📚 Hypothesis Testing for the Mean with Known Variance | 已知方差的均值假设检验
In IB Mathematics, hypothesis testing for a population mean when the population variance is known is a core topic in statistics. This procedure, often called a z-test, allows us to make decisions about a population parameter based on sample evidence. Here we will explore the reasoning, steps, and common pitfalls.
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一个核心统计专题。该方法通常称为z检验，它使我们能够基于样本证据对总体参数做出决策。下面我们将深入探讨其原理、步骤和常见易错点。
1. The Purpose of Hypothesis Testing | 假设检验的目的
Hypothesis testing is a formal procedure for deciding whether a claim about a population is supported by sample data. In the known-variance case, we assume σ² is known, so the sampling distribution of the sample mean follows a normal distribution exactly if the population is normal, or approximately by the Central Limit Theorem for large samples.
假设检验是一种正式程序，用于判断关于总体的某个主张是否得到样本数据的支持。在方差已知的情形下，我们假设σ²已知，因此样本均值的抽样分布若总体为正态则精确服从正态分布，若样本量较大则根据中心极限定理近似服从正态分布。
The key idea is to compare the observed sample mean with the claimed population mean, taking into account sampling variability. If the observed value is very unlikely under the claim, we reject the claim.
核心思想是比较观测到的样本均值与所声称的总体均值，同时考虑抽样变异性。如果观测值在该主张下极不可能出现，则拒绝该主张。
2. Null and Alternative Hypotheses | 零假设与备择假设
difference. For a mean test, we typically write H₀: μ = μ₀, where μ₀ is a specified value.
零假设用H₀表示，是“无效应”或“无差异”的陈述。对于均值检验，我们通常写为 H₀: μ = μ₀，其中μ₀是特定值。
The alternative hypothesis, denoted H₁ or Hₐ, is what we suspect to be true. It can be two-tailed (μ ≠ μ₀), right-tailed (μ > μ₀), or left-tailed (μ < μ₀). The choice depends on the research question.
备择假设用H₁或Hₐ表示，是我们怀疑为真的情况。它可以是双尾（μ ≠ μ₀）、右尾（μ > μ₀）或左尾（μ < μ₀）。选择取决于研究问题。
3. The Test Statistic for Known Variance | 已知方差下的检验统计量
When σ is known, the test statistic is the z-score of the sample mean under H₀:
当σ已知时，检验统计量是H₀下样本均值的z分数：
z = (x̄ − μ₀) / (σ / √n)
Here x̄ is the sample mean, μ₀ is the hypothesized mean, σ is the population standard deviation, and n is the sample size. The denominator σ/√n is the standard error of the mean.
其中x̄是样本均值，μ₀是假设均值，σ是总体标准差，n是样本量。分母σ/√n是均值的标准误。
This z-statistic measures how many standard errors the sample mean is away from the claimed mean. If |z| is large, the evidence against H₀ is strong.
该z统计量衡量样本均值偏离声称均值多少个标准误。如果|z|很大，则反对H₀的证据很强。
4. Significance Level and Critical Values | 显著性水平与临界值
The significance level, denoted α, is the probability of rejecting H₀ when it is actually true. Common choices are 0.05, 0.01, and 0.10. This value is set before the test.
显著性水平用α表示，是当H₀实际为真时拒绝H₀的概率。常见选择为0.05、0.01和0.10。该值在检验前设定。
he critical values are ±z(α/2); in a one-tailed test, we use z(α) on the appropriate side.
对于给定的α，我们从标准正态分布中查找临界z值。在双尾检验中，临界值为±z(α/2)；在单尾检验中，我们在相应侧使用z(α)。
iled, critical values are ±1.96. If the test is right-tailed, the critical value is 1.645.
例如，若α = 0.05且检验为双尾，临界值为±1.96。若检验为右尾，临界值为1.645。
5. One-Tailed vs Two-Tailed Tests | 单尾与双尾检验
A two-tailed test checks for any difference from μ₀, either larger or smaller. The rejection region is split equally in both tails of the normal distribution.
双尾检验检查与μ₀的任何差异，无论更大还是更小。拒绝区域在正态分布的两尾中均分。
A one-tailed test checks for a directional difference. A right-tailed test has H₁: μ > μ₀, and the rejection region is entirely in the upper tail. A left-tailed test has H₁: μ < μ₀, with rejection in the lower tail.
单尾检验检查方向性差异。右尾检验具有H₁: μ > μ₀，拒绝区域完全在上尾。左尾检验具有H₁: μ < μ₀，拒绝区域在下尾。
Choosing between one- and two-tailed tests must be based on the context, not after seeing the data. Using a one-tailed test when unsure increases the risk of unjustified conclusions.
选择单尾还是双尾必须基于背景，而不是看到数据后决定。不确定时使用单尾检验会增加得出不公正结论的风险。
6. The p-Value Method | p值方法
The p-value is the probability of obtaining a test statistic at least as extreme as the observed one, assuming H₀ is true. A small p-value indicates that the observed sample is unlikely under H₀.
p值是在假设H₀为真的前提下，获得至少与观测值一样极端的检验统计量的概率。较小的p值表明观测样本在H₀下不太可能发生。
, compare the p-value with α: if p-value ≤ α, reject H₀; if p-value > α, do not reject H₀.
进行判断时，将p值与α比较：若p值 ≤ α，则拒绝H₀；若p值 > α，则不拒绝H₀。
For a two-tailed z-test, the p-value is 2 × P(Z ≥ |z|). For a right-tailed test, it is P(Z ≥ z); for a left-tailed test, P(Z ≤ z).
对于双尾z检验，p值为2 × P(Z ≥ |z|)。对于右尾检验，为P(Z ≥ z)；对于左尾检验，为P(Z ≤ z)。
7. Step-by-Step Procedure | 逐步检验流程
Step 1: State the null and alternative hypotheses.
步骤1：写出零假设和备择假设。
Step 2: Choose the significance level α.
步骤2：选择显著性水平α。
Step 3: Calculate the z-statistic using the sample data.
步骤3：使用样本数据计算z统计量。
Step 4: Determine the critical value(s) or compute the p-value.
步骤4：确定临界值或计算p值。
c falls in the rejection region (or p-value ≤ α), reject H₀. Otherwise, fail to reject H₀.
步骤5：比较并得出结论。若检验统计量落入拒绝区域（或p值 ≤ α），则拒绝H₀；否则不拒绝H₀。
Always state the conclusion in the context of the problem. Avoid saying “accept H₀” — we only say there is insufficient evidence to reject it.
始终结合问题背景陈述结论。避免说“接受H₀”——我们只说没有足够证据拒绝它。
8. Type I and Type II Errors | 第一类与第二类错误
A Type I error occurs when we reject a true null hypothesis. Its probability is exactly α, the significance level.
第一类错误发生在拒绝了一个真实的零假设时。其概率恰好为α，即显著性水平。
A Type II error occurs when we fail to reject a false null hypothesis. Its probability is denoted β. The value of β depends on the true mean, sample size, and α.
第二类错误发生在未能拒绝一个错误的零假设时。其概率记为β。β的值取决于真实均值、样本量和α。
The power of a test is 1 − β, the probability of correctly rejecting a false null hypothesis. Power increases with sample size and with the distance between the true mean and μ₀.
检验的功效为1 − β，即正确拒绝错误零假设的概率。功效随着样本量以及真实均值与μ₀之间距离的增大而增大。
9. Worked Example | 实例分析
A factory claims that the mean weight of its cereal boxes is 500 g. The population standard deviation is known to be 12 g. A sample of 36 boxes has a mean weight of 496 g. Test at the 5% significance level whether the mean weight is different from 500 g.
某工厂声称其麦片盒的平均重量为500克。已知总体标准差为12克。抽取36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不同于500克。
Solution: H₀: μ = 500, H₁: μ ≠ 500. α = 0.05. The test statistic is:
解：H₀: μ = 500，H₁: μ ≠ 500。α = 0.05。检验统计量为：
z = (496 − 500) / (12 / √36) = −4 / 2 = −2
6. Since −2 < −1.96, the z-statistic lies in the rejection region. Therefore we reject H₀.
在α = 0.05的双尾检验中，临界值为±1.96。由于−2 < −1.96，z统计量落入拒绝区域。因此我们拒绝H₀。
Conclusion: There is sufficient evidence that the mean weight is different from 500 g.
结论：有足够证据表明平均重量不同于500克。
is 2 × P(Z ≥ 2) = 2 × 0.0228 = 0.0456, which is less than 0.05, giving the same decision.
或者，p值为2 × P(Z ≥ 2) = 2 × 0.0228 = 0.0456，小于0.05，得到相同结论。
10. Confidence Interval Connection | 与置信区间的联系
A two-tailed hypothesis test at significance level α is equivalent to checking whether the hypothesized mean μ₀ lies inside a (1 − α) confidence interval for the mean.
显著性水平为α的双尾检验等价于检查假设均值μ₀是否落在均值的(1 − α)置信区间内。
The confidence interval for a known variance is x̄ ± z(α/2) × σ/√n. If μ₀ is outside this interval, we reject H₀; if inside, we do not reject.
已知方差的置信区间为 x̄ ± z(α/2) × σ/√n。若μ₀在该区间之外，则拒绝H₀；若在区间内，则不拒绝。
This connection helps interpret results: the confidence interval shows a range of plausible values for μ, while the hypothesis test gives a binary decision.
这种联系有助于解释结果：置信区间显示μ的合理取值范围，而假设检验给出二元决策。
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
Mistake 1: Using the sample standard deviation when the population standard deviation is known. This test requires σ, not s. If σ is unknown, a t-test would be needed instead.
常见错误1：在已知总体标准差时误用样本标准差。本检验要求σ，而非s。若σ未知，则需使用t检验。
Mistake 2: Confusing the rejection region for one- and two-tailed tests. Always draw a diagram to visualise which tail(s) contain the rejection region.
常见错误2：混淆单尾和双尾检验的拒绝区域。务必画图来可视化拒绝区域所在的尾部。
Mistake 3: Stating “accept H₀” when failing to reject. Correct phrasing is “there is insufficient evidence to reject H₀.”
常见错误3：当不拒绝H₀时说“接受H₀”。正确表述是“没有足够证据拒绝H₀”。
Tip: When calculating z, keep at least three decimal places and round only at the end. Use the normal distribution table carefully, and check whether the given z value is the critical value or the test statistic.
技巧：计算z时至少保留三位小数，最后再四舍五入。使用正态分布表时要仔细，并检查给定的z值是临界值还是检验统计量。
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