8.5: 章节公式回顾
- Page ID
- 204238
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)总体标准差的置信区间未知、小样本案例
\(s\)= 样本值的标准差。
\(t=\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\)是 t 分数的公式，用于衡量学生 t 分布中度量与总体均值的距离
\(df = n - 1\); 学生 t 分布的自由度，其中\(n\)表示样本的大小
\(T \sim t_{d f}\)随机变量具有学生的 t 分布，自由度为 df\(T\)
单一均值、总体标准差未知和样本数量小于 30 Student's t 的置信区间的一般形式由下式给出：\(\overline{x}-t_{\mathrm{v}, \alpha}\left(\frac{s}{\sqrt{n}}\right) \leq \mu \leq \overline{x}+t_{\mathrm{v}, \alpha}\left(\frac{s}{\sqrt{n}}\right)\)
总体比例的置信区间
\(p^{\prime}=\frac{x}{n}\)其中，\(x\)表示样本中的成功次数，\(n\)代表样本数量。 变量 p′ 是样本比例，用作真实总体比率的点估计值。
\(q^{\prime}=1-p^{\prime}\)
该变量\(p^{\prime}\)具有二项式分布，可以近似于此处显示的正态分布。 真实总体比率的置信区间由以下公式给出：
\(\mathrm{p}^{\prime}-Z_{\alpha} \sqrt{\frac{\mathrm{p}^{\prime} \mathrm{q}^{\prime}}{n}} \leq p \leq \mathrm{p}^{\prime}+Z_{\alpha} \sqrt{\frac{\mathrm{p}^{\prime} \mathrm{q}^{\prime}}{n}}\)
\(n=\frac{Z_{\frac{\alpha}{2}}^{2} p^{\prime} q^{\prime}}{e^{2}}\)提供估算总体比例\(p\)、置信\(1 - \alpha\)度和误差幅度进行抽样所需的观测值数量\(e\)。 其中\(e\) = 实际总体比例和样本比率之间的可接受差值。
计算样本数量 n：连续和二进制随机变量
\(n=\frac{Z^{2} \sigma^{2}}{(\overline{x}-\mu)^{2}}\)= 用于确定连续随机变量在给定置信水平下达到所需误差幅度所需的样本数量 (\(n\)) 的公式
\(n=\frac{Z_{\alpha}^{2} \mathrm{pq}}{e^{2}}\)= 如果随机变量为二进制，则用于确定样本数量的公式