Create Z检验案例

notebook_必备数学基础/假设检验章节/.ipynb_checkpoints/假设检验-checkpoint.ipynb

@@ -268,6 +268,137 @@
     "$$"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Z检验实例\n",
+    "### Z检验实例1："
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "研究正常人与高血压患者胆固醇含量（mg%）的资料如下，试比较两组血清胆固醇含量有无差别。\n",
+    "<br>\n",
+    "正常人组 \n",
+    "$$\n",
+    "n_1 = 506,\\overline{X}_1 = 180.6,S_1 = 34.2\n",
+    "$$\n",
+    "样本数506，均值1800.6，标准差34.2\n",
+    "<br>\n",
+    "<br>\n",
+    "高血压组\n",
+    "$$\n",
+    "n_2 = 142,\\overline{X}_2 = 223.6,S_2 = 45.8\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "建立检验假设，确定检验水平\n",
+    "<ul>\n",
+    "<li>H0: μ1 = μ2；认为没有差别\n",
+    "<li>H1: μ1 ≠μ2；认为有区别\n",
+    "<li>α = 0.05   ；有5%的置信空间，即误差在这个范围内是允许的\n",
+    "</li>\n",
+    "</ul>\n",
+    "计算统计量Z\n",
+    "<ul>\n",
+    "    <li>将已知数据代入公式</li>\n",
+    "$$\n",
+    "Z = \\frac{|180.6-223.6|}{\\sqrt{34.2^2/506+45.8^2/142}} = 10.40\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "这里是双侧，即二分之一α，1-α/2=0.975，查表\n",
+    "<br>网上搜索：统计分布临界值\n",
+    "<img src=\"assets/20201115111943.png\" width=\"70%\">\n",
+    "<br><br>\n",
+    "1.9+0.6=1.96，统计量为10.4比1.96大，意味着面积肯定小于1.96临界值（α/2）的面积\n",
+    "<img src=\"assets/20201115112520.png\" width=\"70%\">\n",
+    "根据双侧检验，若p值 < α/2，拒绝H0"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "确认P值，作为判断结论；本例Z=10.40>1.96（查表得0.975对应值），故P<0.05，按α=0.05水准拒绝H0，接受H1，可以认为正常人与高血压患者的血清胆固醇含量有差别，高血压患者高于正常人。\n",
+    "<br>\n",
+    "**注意：我们的第一反应可能是不应该越小表示差异也越小吗，其实是越小于α/2，表示两者值的越偏离，因为我们已经假定了对比值A在中间（H0值），那么对比A（H0）的B（H1）应该越趋向中间（越高），才表示相似**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Z检验实例2："
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "某机床厂加工一种零件，根据经验知道，该厂加工零件的椭圆度近似服从正态分布，其总体均值为p=0.081mm，总体标准差为=0.025。今换一种新机床进行加抽取n=200个零件进行检验，得到的椭圆度为0.076mm。试问新机床加工零件的椭圆度的均值与以前有无显著差异?(a=0.05)"
+   ]
+  },
+  {
+   "attachments": {},
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "H0: μ = 0.081；H1：μ ≠ 0.081； α = 0.05； n = 200\n",
+    "<br>**检验统计量**\n",
+    "$$\n",
+    "z = \\frac{\\overline{x}-μ_0}{σ/\\sqrt{n}} \n",
+    "= \n",
+    "\\frac{0.076-0.081}{0.025/\\sqrt{200}} = -2.83\n",
+    "$$\n",
+    "<br>\n",
+    "决策：\n",
+    "<br>-2.83在-1.96左侧，也就是p值的面积小于α/2，在α = 0.05的水平上拒绝H0\n",
+    "<br>结论：\n",
+    "<br>有证据表明新机床加工的零件的椭圆度与以前有显著差异\n",
+    "<img src=\"assets/20201115151902.png\" width=\"50%\">"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Z检验实例3：\n",
+    "根据过去大量资料，某厂生产的灯泡的使用寿命服从正态分布N~(1020,100^2)。现从最近生产的一批产品中随机抽取16只,测得样本平均寿命为1080小时。试在005的显著性水平下判断这批产品的使用寿命是否有显著提高?(a=0.05)"
+   ]
+  },
+  {
+   "attachments": {},
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "H0: μ ≤ 1020；H1：μ ＞ 1020； α = 0.05； n = 16\n",
+    "<br>**检验统计量（单侧）**\n",
+    "$$\n",
+    "z = \\frac{\\overline{x}-μ_0}{σ/\\sqrt{n}} \n",
+    "= \n",
+    "\\frac{1080-1020}{100/\\sqrt{14}} = 2.4\n",
+    "$$\n",
+    "1-0.05=0.95，其临界值没有，相近的是0.9505和0.9495，那么把它们两相加除以2作为0.95的临界值，(1.6+1.6)/2+(0.04+0.05)/2=1.645\n",
+    "<br>\n",
+    "决策：\n",
+    "<br>2.4在1.645右侧，也就是p值的面积小于α，在α = 0.05的水平上拒绝H0\n",
+    "<br>结论：\n",
+    "<br>有证据表明新生产的灯泡的使用寿命有显著提高\n",
+    "<img src=\"assets/20201115152913.png\" width=\"50%\">"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": null,

notebook_必备数学基础/假设检验章节/假设检验.ipynb

@@ -268,6 +268,137 @@
     "$$"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Z检验实例\n",
+    "### Z检验实例1："
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "研究正常人与高血压患者胆固醇含量（mg%）的资料如下，试比较两组血清胆固醇含量有无差别。\n",
+    "<br>\n",
+    "正常人组 \n",
+    "$$\n",
+    "n_1 = 506,\\overline{X}_1 = 180.6,S_1 = 34.2\n",
+    "$$\n",
+    "样本数506，均值1800.6，标准差34.2\n",
+    "<br>\n",
+    "<br>\n",
+    "高血压组\n",
+    "$$\n",
+    "n_2 = 142,\\overline{X}_2 = 223.6,S_2 = 45.8\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "建立检验假设，确定检验水平\n",
+    "<ul>\n",
+    "<li>H0: μ1 = μ2；认为没有差别\n",
+    "<li>H1: μ1 ≠μ2；认为有区别\n",
+    "<li>α = 0.05   ；有5%的置信空间，即误差在这个范围内是允许的\n",
+    "</li>\n",
+    "</ul>\n",
+    "计算统计量Z\n",
+    "<ul>\n",
+    "    <li>将已知数据代入公式</li>\n",
+    "$$\n",
+    "Z = \\frac{|180.6-223.6|}{\\sqrt{34.2^2/506+45.8^2/142}} = 10.40\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "这里是双侧，即二分之一α，1-α/2=0.975，查表\n",
+    "<br>网上搜索：统计分布临界值\n",
+    "<img src=\"assets/20201115111943.png\" width=\"70%\">\n",
+    "<br><br>\n",
+    "1.9+0.6=1.96，统计量为10.4比1.96大，意味着面积肯定小于1.96临界值（α/2）的面积\n",
+    "<img src=\"assets/20201115112520.png\" width=\"70%\">\n",
+    "根据双侧检验，若p值 < α/2，拒绝H0"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "确认P值，作为判断结论；本例Z=10.40>1.96（查表得0.975对应值），故P<0.05，按α=0.05水准拒绝H0，接受H1，可以认为正常人与高血压患者的血清胆固醇含量有差别，高血压患者高于正常人。\n",
+    "<br>\n",
+    "**注意：我们的第一反应可能是不应该越小表示差异也越小吗，其实是越小于α/2，表示两者值的越偏离，因为我们已经假定了对比值A在中间（H0值），那么对比A（H0）的B（H1）应该越趋向中间（越高），才表示相似**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Z检验实例2："
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "某机床厂加工一种零件，根据经验知道，该厂加工零件的椭圆度近似服从正态分布，其总体均值为p=0.081mm，总体标准差为=0.025。今换一种新机床进行加抽取n=200个零件进行检验，得到的椭圆度为0.076mm。试问新机床加工零件的椭圆度的均值与以前有无显著差异?(a=0.05)"
+   ]
+  },
+  {
+   "attachments": {},
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "H0: μ = 0.081；H1：μ ≠ 0.081； α = 0.05； n = 200\n",
+    "<br>**检验统计量**\n",
+    "$$\n",
+    "z = \\frac{\\overline{x}-μ_0}{σ/\\sqrt{n}} \n",
+    "= \n",
+    "\\frac{0.076-0.081}{0.025/\\sqrt{200}} = -2.83\n",
+    "$$\n",
+    "<br>\n",
+    "决策：\n",
+    "<br>-2.83在-1.96左侧，也就是p值的面积小于α/2，在α = 0.05的水平上拒绝H0\n",
+    "<br>结论：\n",
+    "<br>有证据表明新机床加工的零件的椭圆度与以前有显著差异\n",
+    "<img src=\"assets/20201115151902.png\" width=\"50%\">"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Z检验实例3：\n",
+    "根据过去大量资料，某厂生产的灯泡的使用寿命服从正态分布N~(1020,100^2)。现从最近生产的一批产品中随机抽取16只,测得样本平均寿命为1080小时。试在005的显著性水平下判断这批产品的使用寿命是否有显著提高?(a=0.05)"
+   ]
+  },
+  {
+   "attachments": {},
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "H0: μ ≤ 1020；H1：μ ＞ 1020； α = 0.05； n = 16\n",
+    "<br>**检验统计量（单侧）**\n",
+    "$$\n",
+    "z = \\frac{\\overline{x}-μ_0}{σ/\\sqrt{n}} \n",
+    "= \n",
+    "\\frac{1080-1020}{100/\\sqrt{14}} = 2.4\n",
+    "$$\n",
+    "1-0.05=0.95，其临界值没有，相近的是0.9505和0.9495，那么把它们两相加除以2作为0.95的临界值，(1.6+1.6)/2+(0.04+0.05)/2=1.645\n",
+    "<br>\n",
+    "决策：\n",
+    "<br>2.4在1.645右侧，也就是p值的面积小于α，在α = 0.05的水平上拒绝H0\n",
+    "<br>结论：\n",
+    "<br>有证据表明新生产的灯泡的使用寿命有显著提高\n",
+    "<img src=\"assets/20201115152913.png\" width=\"50%\">"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": null,