Two-Tailed Tests | 双尾检验

📚 Two-Tailed Tests | 双尾检验

In Edexcel A Level Mathematics, a two-tailed test is used when we are interested in whether a population parameter has changed in either direction, not just increased or decreased. It appears in hypothesis testing for both binomial and normal distributions, and it requires splitting the significance level equally between the two tails of the sampling distribution.

在 Edexcel A Level 数学中，当我们关心某个总体参数是否在任意方向上发生变化、而不仅仅是增大或减小时，使用双尾检验。它出现在二项分布与正态分布的假设检验中，需要把显著性水平平均分配到抽样分布的两个尾部。

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1. What Is a Two-Tailed Test? | 什么是双尾检验？

A two-tailed test is a hypothesis test in which the alternative hypothesis states that the population parameter is not equal to a specified value. This means we reject the null hypothesis if the test statistic is either unusually small or unusually large.

双尾检验是一种假设检验，其中备择假设表明总体参数不等于某个指定值。这意味着如果检验统计量异常小或异常大，我们都会拒绝原假设。

The rejection region is split into both tails of the sampling distribution, hence the name 'two-tailed' nst H₁: p ≠ 0.5 is two-tailed because both p < 0.5 and p > 0.5 provide evidence against H₀.

拒绝区域被拆分为抽样分布的两个尾部，因此称为“双尾”。例如，检验 H₀: p = 0.5 与 H₁: p ≠ 0.5 就是双尾检验，因为 p < 0.5 和 p > 0.5 都能提供反对 H₀ 的证据。

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2. Stating the Hypotheses | 陈述假设

H₀: μ = 50. The alternative hypothesis uses the not-equal sign: H₁: p ≠ 0.3 or H₁: μ ≠ 50.

对于双尾检验，原假设总是包含等号，例如 H₀: p = 0.3 或 H₀: μ = 50。备择假设使用不等号：H₁: p ≠ 0.3 或 H₁: μ ≠ 50。

You must define p or μ clearly. For a binomial test, p is the probability of success in a single trial. For a normal test, μ is the population mean. Do not write H₁ as 'p > 0.3 and p < 0.3'; it is simply p ≠ 0.3.

你必须清楚地定义 p 或 μ。对于二项分布检验，p 是单次试验中成功的概率；对于正态分布检验，μ 是总体均值。不要将 H₁ 写成“p > 0.3 且 p < 0.3”，它只是 p ≠ 0.3。

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3. Significance Level and Tail Probabilities | 显著性水平与尾部概率

The significance level, usually 5% or 1%, is the maximum probability of rejecting H₀ when it is actually true. In a two-tailed test, this total significance level is divided equally between the lower tail and the upper tail.

显著性水平（通常为 5% 或 1%）是当 H₀ 实际为真时拒绝 H₀ 的最大概率。在双尾检验中，这个总显著性水平被平均分配到下尾和上尾。

At the 5% level, each tail carries 2.5%. You therefore look for the lower 2.5% and upper 2.5% of the sampling distribution to form the critical region.

在 5% 显著性水平下，每个尾部占 2.5%。因此你需要找出抽样分布最低的 2.5% 和最高的 2.5% 来构成临界区域。

P(lower tail) = P(upper tail) = α / 2

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4. Critical Regions for Binomial Tests | 二项分布检验的临界区域

For a binomial test with n trials and null value p₀, the critical region consists of the smallest possible values of X in the lower tail and the largest possible values of X in the upper tail.

对于 n 次试验、原假设值为 p₀ 的二项分布检验，临界区域由下尾中 X 尽可能小的取值和上尾中 X 尽可能大的取值组成。

ecause the binomial distribution is discrete, the tail probability may not be exactly α/2.

若 P(X ≤ a) ≤ α/2，则 X ≤ a 为下临界区域；若 P(X ≥ b) ≤ α/2，则 X ≥ b 为上临界区域。由于二项分布是离散的，尾部概率可能并不恰好等于 α/2。

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5. Using the P-Value Method | 使用P值法

The p-value is the probability of obtaining a result at least as extreme as the observed test statistic, in the direction of the alternative hypothesis. In a two-tailed binomial test, the p-value is twice the probability in the relevant tail.

P 值是在备择假设方向上，得到至少与观测检验统计量同样极端结果的概率。在双尾二项检验中，P 值是相关尾部概率的两倍。

For example, if X ~ p-value is 2 × P(X ≥ 15). If this p-value is less than the significance level, reject H₀.

例如，若 X ~ B(20, 0.5) 且观测值为 15，上尾概率为 P(X ≥ 15)。由于检验是双尾的，P 值为 2 × P(X ≥ 15)。如果该 P 值小于显著性水平，则拒绝 H₀。

A common error is to compare the one-tail probability with α instead of doubling it. Always double the smaller tail probability in a two-tailed test.

常见错误是将单尾概率与 α 比较，而不是将其加倍。在双尾检验中，始终要把较小的尾部概率加倍。

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6. Worked Example: Testing a Coin | 实例：检验一枚硬币

A coin is tossed 20 times and lands heads 15 times. Test at the 5% hether the coin is biased. Let X = number of heads, and let p be the probability of heads.

一枚硬币抛掷 20 次，其中 15 次正面朝上。在 5% 显著性水平下检验硬币是否有偏。设 X = 正面次数，p = 正面概率。

The hypotheses are H₀: p = 0.5 and H₁: p ≠ 0.5. Under H₀, X ~ B(20, 0.5). The observed value is 15.

假设为 H₀: p = 0.5 与 H₁: p ≠ 0.5。在 H₀ 下，X ~ B(20, 0.5)。观测值为 15。

(X = 20) = 0.0207 to 3 significant figures. The two-tailed p-value is 2 × 0.0207 = 0.0414.

我们计算 P(X ≥ 15) = P(X = 15) + ... + P(X = 20) = 0.0207（保留三位有效数字）。双尾 P 值为 2 × 0.0207 = 0.0414。

Since 0.0414 < 0.05, we reject H₀. There is sufficient evidence at the 5% level to suggest the coin is biased.

因为 0.0414 < 0.05，我们拒绝 H₀。在 5% 显著性水平下有足够证据表明硬币有偏。

Using critical regions: for X ~ B(20, 0.5), P(X ≤ 5) = 0.0207 and P(X ≥ 15) = 0.0207. The critical region is X ≤

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