A-Level Maths: Using the Inverse Normal Distribution Function | A-Level数学：逆正态分布函数的使用

📚 A-Level Maths: Using the Inverse Normal Distribution Function | A-Level数学：逆正态分布函数的使用

The inverse normal distribution function is a key tool in A-Level Mathematics. It allows us to work backwards from a given probability to find the corresponding value of a normally distributed random variable.

逆正态分布函数是A-Level数学中的一个关键工具。它使我们可以从已知概率反推出服从正态分布的随机变量所对应的取值。

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1. What is the Inverse Normal Distribution Function | 什么是逆正态分布函数

The normal distribution function \(N(\mu, \sigma^2)\) gives the probability that a random variable \(X\) is less than or equal to a value \(x\), written as \(P(X \le x)\). The inverse normal distribution function does the reverse: given a probability \(p\), it finds the value \(x\) such that \(P(X \le x) = p\).

正态分布函数 \(N(\mu, \sigma^2)\) 给出随机变量 \(X\) 小于等于某个值 \(x\) 的概率，记作 \(P(X \le x)\)。逆正态分布函数则相反：给定概率 \(p\)，求满足 \(P(X \le x) = p\) 的 \(x\) 值。

In A-Level, this is often denoted by \(\Phi^{-1}(p)\) for the standard normal distribution. For a general normal distribution, the inverse function is usually written as \(\Phi^{-1}\left(\frac{x-\mu}{\sigma}\right)\).

在A-Level中，对于标准正态分布，通常记作 \(\Phi^{-1}(p)\)。对于一般正态分布，逆函数通常写作 \(\Phi^{-1}\left(\frac{x-\mu}{\sigma}\right)\)。

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2. Standard Normal Distribution and z-values | 标准正态分布与z值

The standard normal distribution has mean \(\mu = 0\) and standard deviation \(\sigma = 1\). Its variable is called \(Z\). The probability \(P(Z \le z)\) is given by the area under the standard normal curve to the left of \(z\).

标准正态分布的均值为 \(\mu = 0\)，标准差为 \(\sigma = 1\)，其变量记为 \(Z\)。概率 \(P(Z \le z)\) 是标准正态曲线下 \(z\) 左侧的面积。

In inverse normal calculations, we first find the \(z\)-value that corresponds to a given probability, then convert to the original variable \(x\) using the formula \(x = \mu + z\sigma\).

在逆正态计算中，我们首先找到与给定概率对应的 \(z\) 值，然后使用公式 \(x = \mu + z\sigma\) 将其转换为原始变量 \(x\)。

\(x = \mu + z\sigma\)

Here \(z\) is the z-score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

其中 \(z\) 是Z分数，\(\mu\) 是均值，\(\sigma\) 是标准差。

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3. Steps from Probability to z-value | 从概率到z值的步骤

To find the \(z\)-value for a given left-tail probability \(p\), follow these steps:

对于给定的左尾概率 \(p\)，求 \(z\) 值的步骤如下：

• Identify whether the given probability is a left-tail, right-tail, or two-tailed probability.

  判断给定概率是左尾、右尾还是双尾概率。

• If it is a left-tail probability, use \(p\) directly.

  如果是左尾概率，直接使用 \(p\)。

• If it is a right-tail probability, use \(1 – p\) as the left-tail probability.

  如果是右尾概率，使用 \(1 – p\) 作为左尾概率。

• If it is a two-tailed probability, split the probability equally: use \( \frac{p}{2} \) for each tail.

  如果是双尾概率，将概率均分：每个尾部使用 \( \frac{p}{2} \)。

Once the correct left-tail probability is known, use the inverse normal function or table to find the \(z\)-value.

一旦知道了正确的左尾概率，就可以使用逆正态函数或表格找到对应的 \(z\) 值。

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4. Using Standard Normal Tables | 使用标准正态分布表

Standard normal tables usually give the area to the left of a positive \(z\)-value. To use them inversely, look for the given probability inside the table and read the corresponding \(z\)-value from the margins.

标准正态分布表通常给出某个正 \(z\) 值左侧的面积。若要逆向使用，就在表中查找给定的概率，然后从表边读出对应的 \(z\) 值。

For example, if \(P(Z < z) = 0.975\), the table shows \(z = 1.96\). This is a well-known critical value used in confidence intervals.

例如，如果 \(P(Z < z) = 0.975\)，查表可得 \(z = 1.96\)。这是一个在置信区间中常用的关键值。

In A-Level exam formulas, the percentage points of the normal distribution are often given in a separate table. Ensure you know whether your table lists left-tail probabilities or right-tail probabilities.

在A-Level考试公式中，正态分布的百分点通常会单独列成一张表。请务必了解你的表给出的是左尾概率还是右尾概率。

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5. Using a Calculator | 使用计算器

Most modern scientific calculators have an inverse normal function. On the Casio ClassWiz, it is usually accessed via MENU > DISTRIBUTION > Inverse Normal, or by pressing SHIFT + DISTR and selecting “Inverse”.

大多数现代科学计算器都有逆正态函数。在卡西欧ClassWiz上，通常通过 MENU > DISTRIBUTION > Inverse Normal 访问，或者按 SHIFT + DISTR 并选择 “Inverse”。

When using the calculator, you must enter the tail setting: left tail, right tail, or central (two-tailed). For a left-tail probability \(p\), the calculator directly gives the \(z\)-value.

使用计算器时，必须输入尾部设置：左尾、右尾或中央（双尾）。对于左尾概率 \(p\)，计算器直接给出 \(z\) 值。

For a two-tailed calculation, if the input probability is the total shaded central area, the calculator returns the value \(a\) such that \(P(-a < Z < a) = p\). If you want a single tail probability \(\alpha\), use the left-tail or right-tail mode.

对于双尾计算，如果输入的概率是中央总面积，计算器返回满足 \(P(-a < Z < a) = p\) 的 \(a\) 值。如果你想要单尾概率 \(\alpha\)，请使用左尾或右尾模式。

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6. General Normal Distribution: From z to x | 一般正态分布：从z值到x值

For a random variable \(X \sim N(\mu, \sigma^2)\), the relationship between \(X\) and \(Z\) is \(Z = \frac{X – \mu}{\sigma}\). Therefore, if we know the \(z\)-value, we can find \(x\) by rearranging: \(x = \mu + z\sigma\).

对于随机变量 \(X \sim N(\mu, \sigma^2)\)，\(X\) 与 \(Z\) 的关系为 \(Z = \frac{X – \mu}{\sigma}\)。因此，如果已知 \(z\) 值，可以通过移项得到 \(x = \mu + z\sigma\)。

Consider the problem: \(X \sim N(50, 10^2)\). Find the value of \(x\) such that \(P(X < x) = 0.9\).

考虑问题：\(X \sim N(50, 10^2)\)。求满足 \(P(X < x) = 0.9\) 的 \(x\) 值。

First find \(z = \Phi^{-1}(0.9) \approx 1.2816\). Then \(x = 50 + 1.2816 \times 10 = 62.816\).

首先求出 \(z = \Phi^{-1}(0.9) \approx 1.2816\)。然后 \(x = 50 + 1.2816 \times 10 = 62.816\)。

\(P(X < x) = 0.9 \Rightarrow x = \mu + z\sigma = 50 + 1.2816 \times 10 = 62.816\)

Rounding should be done at the end of the calculation, unless the question states otherwise.

除非题目另有说明，否则应在计算结束时再进行四舍五入。

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7. Handling Tail and Two-Tailed Probabilities | 处理尾部和双侧概率

Some questions give probabilities in the right tail, such as \(P(X > a) = 0.05\). To use the inverse normal function, convert this to a left-tail probability: \(P(X < a) = 1 - 0.05 = 0.95\).

有些题目给出的是右尾概率，例如 \(P(X > a) = 0.05\)。为了使用逆正态函数，应将其转换为左尾概率：\(P(X < a) = 1 - 0.05 = 0.95\)。

For two-tailed questions like \(P(|Z| > k) = 0.05\), this means \(P(Z < -k) + P(Z > k) = 0.05\). By symmetry, each tail has probability \(0.025\).

对于双尾问题，如 \(P(|Z| > k) = 0.05\)，意味着 \(P(Z < -k) + P(Z > k) = 0.05\)。根据对称性，每条尾部概率为 \(0.025\)。

To find \(k\), use the left-tail probability \(0.025\). The inverse normal gives \(z = -1.96\) for the left tail, so by symmetry \(k = 1.96\). Alternatively, using the left-tail probability \(0.975\) gives directly \(z = 1.96\).

要求 \(k\)，使用左尾概率 \(0.025\)。逆正态函数给出左尾 \(z = -1.96\)，根据对称性 \(k = 1.96\)。或者使用左尾概率 \(0.975\) 直接得到 \(z = 1.96\)。

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8. Common Exam Question Types | 常见考试题型

Exam questions often provide the mean and standard deviation, then ask for the value of a variable that is exceeded by a certain percentage of the population. For example: “Given that \(P(X > a) = 0.15\), find \(a\).”

考试题目通常给出均值和标准差，然后要求求出某个变量值，使得总体中一定百分比超过该值。例如：“已知 \(P(X > a) = 0.15\)，求 \(a\)。”

Another common type involves central intervals: find the symmetric values \(a\) and \(b\) such that the central area is 0.9, i.e. \(P(a < X < b) = 0.9\) and \(a = \mu - k\), \(b = \mu + k\).

另一种常见题型涉及中央区间：求对称值 \(a\) 和 \(b\)，使得中央面积为0.9，即 \(P(a < X < b) = 0.9\)，且 \(a = \mu - k\)，\(b = \mu + k\)。

For the central area 0.9, the two tails together have probability 0.1, so each tail is 0.05. Thus \(z = \Phi^{-1}(0.95) = 1.6449\), giving \(k = 1.6449\sigma\).

对于中央面积0.9，两条尾部总共概率为0.1，因此每条尾部为0.05。所以 \(z = \Phi^{-1}(0.95) = 1.6449\)，给出 \(k = 1.6449\sigma\)。

Some questions ask for the value of the mean or standard deviation instead of the variable. In that case, substitute the known probability and z-value into the equation \(x = \mu + z\sigma\) and solve for the unknown parameter.

有些题目要求的是均值或标准差，而不是变量值。这种情况下，将已知概率和z值代入方程 \(x = \mu + z\sigma\)，然后解出未知参数。

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9. Common Mistakes and Tips | 易错点与技巧

Mistake 1: Using the wrong tail probability. If the question gives \(P(X > a) = 0.05\), using 0.05 directly in the inverse normal function will give a negative z-value, which is wrong. Always use \(1 – 0.05 = 0.95\) for a right-tail bound.

易错点1：使用了错误的尾部概率。如果题目给出 \(P(X > a) = 0.05\)，直接在逆正态函数中使用0.05会得到负的z值，这是错误的。对于右尾边界，始终使用 \(1 – 0.05 = 0.95\)。

Mistake 2: Confusing the standard deviation with the variance. In \(X \sim N(\mu, \sigma^2)\), the standard deviation is \(\sigma\), not \(\sigma^2\). If the question says variance is 16, then \(\sigma = \sqrt{16} = 4\).

易错点2：混淆标准差与方差。在 \(X \sim N(\mu, \sigma^2)\) 中，标准差是 \(\sigma\)，而不是 \(\sigma^2\)。如果题目说方差为16，则 \(\sigma = \sqrt{16} = 4\)。

Mistake 3: Forgetting to round to an appropriate degree of accuracy. A-Level questions often specify “give your answer to 3 significant figures” or “3 decimal places”.

易错点3：忘记按题目要求的精度四舍五入。A-Level题目通常会指定“保留3位有效数字”或“保留3位小数”。

Tip: Always draw a sketch of the normal curve and shade the relevant region. This helps you determine whether the z-value should be positive or negative.

技巧：总是画一条正态曲线示意图并标出相关区域。这有助于判断z值应为正还是负。

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10. Practice and Summary | 练习与总结

Let us summarise the key procedure:

让我们总结关键步骤：

• Determine the relationship between the given probability and the required variable: left-tail, right-tail, or two-tailed.

  确定给定概率与所求变量之间的关系：左尾、右尾还是双尾。

• Convert the probability to a left-tail probability if necessary.

  如有必要，将概率转换为左尾概率。

• Use the inverse normal function or table to find the corresponding \(z\)-value.

  使用逆正态函数或表格找到对应的 \(z\) 值。

• Use \(x = \mu + z\sigma\) to find the required value, or substitute to solve for \(\mu\) or \(\sigma\).

  使用 \(x = \mu + z\sigma\) 求所需值，或代入方程求解 \(\mu\) 或 \(\sigma\)。

Try this question: A machine fills bags with mass \(X \sim N(500, 15^2)\) grams. Find the value \(a\) such that 5% of bags have mass exceeding \(a\).

试试这道题：一台机器填充的袋子质量 \(X \sim N(500, 15^2)\) 克。求 \(a\) 的值，使得5%的袋子质量超过 \(a\)。

Solution: \(P(X > a) = 0.05\), so \(P(X < a) = 0.95\). From the inverse normal, \(z = 1.6449\). Thus \(a = 500 + 1.6449 \times 15 = 524.67\) grams.

解答：\(P(X > a) = 0.05\)，所以 \(P(X < a) = 0.95\)。由逆正态函数得 \(z = 1.6449\)。因此 \(a = 500 + 1.6449 \times 15 = 524.67\) 克。

The inverse normal distribution function is a powerful technique. With practice, you will become confident in reversing the normal distribution to solve a wide range of A-Level probability questions.

逆正态分布函数是一种强大的方法。通过练习，你将能够熟练地逆向使用正态分布，解决各种A-Level概率问题。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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