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抽样分布

AP 统计学 · 第 5 主题

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Sampling Distributions

Two thousand students go to this school, and some of them cycle here. We want to know what share. Nobody can ask all two thousand, so we ask a random hundred.…

英文讲解 · 内嵌中英文字幕

讲义 双页 词汇表 词汇测验 4 双页
5.1

统计学导论：为什么我的样本与你的不同？
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

VAR-1
Given that variation may be random or not, conclusions are uncertain.

VAR-1.G
Identify questions suggested by variation in statistics for samples collected from the same population. [Skill 1.A]

  • VAR-1.G.1 Variation in statistics for samples taken from the same population may be random or not.

来源:美国大学理事会 AP 课程与考试说明

一个统计量(statistic)(像一个样本均值 $\bar{x}$ 或样本比例 $\hat{p}$)从一个样本计算并变化——样本到样本——这是抽样变异(sampling variability)。一个参数(parameter)($\mu$ 或 $p$)是关于总体的固定真相。抽样分布(sampling distribution)是一个统计量在一个给定大小的所有可能样本上的分布——它是从一个样本到推断的桥梁。

词汇表 训练
英文 中文 拼音
statistic/stəˈtɪstɪk/ 统计量 tǒng jì liàng
sampling variability/ˈsæmplɪŋ ˌveərɪəˈbɪlɪti/ 抽样变异 chōu yàng biàn yì
parameter/pəˈræmɪtə/ 参数 cān shù
sampling distribution/ˈsæmplɪŋ ˌdɪstrɪˈbjuːʃn/ 抽样分布 chōu yàng fēn bù
练习卷 双页
5.2

再探正态分布
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

VAR-6
The normal distribution may be used to model variation.

VAR-6.A
Calculate the probability that a particular value lies in a given interval of a normal distribution. [Skill 3.A]

  • VAR-6.A.1 A continuous random variable is a variable that can take on any value within a specified domain. Every interval within the domain has a probability associated with it.
  • VAR-6.A.2 A continuous random variable with a normal distribution is commonly used to describe populations. The distribution of a normal random variable can be described by a normal, or "bell-shaped," curve.
  • VAR-6.A.3 The area under a normal curve over a given interval represents the probability that a particular value lies in that interval.
    • Illustrative examples for VAR-6.A: Continuous random variable: If one looks at a clock at a random time, the probability that the minute hand is between the 3 and the 6 is one fourth.

VAR-6.B
Determine the interval associated with a given area in a normal distribution. [Skill 3.A]

  • VAR-6.B.1 The boundaries of an interval associated with a given area in a normal distribution can be determined using $z$-scores or technology, such as a calculator, a standard normal table, or computer-generated output.
  • VAR-6.B.2 Intervals associated with a given area in a normal distribution can be determined by assigning appropriate inequalities to the boundaries of the intervals:
    • a. $P(X < x_a) = \dfrac{p}{100}$ means that the lowest $p\%$ of values lie to the left of $x_a$.
    • b. $P(x_a < X < x_b) = \dfrac{p}{100}$ means that $p\%$ of values lie between $x_a$ and $x_b$.
    • c. $P(X > x_b) = \dfrac{p}{100}$ means that the highest $p\%$ of values lie to the right of $x_b$.
    • d. To determine the most extreme $p\%$ of values requires dividing the area associated with $p\%$ into two equal areas on either extreme of the distribution: $P(X < x_a) = \dfrac{1}{2}\dfrac{p}{100}$ and $P(X > x_b) = \dfrac{1}{2}\dfrac{p}{100}$ means that half of the $p\%$ most extreme values lie to the left of $x_a$ and half of the $p\%$ most extreme values lie to the right of $x_b$.

VAR-6.C
Determine the appropriateness of using the normal distribution to approximate probabilities for unknown distributions. [Skill 3.C]

  • VAR-6.C.1 Normal distributions are symmetrical and "bell-shaped." As a result, normal distributions can be used to approximate distributions with similar characteristics.

来源:美国大学理事会 AP 课程与考试说明

正态分布

对于足够大的样本,许多抽样分布近似正态。那让我们能用一个中心(它的均值)、一个散布(它的标准误(standard error))和一个正态形状描述一个统计量——然后计算一个给定的样本结果有多可能。

探索

Use the normal curve to find a proportion

A normal model turns a range of values into an area = a proportion. Shade a band to read off the fraction of samples falling within it (the 68-95-99.7 rule).

词汇表 训练
英文 中文 拼音
standard error/ˈstændəd ˈerə/ 标准误 biāo zhǔn wù
练习卷 双页
5.3

中心极限定理
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

UNC-3
Probabilistic reasoning allows us to anticipate patterns in data.

UNC-3.H
Estimate sampling distributions using simulation. [Skill 3.C]

  • UNC-3.H.1 A sampling distribution of a statistic is the distribution of values for the statistic for all possible samples of a given size from a given population.
  • UNC-3.H.2 The central limit theorem (CLT) states that when the sample size is sufficiently large, a sampling distribution of the mean of a random variable will be approximately normally distributed.
  • UNC-3.H.3 The central limit theorem requires that the sample values are independent of each other and that $n$ is sufficiently large.
  • UNC-3.H.4 A randomization distribution is a collection of statistics generated by simulation assuming known values for the parameters. For a randomized experiment, this means repeatedly randomly reallocating/reassigning the response values to treatment groups.
  • UNC-3.H.5 The sampling distribution of a statistic can be simulated by generating repeated random samples from a population.

来源:美国大学理事会 AP 课程与考试说明

中心极限定理

中心极限定理(Central Limit Theorem,CLT):对于一个样本均值,若样本量 $n$ 足够大(一个常见规则是 $n\ge 30$),$\bar{x}$ 的抽样分布近似正态,无论总体的形状。$n$ 越大,越正态而分布越紧。

The sample mean is nearly normal whatever the shape of the population
无论总体的形状如何样本均值都几乎正态
探索

Watch a sampling distribution turn normal

The Central Limit Theorem: for a large enough sample, the distribution of the sample mean is approximately normal — whatever the shape of the population.

词汇表 训练
英文 中文 拼音
Central Limit Theorem/ˈsentrəl ˈlɪmɪt ˈθɪərəm/ 中心极限定理 zhōng xīn jí xiàn dìng lǐ
练习卷 双页
5.4

有偏与无偏点估计
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

UNC-3
Probabilistic reasoning allows us to anticipate patterns in data.

UNC-3.I
Explain why an estimator is or is not unbiased. [Skill 4.B]

  • UNC-3.I.1 When estimating a population parameter, an estimator is unbiased if, on average, the value of the estimator is equal to the population parameter.

UNC-3.J
Calculate estimates for a population parameter. [Skill 3.B]

  • UNC-3.J.1 When estimating a population parameter, an estimator exhibits variability that can be modeled using probability.
  • UNC-3.J.2 A sample statistic is a point estimator of the corresponding population parameter.

来源:美国大学理事会 AP 课程与考试说明

一个统计量是无偏(unbiased)的,若它的抽样分布的均值等于这个参数——它平均而言正确。偏差是关于中心偏了;变异性(variability)是关于散布。一个好的估计量既无偏(中心正确)又低变异性(精确);更大的样本减少变异性但不修复来自差劲抽样的偏差。

Four sampling distributions crossing bias with variability, against the true parameter
偏差和变异性是两种不同的毛病。只有左上的估计量既以 $\theta$ 为中心又很集中;左下的那个虽然精确却一贯错误,再多的数据也修复不了。
词汇表 训练
英文 中文 拼音
unbiased/ʌnˈbaɪəst/ 无偏 wú piān
练习卷 双页
5.5

样本比例的抽样分布
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

UNC-3
Probabilistic reasoning allows us to anticipate patterns in data.

UNC-3.K
Determine parameters of a sampling distribution for sample proportions. [Skill 3.B]

  • UNC-3.K.1 For independent samples (sampling with replacement) of a categorical variable from a population with population proportion, $p$, the sampling distribution of the sample proportion, $\hat{p}$, has a mean, $\mu_{\hat{p}} = p$ and a standard deviation, $\sigma_{\hat{p}} = \sqrt{\dfrac{p(1-p)}{n}}$.
  • UNC-3.K.2 If sampling without replacement, the standard deviation of the sample proportion is smaller than what is given by the formula above. If the sample size is less than 10% of the population size, the difference is negligible.

UNC-3.L
Determine whether a sampling distribution for a sample proportion can be described as approximately normal. [Skill 3.C]

  • UNC-3.L.1 For a categorical variable, the sampling distribution of the sample proportion, $\hat{p}$, will have an approximate normal distribution, provided the sample size is large enough: $np \geq 10$ and $n(1-p) \geq 10$

UNC-3.M
Interpret probabilities and parameters for a sampling distribution for a sample proportion. [Skill 4.B]

  • UNC-3.M.1 Probabilities and parameters for a sampling distribution for a sample proportion should be interpreted using appropriate units and within the context of a specific population.

来源:美国大学理事会 AP 课程与考试说明

对于来自一个 SRS 的样本比例 $\hat{p}$:均值是 $p$(无偏),而标准差是

$$\sigma_{\hat p}=\sqrt{\frac{p(1-p)}{n}}.$$
这个散布有两个名字:它是抽样分布的标准差;一旦你必须从样本估计它(用 $\hat p$ 代替 $p$),它就叫标准误——这正是后面的推断单元所做的。 它在 $np\ge 10$ 和 $n(1-p)\ge 10$(大计数(Large Counts)条件)时近似正态,而 $10\%$ 条件($n\le 0.10N$)使观测保持近独立。

Worked example. 假设 $40\%$ 的选民赞成一项措施($p=0.4$)而你抽样 $n=100$。标准误是 $\sigma_{\hat p}=\sqrt{\dfrac{0.4(0.6)}{100}}=0.049$。一个样本给出 $\hat{p}>0.5$ 的机会是 $z=\dfrac{0.5-0.4}{0.049}=2.04$,所以 $P(\hat p>0.5)\approx0.02$ ——样本里的一个多数会令人惊讶。

练习卷 双页
5.6

样本比例之差的抽样分布
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

UNC-3
Probabilistic reasoning allows us to anticipate patterns in data.

UNC-3.N
Determine parameters of a sampling distribution for a difference in sample proportions. [Skill 3.B]

  • UNC-3.N.1 For a categorical variable, when randomly sampling with replacement from two independent populations with population proportions $p_1$ and $p_2$, the sampling distribution of the difference in sample proportions $\hat{p}_1 - \hat{p}_2$ has mean, $\mu_{\hat{p}_1 - \hat{p}_2} = p_1 - p_2$ and standard deviation, $\sigma_{\hat{p}_1 - \hat{p}_2} = \sqrt{\dfrac{p_1(1-p_1)}{n_1} + \dfrac{p_2(1-p_2)}{n_2}}$.
  • UNC-3.N.2 If sampling without replacement, the standard deviation of the difference in sample proportions is smaller than what is given by the formula above. If the sample sizes are less than 10% of the population sizes, the difference is negligible.

UNC-3.O
Determine whether a sampling distribution for a difference of sample proportions can be described as approximately normal. [Skill 3.C]

  • UNC-3.O.1 The sampling distribution of the difference in sample proportions $\hat{p}_1 - \hat{p}_2$ will have an approximate normal distribution provided the sample sizes are large enough: $n_1 p_1 \geq 10, n_1(1-p_1) \geq 10, n_2 p_2 \geq 10, n_2(1-p_2) \geq 10$.

UNC-3.P
Interpret probabilities and parameters for a sampling distribution for a difference in proportions. [Skill 4.B]

  • UNC-3.P.1 Parameters for a sampling distribution for a difference of proportions should be interpreted using appropriate units and within the context of a specific populations.

来源:美国大学理事会 AP 课程与考试说明

对于来自两个独立样本的 $\hat{p}_1-\hat{p}_2$:均值是 $p_1-p_2$,而因为样本独立方差相加:

$$\sigma_{\hat p_1-\hat p_2}=\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}}.$$
它在大计数条件在两个样本里都成立时近似正态。

练习卷 双页
5.7

样本均值的抽样分布
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

UNC-3
Probabilistic reasoning allows us to anticipate patterns in data.

UNC-3.Q
Determine parameters for a sampling distribution for sample means. [Skill 3.B]

  • UNC-3.Q.1 For a numerical variable, when random sampling with replacement from a population with mean $\mu$ and standard deviation, $\sigma$, the sampling distribution of the sample mean has mean $\mu_{\bar{x}} = \mu$ and standard deviation $\sigma_{\bar{x}} = \dfrac{\sigma}{\sqrt{n}}$.
  • UNC-3.Q.2 If sampling without replacement, the standard deviation of the sample mean is smaller than what is given by the formula above. If the sample size is less than 10% of the population size, the difference is negligible.

UNC-3.R
Determine whether a sampling distribution of a sample mean can be described as approximately normal. [Skill 3.C]

  • UNC-3.R.1 For a numerical variable, if the population distribution can be modeled with a normal distribution, the sampling distribution of the sample mean, $\bar{x}$, can be modeled with a normal distribution.
  • UNC-3.R.2 For a numerical variable, if the population distribution cannot be modeled with a normal distribution, the sampling distribution of the sample mean, $\bar{x}$, can be modeled approximately by a normal distribution, provided the sample size is large enough, e.g., greater than or equal to 30.

UNC-3.S
Interpret probabilities and parameters for a sampling distribution for a sample mean. [Skill 4.B]

  • UNC-3.S.1 Probabilities and parameters for a sampling distribution for a sample mean should be interpreted using appropriate units and within the context of a specific population.

来源:美国大学理事会 AP 课程与考试说明

对于来自一个 SRS 的样本均值 $\bar{x}$:均值是 $\mu$(无偏),而标准差是

$$\sigma_{\bar x}=\frac{\sigma}{\sqrt{n}}.$$
它的形状是正态的,若总体是正态的,或对大的 $n$ 由 CLT 近似正态。注意散布像 $\sqrt{n}$ 那样缩小——把样本变四倍使标准误减半。

Worked example. 一个总体有 $\mu=70$ 和 $\sigma=12$。对于 $n=36$ 的样本,$\bar{x}$ 的抽样分布以 $70$ 为中心带标准误 $\dfrac{12}{\sqrt{36}}=2$。一个样本均值超过 $73$ 的机会是 $z=\dfrac{73-70}{2}=1.5$,所以 $P(\bar x>73)\approx0.067$。

The sampling distribution of the mean narrows and becomes more normal as n grows
左边的总体强烈偏斜,但 $\bar{x}$ 的每个抽样分布都以 $\mu$ 为中心。更大的 $n$ 使标准误 $\sigma/\sqrt{n}$ 更小,所以曲线更高更窄——而且也更对称:在 $n=2$ 时仍明显偏斜,到 $n=30$ 时几乎正好是正态(虚线)。
练习卷 双页
5.8

样本均值之差的抽样分布
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

UNC-3
Probabilistic reasoning allows us to anticipate patterns in data.

UNC-3.T
Determine parameters of a sampling distribution for a difference in sample means. [Skill 3.B]

  • UNC-3.T.1 For a numerical variable, when randomly sampling with replacement from two independent populations with population means $\mu_1$ and $\mu_2$ and population standard deviations $\sigma_1$ and $\sigma_2$, the sampling distribution of the difference in sample means $\bar{x}_1 - \bar{x}_2$ has mean $\mu_{(\bar{x}_1 - \bar{x}_2)} = \mu_1 - \mu_2$ and standard deviation, $\sigma_{(\bar{x}_1 - \bar{x}_2)} = \sqrt{\dfrac{\sigma_1^2}{n_1} + \dfrac{\sigma_2^2}{n_2}}$.
  • UNC-3.T.2 If sampling without replacement, the standard deviation of the difference in sample means is smaller than what is given by the formula above. If the sample sizes are less than 10% of the population sizes, the difference is negligible.

UNC-3.U
Determine whether a sampling distribution of a difference in sample means can be described as approximately normal. [Skill 3.C]

  • UNC-3.U.1 The sampling distribution of the difference in sample means $\bar{x}_1 - \bar{x}_2$ can be modeled with a normal distribution if the two population distributions can be modeled with a normal distribution.
  • UNC-3.U.2 The sampling distribution of the difference in sample means $\bar{x}_1 - \bar{x}_2$ can be modeled approximately by a normal distribution if the two population distributions cannot be modeled with a normal distribution but both sample sizes are greater than or equal to 30.

UNC-3.V
Interpret probabilities and parameters for a sampling distribution for a difference in sample means. [Skill 4.B]

  • UNC-3.V.1 Probabilities and parameters for a sampling distribution for a difference of sample means should be interpreted using appropriate units and within the context of a specific populations.

来源:美国大学理事会 AP 课程与考试说明

对于来自两个独立样本的 $\bar{x}_1-\bar{x}_2$:均值是 $\mu_1-\mu_2$,而(独立,所以方差相加)

$$\sigma_{\bar x_1-\bar x_2}=\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}.$$
这是下面几个单元里两样本推断的基础。

练习卷 双页
5.8

考试技巧
  • 一个抽样分布是一个统计量在许多样本上的分布,以真参数为中心。
  • 中心极限定理:对于一个足够大的样本样本均值近似正态,即使总体不是。
  • 更大的样本给更少的变异性(一个更小的标准误)。
  • 在用一个正态模型前检查条件(随机、独立/10%、足够大)。
  • 弄清什么变化——统计量——对固定参数。

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