剑桥 IGCSE IGCSE, A-Level & AP 练习单预览

分类数据的推断：比例

AP 统计学 · 第 6 主题

训练
讲义 词汇表
6.1

统计学导论：为什么要用正态？
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

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

VAR-1.H
Identify questions suggested by variation in the shapes of distributions of samples taken from the same population. [Skill 1.A]

  • VAR-1.H.1 Variation in shapes of data distributions may be random or not.

来源:美国大学理事会 AP 课程与考试说明

因为一个样本比例 $\hat{p}$ 近似正态分布(当条件成立时),我们能以标准误测量一个样本结果离一个宣称的值多远,并把那转成一个概率。这正是使推断(inference)——从一个样本得出关于一个总体的结论——成为可能的东西。

词汇表 训练
英文 中文 拼音
inference 推断 tuī duàn
练习卷
6.2

构建总体比例的置信区间
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

UNC-4
An interval of values should be used to estimate parameters, in order to account for uncertainty.

UNC-4.A
Identify an appropriate confidence interval procedure for a population proportion. [Skill 1.D]

  • UNC-4.A.1 The appropriate confidence interval procedure for a one-sample proportion for one categorical variable is a one sample $z$-interval for a proportion.

UNC-4.B
Verify the conditions for calculating confidence intervals for a population proportion. [Skill 4.C]

  • UNC-4.B.1 In order to make assumptions necessary for inference on population proportions, means, and slopes, we must check for independence in data collection methods and for selection of the appropriate sampling distribution.
  • UNC-4.B.2 In order to calculate a confidence interval to estimate a population proportion, $p$, we must check for independence and that the sampling distribution is approximately normal.
    • a. To check for independence:
      • i. Data should be collected using a random sample or a randomized experiment.
      • ii. When sampling without replacement, check that $n \leq 10\%N$, where $N$ is the size of the population.
    • b. To check that the sampling distribution of $\hat{p}$ is approximately normal (shape):
      • i. For categorical variables, check that both the number of successes, $n\hat{p}$, and the number of failures, $n(1-\hat{p})$ are at least 10 so that the sample size is large enough to support an assumption of normality.

UNC-4.C
Determine the margin of error for a given sample size and an estimate for the sample size that will result in a given margin of error for a population proportion. [Skill 3.D]

  • UNC-4.C.1 Based on sample data, the standard error of a statistic is an estimate for the standard deviation for the statistic. The standard error of $\hat{p}$ is $SE_{\hat{p}} = \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$.
  • UNC-4.C.2 A margin of error gives how much a value of a sample statistic is likely to vary from the value of the corresponding population parameter.
  • UNC-4.C.3 For categorical variables, the margin of error is the critical value ($z^*$) times the standard error (SE) of the relevant statistic, which equals $z^* \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$ for a one sample proportion.
  • UNC-4.C.4 The formula for margin of error can be rearranged to solve for $n$, the minimum sample size needed to achieve a given margin of error. For this purpose, use a guess for $\hat{p}$ or use $\hat{p} = 0.5$ in order to find an upper bound for the sample size that will result in a given margin of error.

UNC-4.D
Calculate an appropriate confidence interval for a population proportion. [Skill 3.D]

  • UNC-4.D.1 In general, an interval estimate can be constructed as point estimate ± (margin of error). For a one-sample proportion, the interval estimate is $\hat{p} \pm z^* \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$.
    • Clarifying statement: Formulas for interval estimates do not appear explicitly on the AP Statistics Formula Sheet provided with the AP Statistics Exam. However, these formulas do not need to be memorized, as they can be constructed based on the general test statistic formula and the relevant standard error formulas that are provided on the formula sheet.
  • UNC-4.D.2 Critical values represent the boundaries encompassing the middle C% of the standard normal distribution, where C% is an approximate confidence level for a proportion.

UNC-4.E
Calculate an interval estimate based on a confidence interval for a population proportion. [Skill 3.D]

  • UNC-4.E.1 Confidence intervals for population proportions can be used to calculate interval estimates with specified units.

来源:美国大学理事会 AP 课程与考试说明

置信区间的含义

一个置信区间(confidence interval)把参数估计为一个范围:统计量 $\pm$ 误差幅度(margin of error)。

$$\hat{p}\pm z^{*}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}.$$
$z^{*}$ 是置信水平(confidence level)的临界值(例如 95% 是 $1.96$)。条件:随机样本、大计数(Large Counts)($n\hat p\ge 10$ 和 $n(1-\hat p)\ge 10$),以及 10% 条件。解释它:"我们有 95% 的信心,……的真比例在……和……之间。"解释这个水平:"在 95% 的样本里,这个方法产生一个捕获真比例的区间。"

Over many samples, about 95% of 95% confidence intervals capture the true proportion
在许多样本上,约 95% 的 95% 置信区间捕获真比例

Worked example. 在一个 $200$ 人的随机样本里,$120$ 个支持一项政策,所以 $\hat{p}=0.60$。一个 $95\%$ 区间用 $z^*=1.96$:

$$0.60\pm1.96\sqrt{\frac{0.60(0.40)}{200}}=0.60\pm0.068=(0.532,\ 0.668).$$
我们有 $95\%$ 的信心,支持者的真比例在 $53.2\%$ 和 $66.8\%$ 之间。

A 95% confidence interval reaches 1.96 standard errors each side of the estimate
一个 95% 置信区间在估计值的每一侧伸出 1.96 个标准误

选择样本量。 要让误差幅度不大于一个目标 $m$,令 $z^{*}\sqrt{\dfrac{\hat p(1-\hat p)}{n}}\le m$ 并解出 $n$。当你没有 $\hat p$ 的估计时,用 $\hat p=0.5$:它使 $\hat p(1-\hat p)$ 尽可能大,给出安全的(最大的)所需样本量。总是把结果向上取整到下一个整人数。

Worked example. 要一个误差幅度至多 $0.03$ 的 $95\%$ 区间,你必须调查多少人?用 $\hat p=0.5$ 和 $z^*=1.96$:

$$n=\frac{(z^*)^2\,\hat p(1-\hat p)}{m^2}=\frac{1.96^2(0.5)(0.5)}{0.03^2}=\frac{0.9604}{0.0009}\approx1067.1,$$
所以你调查 $1068$ 人(总是向上取整,因为 $1067$ 会让误差幅度稍微太大)。

词汇表 训练
英文 中文 拼音
confidence interval 置信区间 zhì xìn qū jiān
margin of error 误差幅度 wù chā fú dù
confidence level 置信水平 zhì xìn shuǐ píng
练习卷
6.3

基于总体比例置信区间论证结论
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

UNC-4
An interval of values should be used to estimate parameters, in order to account for uncertainty.

UNC-4.F
Interpret a confidence interval for a population proportion. [Skill 4.B]

  • UNC-4.F.1 A confidence interval for a population proportion either contains the population proportion or it does not, because each interval is based on random sample data, which varies from sample to sample.
  • UNC-4.F.2 We are C% confident that the confidence interval for a population proportion captures the population proportion.
  • UNC-4.F.3 In repeated random sampling with the same sample size, approximately C% of confidence intervals created will capture the population proportion.
  • UNC-4.F.4 Interpreting a confidence interval for a one-sample proportion should include a reference to the sample taken and details about the population it represents.
    • Illustrative examples for UNC-4.F.4: For interpreting a 99% confidence interval of (0.268, 0.292), based on the proportion of a nationally representative sample of twelfth-grade students who answered a particular multiple choice question correctly: "We are 99 percent confident that the interval from 0.268 to 0.292 contains the population proportion of all United States twelfth-grade students who would answer this question correctly" (2011 FRQ 6(a)).

UNC-4.G
Justify a claim based on a confidence interval for a population proportion. [Skill 4.D]

  • UNC-4.G.1 A confidence interval for a population proportion provides an interval of values that may provide sufficient evidence to support a particular claim in context.

UNC-4.H
Identify the relationships between sample size, width of a confidence interval, confidence level, and margin of error for a population proportion. [Skill 4.A]

  • UNC-4.H.1 When all other things remain the same, the width of the confidence interval for a population proportion tends to decrease as the sample size increases. For a population proportion, the width of the interval is proportional to $\dfrac{1}{\sqrt{n}}$.
  • UNC-4.H.2 For a given sample, the width of the confidence interval for a population proportion increases as the confidence level increases.
  • UNC-4.H.3 The width of a confidence interval for a population proportion is exactly twice the margin of error.

来源:美国大学理事会 AP 课程与考试说明

要判断一个宣称的值:若它落在区间内,数据与它一致;若它落在外,数据给出反对它的证据。基于这些可信的值是否包含这个宣称、在上下文里得出结论。

练习卷
6.4

建立总体比例的检验
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

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

VAR-6.D
Identify the null and alternative hypotheses for a population proportion. [Skill 1.F]

  • VAR-6.D.1 The null hypothesis is the situation that is assumed to be correct unless evidence suggests otherwise, and the alternative hypothesis is the situation for which evidence is being collected.
  • VAR-6.D.2 For hypotheses about parameters, the null hypothesis contains an equality reference (=, ≥, or ≤), while the alternative hypothesis contains a strict inequality (<, >, or ≠). The type of inequality in the alternative hypothesis is based on the question of interest. Alternative hypotheses with < or > are called one-sided, and alternative hypotheses with ≠ are called two-sided. Although the null hypothesis for a one-sided test may include an inequality symbol, it is still tested at the boundary of equality.
  • VAR-6.D.3 The null hypothesis for a population proportion is: $H_0 : p = p_0$, where $p_0$ is the null hypothesized value for the population proportion.
  • VAR-6.D.4 A one-sided alternative hypothesis for a proportion is either $H_a : p < p_0$ or $H_a : p > p_0$. A two-sided alternate hypothesis is $H_a : p_1 \neq p_2$.
  • VAR-6.D.5 For a one-sample $z$-test for a population proportion, the null hypothesis specifies a value for the population proportion, usually one indicating no difference or effect.

VAR-6.E
Identify an appropriate testing method for a population proportion. [Skill 1.E]

  • VAR-6.E.1 For a single categorical variable, the appropriate testing method for a population proportion is a one-sample $z$-test for a population proportion.

VAR-6.F
Verify the conditions for making statistical inferences when testing a population proportion. [Skill 4.C]

  • VAR-6.F.1 In order to make statistical inferences when testing a population proportion, we must check for independence and that the sampling distribution is approximately normal:
    • a. To check for independence:
      • i. Data should be collected using a random sample or a randomized experiment.
      • ii. When sampling without replacement, check that $n \leq 10\%N$.
    • b. To check that the sampling distribution of $\hat{p}$ is approximately normal (shape):
      • i. Assuming that $H_0$ is true $(p = p_0)$, verify that both the number of successes, $np_0$, and the number of failures, $n(1-p_0)$ are at least 10 so that that the sample size is large enough to support an assumption of normality.

来源:美国大学理事会 AP 课程与考试说明

一个显著性检验(significance test)权衡反对一个宣称的证据。陈述一个关于参数 $p$ 的原假设(null hypothesis)$H_0$ 和一个备择假设(alternative hypothesis)$H_a$:

$$H_0: p=p_0 \qquad H_a: p\neq p_0 \ (\text{or } <,\, >).$$
检查相同的条件(随机、用 $p_0$ 的大计数、10%)。检验统计量(test statistic)数离 $p_0$ 的标准误:
$$z=\frac{\hat p-p_0}{\sqrt{p_0(1-p_0)/n}}.$$

Worked example. 一个公司宣称 $90\%$ 满意度($p_0=0.90$);一个 $100$ 的样本发现 $84$ 个满意($\hat{p}=0.84$)。在 $\alpha=0.05$ 检验 $H_0:p=0.90$ 对 $H_a:p\neq0.90$:

$$z=\frac{0.84-0.90}{\sqrt{0.90(0.10)/100}}=\frac{-0.06}{0.03}=-2.0,$$
给出一个约 $2(0.023)=0.046$ 的双尾 $p$ 值。因为 $0.046<0.05$,拒绝 $H_0$ ——有证据表明真满意率不同于(低于)$90\%$。

A two-tailed 5% test rejects the null hypothesis in the shaded tails
一个双尾 5% 检验在阴影尾里拒绝原假设
词汇表 训练
英文 中文 拼音
significance test 显著性检验 xiǎn zhù xìng jiǎn yàn
null hypothesis 原假设 yuán jiǎ shè
alternative hypothesis 备择假设 bèi zé jiǎ shè
test statistic 检验统计量 jiǎn yàn tǒng jì liàng
练习卷
6.5

解释 p 值
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

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

VAR-6.G
Calculate an appropriate test statistic and $p$-value for a population proportion. [Skill 3.E]

  • VAR-6.G.1 The distribution of the test statistic assuming the null hypothesis is true (null distribution) can be either a randomization distribution or when a probability model is assumed to be true, a theoretical distribution ($z$).
  • VAR-6.G.2 When using a $z$-test, the standardized test statistic can be written: $\text{test statistic} = \dfrac{\text{sample statistic} - \text{null value of the parameter}}{\text{standard deviation of the statistic}}$. This is called a $z$-statistic for proportions.
  • VAR-6.G.3 The test statistic for a population proportion is: $z = \dfrac{\hat{p} - p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}$.
    • Clarifying statement: The formulas for test statistics do not appear explicitly on the AP Statistics Formula Sheet provided with the AP Statistics Exam. However, these formulas do not need to be memorized, as they can be constructed based on the general test statistic formula and the relevant standard error formulas that are provided on the formula sheet.
  • VAR-6.G.4 A $p$-value is the probability of obtaining a test statistic as extreme or more extreme than the observed test statistic when the null hypothesis and probability model are assumed to be true. The significance level may be given or determined by the researcher.

DAT-3
Significance testing allows us to make decisions about hypotheses within a particular context.

DAT-3.A
Interpret the $p$-value of a significance test for a population proportion. [Skill 4.B]

  • DAT-3.A.1 The $p$-value is the proportion of values for the null distribution that are as extreme or more extreme than the observed value of the test statistic. This is:
    • a. The proportion at or above the observed value of the test statistic, if the alternative is >.
    • b. The proportion at or below the observed value of the test statistic, if the alternative is <.
    • c. The proportion less than or equal to the negative of the absolute value of the test statistic plus the proportion greater than or equal to the absolute value of the test statistic, if the alternative is ≠.
  • DAT-3.A.2 An interpretation of the $p$-value of a significance test for a one-sample proportion should recognize that the $p$-value is computed by assuming that the probability model and null hypothesis are true, i.e., by assuming that the true population proportion is equal to the particular value stated in the null hypothesis.

来源:美国大学理事会 AP 课程与考试说明

p 值的含义

$p$ 值(p-value)是得到一个与观测的一样极端或更极端的样本结果的概率,假设 $H_0$ 为真。一个小的 $p$ 值意味着若 $H_0$ 成立数据会令人惊讶——反对 $H_0$ 的证据。它不是 $H_0$ 为真的概率。

探索

A p-value as a tail area

A p-value is the probability, if the null hypothesis were true, of a result at least this extreme — the shaded tail area. Small p-values cast doubt on the null.

词汇表 训练
英文 中文 拼音
p-value P值 P zhí
练习卷
6.6

得出总体比例检验的结论
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

DAT-3
Significance testing allows us to make decisions about hypotheses within a particular context.

DAT-3.B
Justify a claim about the population based on the results of a significance test for a population proportion. [Skill 4.E]

  • DAT-3.B.1 The significance level, $\alpha$, is the predetermined probability of rejecting the null hypothesis given that it is true.
  • DAT-3.B.2 A formal decision explicitly compares the $p$-value to the significance level, $\alpha$. If the $p$-value $\leq \alpha$, reject the null hypothesis. If the $p$-value $> \alpha$, fail to reject the null hypothesis.
  • DAT-3.B.3 Rejecting the null hypothesis means there is sufficient statistical evidence to support the alternative hypothesis. Failing to reject the null means there is insufficient statistical evidence to support the alternative hypothesis.
  • DAT-3.B.4 The conclusion about the alternative hypothesis must be stated in context.
  • DAT-3.B.5 A significance test can lead to rejecting or not rejecting the null hypothesis, but can never lead to concluding or proving that the null hypothesis is true. Lack of statistical evidence for the alternative hypothesis is not the same as evidence for the null hypothesis.
  • DAT-3.B.6 Small $p$-values indicate that the observed value of the test statistic would be unusual if the null hypothesis and probability model were true, and so provide evidence for the alternative. The lower the $p$-value, the more convincing the statistical evidence for the alternative hypothesis.
  • DAT-3.B.7 $p$-values that are not small indicate that the observed value of the test statistic would not be unusual if the null hypothesis and probability model were true, so do not provide convincing statistical evidence for the alternative hypothesis nor do they provide evidence that the null hypothesis is true.
  • DAT-3.B.8 A formal decision explicitly compares the $p$-value to the significance $\alpha$. If the $p$-value $\leq \alpha$, then reject the null hypothesis, $H_0 : p = p_0$. If the $p$-value $> \alpha$, then fail to reject the null hypothesis.
  • DAT-3.B.9 The results of a significance test for a population proportion can serve as the statistical reasoning to support the answer to a research question about the population that was sampled.

来源:美国大学理事会 AP 课程与考试说明

把 $p$ 值与显著性水平(significance level)$\alpha$(常常 $0.05$)比较:

  • $p\le\alpha$:拒绝 $H_0$ ——有令人信服的证据支持 $H_a$。
  • $p>\alpha$:未能拒绝 $H_0$ ——支持 $H_a$ 的证据不够(永不"接受 $H_0$")。

总是在上下文里、连回这个宣称地写结论。

词汇表 训练
英文 中文 拼音
significance level 显著性水平 xiǎn zhù xìng shuǐ píng
练习卷
6.7

进行检验时的潜在错误
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

UNC-5
Probabilities of Type I and Type II errors influence inference.

UNC-5.A
Identify Type I and Type II errors. [Skill 1.B]

  • UNC-5.A.1 A Type I error occurs when the null hypothesis is true and is rejected (false positive).
  • UNC-5.A.2 A Type II error occurs when the null hypothesis is false and is not rejected (false negative).
    • Table of Errors: With Actual Population Value across the top ($H_0$ true; $H_a$ true) and Decision down the side (Reject $H_0$; Fail to Reject $H_0$): Reject $H_0$ when $H_0$ true = Type I Error; Reject $H_0$ when $H_a$ true = Correct Decision; Fail to Reject $H_0$ when $H_0$ true = Correct Decision; Fail to Reject $H_0$ when $H_a$ true = Type II Error.

UNC-5.B
Calculate the probability of a Type I and Type II errors. [Skill 3.A]

  • UNC-5.B.1 The significance level, $\alpha$, is the probability of making a Type I error, if the null hypothesis is true.
  • UNC-5.B.2 The power of a test is the probability that a test will correctly reject a false null hypothesis.
  • UNC-5.B.3 The probability of making a Type II error $= 1 - power$.

UNC-5.C
Identify factors that affect the probability of errors in significance testing. [Skill 4.A]

  • UNC-5.C.1 The probability of a Type II error decreases when any of the following occurs, provided the others do not change:
    • i. Sample size(s) increases.
    • ii. Significance level ($\alpha$) of a test increases.
    • iii. Standard error decreases.
    • iv. True parameter value is farther from the null.

UNC-5.D
Interpret Type I and Type II errors. [Skill 4.B]

  • UNC-5.D.1 Whether a Type I or a Type II error is more consequential depends upon the situation.
  • UNC-5.D.2 Since the significance level, $\alpha$, is the probability of a Type I error, the consequences of a Type I error influence decisions about a significance level.

来源:美国大学理事会 AP 课程与考试说明

第一类与第二类错误
  • 一个第一类错误(Type I error):拒绝一个真的 $H_0$(一个假警报)。它的概率是 $\alpha$。
  • 一个第二类错误(Type II error):未能拒绝一个假的 $H_0$(一个漏检)。它的概率是 $\beta$。
  • 检验效能(power)$=1-\beta$ 是正确地检测一个真实效果的机会。效能随一个更大的样本、一个更大的效果,或一个更大的 $\alpha$ 上升。

在问题的上下文里描述每个错误和它的后果。

探索

Two ways a test can be wrong

A Type I error rejects a true null (false alarm); a Type II error keeps a false null (a miss). Lowering one usually raises the other.

词汇表 训练
英文 中文 拼音
Type I error 第一类错误 dì yī lèi cuò wù
Type II error 第二类错误 dì èr lèi cuò wù
power 检验效能 jiǎn yàn xiào néng
练习卷
6.8

两个比例之差的置信区间
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

UNC-4
An interval of values should be used to estimate parameters, in order to account for uncertainty.

UNC-4.I
Identify an appropriate confidence interval procedure for a comparison of population proportions. [Skill 1.D]

  • UNC-4.I.1 The appropriate confidence interval procedure for a two-sample comparison of proportions for one categorical variable is a two-sample $z$-interval for a difference between population proportions.

UNC-4.J
Verify the conditions for calculating confidence intervals for a difference between population proportions. [Skill 4.C]

  • UNC-4.J.1 In order to calculate confidence intervals to estimate a difference between proportions, we must check for independence and that the sampling distribution is approximately normal:
    • a. To check for independence:
      • i. Data should be collected using two independent, random samples or a randomized experiment.
      • ii. When sampling without replacement, check that $n_1 \leq 10\%N_1$ and $n_2 \leq 10\%N_2$.
    • b. To check that sampling distribution of $\hat{p}_1 - \hat{p}_2$ is approximately normal (shape).
      • i. For categorical variables, check that $n_1\hat{p}_1$, $n_1(1-\hat{p}_1)$, $n_2\hat{p}_2$, and $n_2\left(1-\hat{p}_2\right)$ are all greater than or equal to some predetermined value, typically either 5 or 10.

UNC-4.K
Calculate an appropriate confidence interval for a comparison of population proportions. [Skill 3.D]

  • UNC-4.K.1 For a comparison of proportions, the interval estimate is $(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\dfrac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \dfrac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$.
    • Clarifying statement: Formulas for interval estimates do not appear explicitly on the AP Statistics Formula Sheet provided with the AP Statistics Exam. However, these formulas do not need to be memorized, as they can be constructed based on the general test statistic formula and the relevant standard error formulas that are provided on the formula sheet.

UNC-4.L
Calculate an interval estimate based on a confidence interval for a difference of proportions. [Skill 3.D]

  • UNC-4.L.1 Confidence intervals for a difference in proportions can be used to calculate interval estimates with specified units.

来源:美国大学理事会 AP 课程与考试说明

要比较两个比例,估计 $p_1-p_2$:

$$(\hat p_1-\hat p_2)\pm z^{*}\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+\frac{\hat p_2(1-\hat p_2)}{n_2}}.$$
条件必须在两个样本里成立,而样本必须独立。

练习卷
6.9

基于两个总体比例之差的置信区间论证结论
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

UNC-4
An interval of values should be used to estimate parameters, in order to account for uncertainty.

UNC-4.M
Interpret a confidence interval for a difference of proportions. [Skill 4.B]

  • UNC-4.M.1 In repeated random sampling with the same sample size, approximately C% of confidence intervals created will capture the difference in population proportions.
  • UNC-4.M.2 Interpreting a confidence interval for difference between population proportions should include a reference to the sample taken and details about the population it represents.

UNC-4.N
Justify a claim based on a confidence interval for a difference of proportions. [Skill 4.D]

  • UNC-4.N.1 A confidence interval for difference in population proportions provides an interval of values that may provide sufficient evidence to support a particular claim in context.

来源:美国大学理事会 AP 课程与考试说明

若 $p_1-p_2$ 的区间包含 $0$,数据与没有差异一致;若它整个在 $0$ 之上或之下,有一个差异的证据(在那个方向)。陈述方向和上下文。

练习卷
6.10

建立两个总体比例之差的检验
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

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

VAR-6.H
Identify the null and alternative hypotheses for a difference of two population proportions. [Skill 1.F]

  • VAR-6.H.1 For a two-sample test for a difference of two proportions, the null hypothesis specifies a value of $0$ for the difference in population proportions, indicating no difference or effect.
  • VAR-6.H.2 The null hypothesis for a difference in proportions is: $H_0 : p_1 = p_2$, or $H_0 : p_1 - p_2 = 0$.
  • VAR-6.H.3 A one-sided alternative hypothesis for a difference in proportions is $H_a : p_1 < p_2$, or, $H_a : p_1 > p_2$. A two-sided alternative hypothesis for a difference of proportions is $H_a : p_1 \neq p_2$.

VAR-6.I
Identify an appropriate testing method for the difference of two population proportions. [Skill 1.E]

  • VAR-6.I.1 For a single categorical variable, the appropriate testing method for the difference of two population proportions is a two-sample $z$-test for a difference between two population proportions.

VAR-6.J
Verify the conditions for making statistical inferences when testing a difference of two population proportions. [Skill 4.C]

  • VAR-6.J.1 In order to make statistical inferences when testing a difference between population proportions, we must check for independence and that the sampling distribution is approximately normal:
    • a. To check for independence:
      • i. Data should be collected using two independent, random samples or a randomized experiment.
      • ii. When sampling without replacement, check that $n_1 \leq 10\%N_1$ and $n_2 \leq 10\%N_2$.
    • b. To check that the sampling distribution of $\hat{p}_1 - \hat{p}_2$ is approximately normal (shape):
      • i. For the combined sample, define the combined (or pooled) proportion, $\hat{p}_c = \dfrac{n_1\hat{p}_1 + n_2\hat{p}_2}{n_1 + n_2}$. Assuming that $H_0$ is true $(p_1 - p_2 = 0$ or $p_1 = p_2)$, check that $n_1\hat{p}_c$, $n_1\left(1-\hat{p}_c\right)$, $n_2\hat{p}_c$, and $n_2\left(1-\hat{p}_c\right)$ are all greater than or equal to some predetermined value, typically either 5 or 10.

来源:美国大学理事会 AP 课程与考试说明

假设比较两个比例:$H_0: p_1=p_2$ 对 $H_a: p_1\neq p_2$(或 $<,>$)。因为 $H_0$ 说比例相等,用一个合并(combined,pooled)样本比例 $\hat p_c=\dfrac{\text{total successes}}{\text{total sample size}}$ 来估计共同的 $p$。

词汇表 训练
英文 中文 拼音
combined (pooled) 合并 hé bìng
练习卷
6.11

执行两个总体比例之差的检验
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

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

VAR-6.K
Calculate an appropriate test statistic for the difference of two population proportions. [Skill 3.E]

  • VAR-6.K.1 The test statistic for a difference in proportions is: $z = \dfrac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}_c(1-\hat{p}_c)}\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$, where $\hat{p}_c = \dfrac{n_1\hat{p}_1 + n_2\hat{p}_2}{n_1 + n_2}$.
    • Clarifying statement: The formulas for test statistics do not appear explicitly on the AP Statistics Formula Sheet provided with the AP Statistics Exam. However, these formulas do not need to be memorized, as they can be constructed based on the general test statistic formula and the standard error formulas for each of the relevant test statistics that are provided on the formula sheet.

DAT-3
Significance testing allows us to make decisions about hypotheses within a particular context.

DAT-3.C
Interpret the $p$-value of a significance test for a difference of population proportions. [Skill 4.B]

  • DAT-3.C.1 An interpretation of the $p$-value of a significance test for a difference of two population proportions should recognize that the $p$-value is computed by assuming that the null hypothesis is true, i.e., by assuming that the true population proportions are equal to each other.

DAT-3.D
Justify a claim about the population based on the results of a significance test for a difference of population proportions. [Skill 4.E]

  • DAT-3.D.1 A formal decision explicitly compares the $p$-value to the significance $\alpha$. If the $p\text{-value} \leq \alpha$, then reject the null hypothesis, $H_0 : p_1 = p_2$, or $H_0 : p_1 - p_2 = 0$. If the $p$-value $> \alpha$, then fail to reject the null hypothesis.
  • DAT-3.D.2 The results of a significance test for a difference of two population proportions can serve as the statistical reasoning to support the answer to a research question about the two populations that were sampled.

来源:美国大学理事会 AP 课程与考试说明

合并的两比例 $z$ 统计量:

$$z=\frac{\hat p_1-\hat p_2}{\sqrt{\hat p_c(1-\hat p_c)\left(\frac{1}{n_1}+\frac{1}{n_2}\right)}}.$$
从正态模型求 $p$ 值、与 $\alpha$ 比较,并在上下文里得出结论——与单比例检验相同的四步逻辑。

练习卷
6.11

考试技巧
  • 在任何比例推断前陈述条件(随机、10%、大计数 $np,\,nq\ge10$)。
  • 一个置信区间 = 估计 $\pm$ 误差幅度;"95% 信心"指方法的长期捕获率。
  • 对一个检验,写 $H_0$ 和 $H_a$、计算检验统计量、求 p 值,并与 $\alpha$ 比较。
  • 一个小的 p 值是反对 $H_0$ 的证据;未能拒绝并不证明 $H_0$。
  • 更大的样本缩小误差幅度;一个更高的置信水平加宽它。

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