease.

H₁ : μ < μ₀（下尾或左尾检验）— 检验是否减小。

The choice of alternative hypothesis is determined by the wording of the problem: “different from”, “greater than”, or “less than”.

备择假设的选择由问题的措辞决定：“不同于”、“大于”或“小于”。

2. Significance Level and Type I Error | 显著性水平与第一类错误

The significance level, denoted α, is the probability of rejecting H₀ when H₀ is in fact true. This is known as a Type I error. Common levels in IB are 5% (α = 0.05), 1% (α = 0.01), and 10% (α = 0.10).

显著性水平记作 α，是当原假设实际为真时拒绝原假设的概率，这称为第一类错误。IB中常见的水平为 5%（α = 0.05）、1%（α = 0.01）和 10%（α = 0.10）。

The critical region is the set of values of the test statistic that lead to rejection of H₀. Its boundaries are called critical values, denoted z* or z_c, and are obtained from the standard normal distribution table.

拒绝域是使检验统计量导致拒绝原假设的取值集合，其边界称为临界值，记作 z* 或 z_c，通过标准正态分布表查得。

For a two-tailed test at significance level α, the total area in both tails is α, so each tail has area α/2. For a right-tailed test, the critical value is z_α such that P(Z > z_α) = α. For a left-tailed test, the critical value is -z_α.

对于显著性水平 α 的双尾检验，两尾总面积是 α，因此每尾面积为 α/2。对于右尾检验，临界值为 z_α，满足 P(Z > z_α) = α。对于左尾检验，临界值为 -z_α。

Common critical values: z_0.05 = 1.645, z_0.025 = 1.960, z_0.005 = 2.576

常用临界值：z_0.05 = 1.645，z_0.025 = 1.960，z_0.005 = 2.576

3. The Test Statistic for Known Variance | 方差已知时的检验统计量

When the population variance σ² is known, the sample mean x̄ is normally distributed with mean μ₀ under H₀ and standard deviation σ/√n. Therefore, we standardise x̄ to obtain the z-test statistic.

当总体方差 σ² 已知时，在原假设 H₀ 下，样本均值 x̄ 服从均值为 μ₀、标准差为 σ/√n 的正态分布。因此，对 x̄ 进行标准化得到 z 检验统计量。

z = (x̄ − μ₀) / (σ / √n)

Here, n is the sample size, x̄ is the sample mean, μ₀ is the claimed population mean under H₀, and σ is the known population standard deviation. This statistic follows a standard normal distribution N(0, 1) when H₀ is true.

其中，n 是样本容量，x̄ 是样本均值，μ₀ 是原假设下声称的总体均值，σ 是已知的总体标准差。当原假设为真时，该统计量服从标准正态分布 N(0, 1)。

It is crucial to use the population standard deviation, not the sample standard deviation. If σ is unknown, the t-distribution would be used, which is a different topic.

必须使用总体标准差，而不是样本标准差。如果 σ 未知，则应使用 t 分布，这是另一个不同的主题。

4. Critical Region Approach | 临界值法

In this approach, we compare the calculated test statistic z with the critical value(s). If z falls in the critical region, we reject H₀; otherwise, we fail to reject H₀.

在此方法中，我们将计算出的检验统计量 z 与临界值进行比较。如果 z 落入拒绝域，则拒绝原假设；否则，不能拒绝原假设。

For a two-tailed test with significance level α, we reject H₀ if |z| > z_{α/2}. For a right-tailed test, reject H₀ if z > z_α. For a left-tailed test, reject H₀ if z < −z_α.

对于显著性水平 α 的双尾检验，若 |z| > z_{α/2} 则拒绝 H₀。对于右尾检验，若 z > z_α 则拒绝 H₀。对于左尾检验，若 z < −z_α 则拒绝 H₀。

It is helpful to draw a standard normal curve and shade the critical region before making the decision. This visual step prevents sign errors and helps in interpreting the result.

在做出决定前，画出标准正态曲线并标出拒绝域将很有帮助。这个可视化步骤可以防止符号错误，并有助于解释结果。

Test Type / 检验类型	H₁	Reject H₀ if / 拒绝条件
Two-tailed / 双尾	μ ≠ μ₀	\|z\| > z_{α/2}
Right-tailed / 右尾	μ > μ₀	z > z_α
Left-tailed / 左尾	μ < μ₀	z < −z_α

Always state your conclusion in the context of the problem. “Reject H₀” alone is not a complete answer; you must explain what this means for the original claim.

务必在问题情境中陈述你的结论。单独写“拒绝 H₀”不是完整答案，你必须解释这对原始声称意味着什么。

5. P-value Approach | P值法

The p-value is the probability, assuming H₀ is true, of obtaining a test statistic at least as extreme as the one actually observed. A small p-value indicates that the observed data is unlikely under H₀.

P值是在假设 H₀ 为真的前提下，获得至少与实际观测到的检验统计量一样极端的统计量的概率。P值很小表明观测数据在原假设下不太可能发生。

For a right-tailed test, p = P(Z > z). For a left-tailed test, p = P(Z < z). For a two-tailed test, p = 2 × P(Z > |z|) if z is positive, or equivalently 2 × P(Z < −|z|); note that for symmetric distributions this is simply double the one-tailed probability.

对于右尾检验，p = P(Z > z)。对于左尾检验，p = P(Z < z)。对于双尾检验，p = 2 × P(Z > |z|)（若 z 为正），或等价地 p = 2 × P(Z < −|z|)；注意对于对称分布，这只是一尾概率的两倍。

Decision rule: if p ≤ α, reject H₀; if p > α, do not reject H₀. The p-value approach is often preferred because it gives more information: the smallest significance level at which H₀ would be rejected.

决策规则：若 p ≤ α，拒绝 H₀；若 p > α，不拒绝 H₀。P值法通常更受青睐，因为它提供了更多信息：能够拒绝原假设的最小显著性水平。

In IB examinations, you may use your GDC (e.g., TI-Nspire or Casio) to compute p-values directly from the z statistic. Ensure you understand the syntax and whether the calculator gives one-tailed or two-tailed probabilities.

在IB考试中，你可以使用图形计算器（如 TI-Nspire 或 Casio）直接从 z 统计量计算P值。确保你理解计算器的语法，并区分给出的是单尾还是双尾概率。

6. Worked Example: Two-tailed Test | 例题：双尾检验

Problem: A factory claims that the mean weight of cereal boxes is 500 g. The population standard deviation is known to be 12 g. A sample of 50 boxes has a mean weight of 496 g. Test at the 5% significance level whether the mean weight is different from 500 g.

题目：某工厂声称麦片盒的平均重量为 500 克。已知总体标准差为 12 克。抽取 50 盒样本，平均重量为 496 克。在 5% 显著性水平下检验平均重量是否与 500 克不同。

Solution: Step 1: State the hypotheses. H₀ : μ = 500, H₁ : μ ≠ 500.

解答：第一步：陈述假设。H₀ : μ = 500，H₁ : μ ≠ 500。

Step 2: Compute the test statistic.

第二步：计算检验统计量。

z = (496 − 500) / (12 / √50) = −4 / 1.697 ≈ −2.357

Step 3: Determine the critical value. At α = 0.05, two-tailed, z_{0.025} = 1.960. So the critical region is z < −1.960 or z > 1.960.

第三步：确定临界值。在 α = 0.05 双尾检验中，z_{0.025} = 1.960。因此拒绝域为 z < −1.960 或 z > 1.960。

Step 4: Compare and conclude. Since z ≈ −2.357 < −1.960, z lies in the critical region. We reject H₀.

第四步：比较并得出结论。由于 z ≈ −2.357 < −1.960，z 落入拒绝域。我们拒绝 H₀。

Conclusion: There is sufficient evidence at the 5% level that the mean weight is different from 500 g.

结论：在 5% 显著性水平下有充分证据表明平均重量与 500 克不同。

7. Worked Example: One-tailed Test | 例题：单尾检验

Problem: A light bulb manufacturer claims that the mean lifetime of a bulb is at least 800 hours. The population standard deviation is 40 hours. A sample of 36 bulbs has a mean lifetime of 790 hours. Test at the 1% significance level whether the claim is overstated.

题目：灯泡制造商声称灯泡的平均寿命至少为 800 小时。总体标准差为 40 小时。抽样 36 个灯泡，平均寿命为 790 小时。在 1% 显著性水平下检验该声称是否被夸大。

Since the claim is “at least 800”, the alternative hypothesis should be μ < 800, because we want to test if the true mean is less than 800. Thus H₀ : μ = 800 and H₁ : μ < 800.

由于声称是“至少 800”，备择假设应为 μ < 800，因为我们想检验真实均值是否小于 800。因此 H₀ : μ = 800，H₁ : μ < 800。

z = (790 − 800) / (40 / √36) = −10 / 6.6667 ≈ −1.5

At α = 0.01, left-tailed, the critical value is −z_0.01 = −2.326. Since z ≈ −1.5 > −2.326, we fail to reject H₀.

在 α = 0.01 左尾检验中，临界值为 −z_0.01 = −2.326。由于 z ≈ −1.5 > −2.326，我们不能拒绝 H₀。

Conclusion: There is insufficient evidence at the 1% level to conclude that the mean lifetime is less than 800 hours.

结论：在 1% 显著性水平下，没有足够证据表明平均寿命小于 800 小时。

8. P-value Computation for the Examples | 例题的P值计算

For the two-tailed cereal example, z = −2.357. The one-tailed p-value is P(Z < −2.357) ≈ 0.0092. For two-tailed, p = 2 × 0.0092 ≈ 0.0184. Since 0.0184 < 0.05, we reject H₀. This matches the critical value conclusion.

对于双尾麦片示例，z = −2.357。单尾P值为 P(Z < −2.357) ≈ 0.0092。双尾P值为 p = 2 × 0.0092 ≈ 0.0184。由于 0.0184 < 0.05，我们拒绝 H₀。这与临界值法的结论一致。

For the bulb example, z = −1.5. The p-value for the left-tailed test is P(Z < −1.5) ≈ 0.0668. Since 0.0668 > 0.01, we fail to reject H₀.

对于灯泡示例，z = −1.5。左尾检验的P值为 P(Z < −1.5) ≈ 0.0668。由于 0.0668 > 0.01，我们不能拒绝 H₀。

When using a GDC, you can often compute the p-value directly from the data or from the z statistic. Always double-check whether your calculator computes P(Z < z) or P(Z > z) to avoid off-by-one errors.

使用图形计算器时，通常可以直接从数据或 z 统计量计算P值。始终检查你的计算器计算的是 P(Z < z) 还是 P(Z > z)，以避免方向错误。

9. Relationship Between Confidence Intervals and Two-tailed Tests | 置信区间与双尾检验的关系

There is a direct duality between a two-tailed hypothesis test at significance level α and a (1 − α) × 100% confidence interval for μ. If the value μ₀ lies inside the confidence interval, then we fail to reject H₀; if it lies outside, we reject H₀.

在显著性水平 α 的双尾检验与 μ 的 (1 − α) × 100% 置信区间之间存在直接的对偶关系。如果 μ₀ 位于置信区间内，则不能拒绝 H₀；如果位于区间外，则拒绝 H₀。

For a known variance, the confidence interval for μ is given by:

对于方差已知的情形，μ 的置信区间为：

x̄ ± z_{α/2} × (σ / √n)

In the cereal example, a 95% confidence interval is 496 ± 1.960 × 1.697, i.e. (492.67, 499.33). Since 500 is not in the interval, we reject H₀, consistent with the test result.

在麦片示例中，95% 置信区间为 496 ± 1.960 × 1.697，即 (492.67, 499.33)。由于 500 不在区间内，我们拒绝 H₀，与检验结果一致。

This equivalence is useful in exam questions that ask for both a confidence interval and a hypothesis test. Showing both strengthens your answer.

这种等价性在同时要求置信区间和假设检验的考题中非常有用。将两者都展示出来会使你的答案更有力。

10. Assumptions and Conditions | 假设与条件

To use the z-test for a population mean, the following conditions must hold:

要对总体均值使用 z 检验，必须满足以下条件：

The sample is randomly selected, or at least representative of the population. / 样本是随机选取的，或至少能代表总体。
The population variance σ² is known. This is often assumed in exam problems even when not realistic. / 总体方差 σ² 已知。在考试题目中通常直接假设已知，尽管现实中不一定成立。
Either the population is normally distributed, or the sample size n is large enough (typically n ≥ 30) for the Central Limit Theorem to apply. / 要么总体服从正态分布，要么样本量 n 足够大（通常 n ≥ 30）以使中心极限定理适用。

If the population is normal and σ is known, the z-test is exact for any sample size. If the population is not normal but n is large, the test is approximately valid, and the approximation improves as n increases.

如果总体为正态且 σ 已知，则 z 检验对任意样本量都是精确的。如果总体非正态但 n 较大，则检验近似有效，且近似程度随 n 增大而提高。

Be careful: if the sample size is small and the population is not normal, the z-test may be misleading. In such cases, a non-parametric test may be more appropriate, but that is beyond the IB curriculum.

注意：如果样本量较小且总体非正态，z 检验可能会产生误导。在这种情况下，非参数检验可能更合适，但这超出了IB课程范围。

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

Below are frequent errors made by students when performing hypothesis tests.

以下是学生在进行假设检验时常犯的错误。

Using the sample standard deviation instead of the population standard deviation. The formula requires σ, not s. / 使用样本标准差代替总体标准差。公式中要求 σ，而不是 s。
Mixing up the direction of the alternative hypothesis. Read the question carefully: “greater than” means right-tailed, “less than” means left-tailed, “different” means two-tailed. / 混淆备择假设的方向。仔细阅读题目：“大于”是右尾，“小于”是左尾，“不同”是双尾。
Forgetting to square the standard deviation when given variance. If the variance is σ² = 144, then σ = 12. / 在给出方差时忘记取平方根。如果方差 σ² = 144，则 σ = 12。
Not stating the conclusion in context. Always refer to the original problem: “the mean lifetime” or “the average weight”. / 没有结合情境陈述结论。始终参照原题：“平均寿命”或“平均重量”。
Confusing p ≤ α with p > α. Recheck the inequality: smaller p-values provide stronger evidence against H₀. / 混淆 p ≤ α 与 p > α。再检查不等式：P值越小，反对 H₀ 的证据越强。

In exams, always write out the hypotheses first, then the test statistic formula with values substituted, then the decision rule, and finally a concluding sentence. Proper structure earns method marks even if you make a calculation error.

在考试中，务必先写出假设，然后写出代入数值的检验统计量公式，接着是决策规则，最后是结论句。良好的结构即使在计算出错时也能获得方法分。

12. Summary and Revision Checklist | 总结与复习清单

The z-test for a population mean with known variance is a concise application of the normal distribution. The key steps are: (1) state H₀ and H₁; (2) compute z; (3) find the critical value or p-value; (4) make a decision; (5) write a contextual conclusion.

方差已知时总体均值的 z 检验是正态分布的一个简洁应用。关键步骤是：(1) 陈述 H₀ 和 H₁；(2) 计算 z；(3) 求临界值或P值；(4) 做出决策；(5) 写出结合情境的结论。

Make sure you can:

确保你能够：

Skill / 技能	Confidence / 掌握度
Formulating hypotheses for one-tailed and two-tailed tests / 为单尾和双尾检验提出假设	□
Computing z = (x̄ − μ₀) / (σ/√n) correctly / 正确计算 z = (x̄ − μ₀) / (σ/√n)	□
Finding critical values from the normal table or GDC / 从正态分布表或图形计算器查找临界值	□
Calculating p-values correctly / 正确计算P值	□
Stating conclusions in context / 结合情境陈述结论	□

Review the worked examples, then attempt past-paper questions. The more you practice, the faster you will identify the correct tail and the less likely you are to make sign errors.

复习例题，然后尝试历年真题。练习越多，你就越能快速识别正确的尾部方向，也越不容易出现符号错误。

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Hypothesis Testing for a Population Mean with Known Variance (z-test) | 方差已知时总体均值的假设检验（z检验）