Determining Critical Values in Hypothesis Testing | 假设检验中临界值的确定方法

📚 Determining Critical Values in Hypothesis Testing | 假设检验中临界值的确定方法

In hypothesis testing, the critical value is the boundary that separates the rejection region from the non-rejection region of the null hypothesis. Once you have the test statistic, you compare it to the critical value to decide whether to reject H₀. This article explains how to find those boundaries using significance levels, tail directions, and the correct probability distribution.

在假设检验中，临界值是划分拒绝域与接受域（不拒绝域）的边界。计算出检验统计量后，需要将其与临界值比较，从而决定是否拒绝原假设 H₀。本文将详细讲解如何根据显著性水平、单双尾方向以及正确的概率分布来确定这些边界。

1. What Is a Critical Value? | 什么是临界值？

A critical value is a point on the test statistic’s distribution that corresponds to a specified significance level α. If the test statistic lies beyond this value, the result is statistically significant and H₀ is rejected. The set of all such extreme values is called the critical region or rejection region.

临界值是检验统计量分布上对应给定显著性水平 α 的一个点。如果检验统计量落在该值之外，则结果具有统计学显著性，应拒绝 H₀。所有这类极端值的集合称为临界域或拒绝域。

The critical value depends on three factors: the significance level α, the direction of the test (one-tailed or two-tailed), and the sampling distribution of the test statistic (normal, t, binomial, etc.).

临界值取决于三个因素：显著性水平 α、检验方向（单尾或双尾）、以及检验统计量的抽样分布（正态、t、二项分布等）。

2. Significance Level and Critical Region | 显著性水平与临界域

The significance level α is the probability of making a Type I error, which means rejecting H₀ when it is actually true. Common choices are 0.05, 0.01, and 0.10. The critical region is constructed so that its total area under the probability density curve equals α.

显著性水平 α 是犯第一类错误的概率，即原假设事实上为真时却错误地拒绝它的概率。常用取值有 0.05、0.01 和 0.10。临界域的构造使得其在概率密度曲线下的总面积等于 α。

For a two-tailed test with α = 0.05, the total tail area is 0.05, so each tail gets 0.025. For a one-tailed test, the entire 0.05 goes into one tail. This distinction changes the critical value even when using the same distribution.

对于 α = 0.05 的双尾检验，总尾面积为 0.05，因此每个尾部分得 0.025。对于单尾检验，整个 0.05 都放在一个尾部。即使使用相同的分布，这种区别也会改变临界值。

3. One-Tailed Tests | 单尾检验

A one-tailed test is used when the alternative hypothesis has a direction, such as H₁: μ > μ₀ or H₁: μ < μ₀. For a right-tailed test, the critical region lies in the upper tail; for a left-tailed test, it lies in the lower tail.

当备择假设具有方向时使用单尾检验，例如 H₁: μ > μ₀ 或 H₁: μ < μ₀。右尾检验的临界域位于上尾；左尾检验的临界域位于下尾。

To find the critical value for a right-tailed z-test at α, you solve P(Z ≥ z_critical) = α. For α = 0.05, z_critical = 1.645. For a left-tailed test, z_critical = -1.645 because the normal distribution is symmetric.

对于右尾 z 检验，要求解 P(Z ≥ z_临界) = α 来得到临界值。当 α = 0.05 时，z_临界 = 1.645。对于左尾检验，z_临界 = -1.645，因为正态分布是对称的。

P(Z ≥ z_critical) = α ⇒ z_critical = Φ⁻¹(1 − α)

P(Z ≤ z_critical) = α ⇒ z_critical = Φ⁻¹(α)

4. Two-Tailed Tests | 双尾检验

A two-tailed test is used when the alternative hypothesis is non-directional, such as H₁: μ ≠ μ₀. The critical region is split equally between both tails, so each tail contains α/2 of the probability.

当备择假设无方向时使用双尾检验，例如 H₁: μ ≠ μ₀。临界域等分到两个尾部，因此每个尾部包含 α/2 的概率。

For α = 0.05, each tail has 0.025. The upper critical value satisfies P(Z ≥ z_critical) = 0.025, giving z_critical = 1.96. The lower critical value is -1.96. Thus the rejection region is |Z| > 1.96.

对于 α = 0.05，每个尾部为 0.025。上临界值满足 P(Z ≥ z_临界) = 0.025，得到 z_临界 = 1.96。下临界值为 -1.96。因此拒绝域为 |Z| > 1.96。

P(Z ≥ z_critical) = α/2 and P(Z ≤ -z_critical) = α/2

A common mistake is using 1.645 instead of 1.96 for a two-tailed test at 5% significance. Always remember: for two-tailed tests, divide α by 2 first.

一个常见错误是在 5% 显著性水平的双尾检验中使用 1.645 而不是 1.96。请牢记：对于双尾检验，先将 α 除以 2。

5. Critical Values for the Normal Distribution | 正态分布的临界值

When the population variance is known or the sample size is large, the test statistic follows a standard normal distribution Z ∼ N(0, 1). The critical values are obtained from the standard normal table or calculator.

当总体方差已知或样本量较大时，检验统计量服从标准正态分布 Z ∼ N(0, 1)。临界值可以通过标准正态分布表或计算器求得。

The table below shows common critical values for different significance levels and test directions. These values are worth memorising for the exam.

下表列出了常见显著性水平和检验方向下的临界值。这些数值值得在考试前熟记。

α	One-tailed (right)	One-tailed (left)	Two-tailed
0.10	1.282	-1.282	±1.645
0.05	1.645	-1.645	±1.960
0.01	2.326	-2.326	±2.576

For example, if you conduct a right-tailed test at α = 0.01, the critical value is 2.326. If your test statistic exceeds 2.326, reject H₀.

例如，若在 α = 0.01 下进行右尾检验，临界值为 2.326。如果检验统计量超过 2.326，则拒绝 H₀。

6. Critical Values for the t-Distribution | t 分布的临界值

When the population standard deviation is unknown and the sample size is small (n < 30), use the t-distribution. The critical value depends on the degrees of freedom, df = n - 1. As df increases, the t-distribution approaches the standard normal distribution.

当总体标准差未知且样本量较小（n < 30）时，使用 t 分布。临界值依赖于自由度 df = n - 1。随着自由度增大，t 分布趋近于标准正态分布。

The critical value t_critical is found from a t-table using the appropriate tail probability and df. For a two-tailed test at α = 0.05 with df = 10, look up the column for 0.025 (α/2) and row df = 10, giving t_critical = 2.228.

临界值 t_临界通过 t 分布表查找，需确定对应的尾部概率和自由度。对于 α = 0.05、df = 10 的双尾检验，查找 0.025（α/2）列和 df = 10 行，得到 t_临界 = 2.228。

t_critical = t(α, df) or t_critical = t(α/2, df) for two-tailed tests

Unlike the normal distribution, the t-table is usually one-sided in the sense that it gives the tail probability directly. Be careful to select the correct column: α for one-tailed, α/2 for two-tailed.

与正态分布不同，t 分布表通常直接给出尾概率。请务必选择正确的列：单尾用 α，双尾用 α/2。

7. Critical Values for the Binomial Distribution | 二项分布的临界值

For a hypothesis test about a proportion, the number of successes X follows a binomial distribution X ∼ B(n, p). Because this distribution is discrete, the critical value is the smallest (or largest) integer such that the tail probability is at most α.

对于关于比例的假设检验，成功次数 X 服从二项分布 X ∼ B(n, p)。由于该分布是离散的，临界值是指使得尾部概率不超过 α 的最小（或最大）整数。

For a right-tailed test at significance level α, find the smallest value c such that P(X ≥ c) ≤ α. This c is the critical value. For a left-tailed test, find the largest c such that P(X ≤ c) ≤ α.

对于显著性水平 α 的右尾检验，找到最小的 c 使得 P(X ≥ c) ≤ α。这个 c 就是临界值。对于左尾检验，找到最大的 c 使得 P(X ≤ c) ≤ α。

For a two-tailed binomial test, you need to determine approximate critical regions on both sides. Often the lower and upper bounds are found separately so that the total probability in both tails does not exceed α.

对于双尾二项检验，需要分别确定两侧的近似临界域。通常分别找到下界和上界，使得两侧尾部总概率不超过 α。

Example: If X ∼ B(20, 0.4) and you conduct a right-tailed test at α = 0.05, list the binomial probabilities from the top. Suppose P(X ≥ 12) = 0.056 and P(X ≥ 13) = 0.029. Since 0.029 ≤ 0.05 but 0.056 > 0.05, the critical value is 13.

例如：若 X ∼ B(20, 0.4)，在 α = 0.05 下进行右尾检验，从高到低列出二项概率。假设 P(X ≥ 12) = 0.056，P(X ≥ 13) = 0.029。因为 0.029 ≤ 0.05，而 0.056 > 0.05，所以临界值为 13。

8. Using Statistical Tables and Calculators | 使用统计表与计算器

In A-level exams, you may use a statistical table or a calculator with inverse normal and inverse t functions. The inverse normal function, often labelled INVN or Φ⁻¹, directly gives the z-value for a given left-tail probability.

在 A-level 考试中，你可以使用统计表或带有正态分布和 t 分布反函数的计算器。反正态函数通常标记为 INVN 或 Φ⁻¹，直接给出给定左尾概率对应的 z 值。

For a right-tailed test at α = 0.05, compute INVN(0.95) because the left-tail area is 1 – α = 0.95. The result is 1.645. For a two-tailed test at α = 0.05, compute INVN(0.975) for the upper bound and INVN(0.025) for the lower bound.

对于 α = 0.05 的右尾检验，计算 INVN(0.95)，因为左尾面积为 1 – α = 0.95，结果为 1.645。对于 α = 0.05 的双尾检验，上界计算 INVN(0.975)，下界计算 INVN(0.025)。

For t-distributions, use the inverse t function with degrees of freedom. Some calculators require the tail probability, while others require the cumulative probability. Always check the calculator manual and practise before the exam.

对于 t 分布，使用带自由度的反 t 函数。有些计算器要求输入尾部概率，有些则要求累计概率。务必提前查看计算器说明书并在考前练习。

9. Worked Example: Normal Distribution | 实例：正态分布

A company claims the mean weight of its cereal boxes is 500 g. A sample of 40 boxes has a mean of 495 g. The population standard deviation is known to be 15 g. Test at the 5% significance level whether the mean is less than 500 g (one-tailed left).

某公司声称其麦片盒平均重量为 500 克。抽取 40 盒样本，平均重量为 495 克。已知总体标准差为 15 克。在 5% 显著性水平下检验平均重量是否小于 500 克（左尾检验）。

Step 1: State hypotheses. H₀: μ = 500, H₁: μ < 500. Step 2: Compute the test statistic:

第一步：设立假设。H₀: μ = 500，H₁: μ < 500。第二步：计算检验统计量：

z = (495 − 500) / (15 / √40) = −5 / 2.372 = −2.108

Step 3: Find the critical value. For α = 0.05 left-tailed, z_critical = -1.645. Step 4: Compare. Since -2.108 < -1.645, the test statistic lies in the rejection region. Reject H₀.

第三步：求临界值。对于 α = 0.05 的左尾检验，z_临界 = -1.645。第四步：比较。因为 -2.108 < -1.645，检验统计量落入拒绝域，因此拒绝 H₀。

Thus there is sufficient evidence to support the claim that the mean weight is less than 500 g.

因此有足够证据支持平均重量小于 500 克的说法。

10. Worked Example: Binomial Distribution | 实例：二项分布

A dice manufacturer claims that a die is fair, so P(6) = 1/6. In 30 rolls, a 6 appears 9 times. Test at the 5% significance level whether the die is biased towards showing 6.

某骰子制造商声称骰子是公平的，即 P(6) = 1/6。在 30 次投掷中，出现 6 的次数为 9。在 5% 显著性水平下检验骰子是否偏向于出现 6。

Let X be the number of 6s. H₀: p = 1/6, H₁: p > 1/6. Under H₀, X ∼ B(30, 1/6). This is a right-tailed test. We need the smallest c such that P(X ≥ c) ≤ 0.05.

设 X 为出现 6 的次数。H₀: p = 1/6，H₁: p > 1/6。在 H₀ 下，X ∼ B(30, 1/6)。这是右尾检验。需要找到最小的 c 使得 P(X ≥ c) ≤ 0.05。

Using a binomial table or calculator: P(X ≥ 10) = 0.0684, P(X ≥ 11) = 0.0318. Since 0.0318 ≤ 0.05, the critical value is 11. The observed value 9 is less than 11, so it does not fall in the rejection region.

使用二项分布表或计算器：P(X ≥ 10) = 0.0684，P(X ≥ 11) = 0.0318。因为 0.0318 ≤ 0.05，临界值为 11。观测值 9 小于 11，因此未落入拒绝域。

Conclusion: We do not reject H₀. There is insufficient evidence to say the die is biased towards 6.

结论：不拒绝 H₀。没有足够证据表明骰子偏向于出现 6。

11. Common Mistakes and Pitfalls | 常见错误与陷阱

One frequent error is using the wrong sign or direction. For a left-tailed test, the critical value must be negative; for a right-tailed test, positive. Another error is forgetting to halve α for two-tailed tests.

一个常见错误是使用错误的符号或方向。左尾检验的临界值必须为负，右尾检验必须为正。另一个错误是忘记在双尾检验中将 α 减半。

For discrete distributions like the binomial, do not use the normal approximation blindly. Always check the conditions, such as np and n(1-p) being at least 5, before approximating. Also remember that the binomial critical value is an integer, not a continuous value.

对于二项分布等离散分布，不要盲目使用正态近似。在近似前务必检查条件，例如 np 和 n(1-p) 是否至少为 5。还要记住二项临界值是整数，而不是连续值。

Finally, always write the hypotheses and state the critical value before comparing. In exam marking, the critical value and comparison are key steps; missing them loses marks even if the final conclusion is correct.

最后，在比较前务必写出假设并陈述临界值。在考试评分中，临界值和比较是关键步骤；即使最终结论正确，缺少这些步骤也可能会扣分。

12. Summary | 总结

The critical value is the boundary that determines whether a test statistic is significant. To find it, identify the significance level, determine whether the test is one-tailed or two-tailed, and select the correct probability distribution.

临界值是决定检验统计量是否显著的分界线。确定临界值时，需要明确显著性水平、判断检验是单尾还是双尾，并选择正确的概率分布。

For normal and t-distributions, use inverse probability functions or tables. For binomial distributions, find the smallest integer with tail probability no greater than α. Practising these steps will help you avoid common errors and gain full marks in hypothesis testing questions.

对于正态分布和 t 分布，使用反概率函数或查表。对于二项分布，找到尾部概率不超过 α 的最小整数。多加练习这些步骤，能够帮助你避免常见错误，并在假设检验题目中拿到满分。

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Determining Critical Values in Hypothesis Testing | 假设检验中临界值的确定方法