diff --git a/Visulization/Statistical Test.ipynb b/Visulization/Statistical Test.ipynb
new file mode 100644
index 0000000..48b139d
--- /dev/null
+++ b/Visulization/Statistical Test.ipynb	
@@ -0,0 +1,296 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import pandas as pd\n",
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "from statsmodels.stats.proportion import proportions_ztest\n",
+    "pd.set_option('mode.chained_assignment', None)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# data import\n",
+    "train = pd.read_csv('Training_Dataset.csv')\n",
+    "train_labels = pd.read_csv(\"Labels_Train.csv\")\n",
+    "test = pd.read_csv(\"Repackaged_Benign_Testset.csv\")\n",
+    "test_labels = pd.read_csv(\"Labels_Repackaged_Benign_Test.csv\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "train['label'] = train_labels['label']\n",
+    "test['label'] = test_labels['label']\n",
+    "data = pd.concat([train, test])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Number of benign apps: 4862\n",
+      "Number of malicious apps: 9142\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(\"Number of benign apps:\", data[data['label']==0].shape[0])\n",
+    "print(\"Number of malicious apps:\", data[data['label']==1].shape[0])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "sensor_types = data.iloc[:, -30:-1].columns"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# get number of sensors used\n",
+    "data['num_sensors'] = data[sensor_types].sum(axis=1)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "count    14004.000000\n",
+       "mean         0.384462\n",
+       "std          1.569956\n",
+       "min          0.000000\n",
+       "25%          0.000000\n",
+       "50%          0.000000\n",
+       "75%          0.000000\n",
+       "max         29.000000\n",
+       "Name: num_sensors, dtype: float64"
+      ]
+     },
+     "execution_count": 7,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "data['num_sensors'].describe()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "Text(0.5, 1.0, 'Distribution of number of sensors used (Malicious apps)')"
+      ]
+     },
+     "execution_count": 8,
+     "metadata": {},
+     "output_type": "execute_result"
+    },
+    {
+     "data": {
+      "image/png": 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\n",
+      "text/plain": [
+       "<Figure size 432x288 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "plt.hist(data[data['label']==1]['num_sensors'])\n",
+    "plt.title(\"Distribution of number of sensors used (Malicious apps)\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "Text(0.5, 1.0, 'Distribution of number of sensors used (Benign apps)')"
+      ]
+     },
+     "execution_count": 9,
+     "metadata": {},
+     "output_type": "execute_result"
+    },
+    {
+     "data": {
+      "image/png": "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\n",
+      "text/plain": [
+       "<Figure size 432x288 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "plt.hist(data[data['label']==0]['num_sensors'])\n",
+    "plt.title(\"Distribution of number of sensors used (Benign apps)\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# whether an app uses any sensors\n",
+    "data['use_sensors'] = data['num_sensors']>0"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "label\n",
+       "0    0.137392\n",
+       "1    0.186721\n",
+       "Name: use_sensors, dtype: float64"
+      ]
+     },
+     "execution_count": 11,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "data.groupby('label')['use_sensors'].mean()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Let $p$ be the proportion of apps that use at least one sensor.\n",
+    "\n",
+    "$$H_0: p_1 = p_2 $$\n",
+    "\n",
+    "$$H_a: p_1 > p_2 $$\n",
+    "\n",
+    "$p_1$: malicious apps\n",
+    "\n",
+    "$p_2$: benign apps"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "z_stat: 7.405, p_value: 0.000\n",
+      "Reject the null hypothesis. Proportion of malicious apps using sensors is higher.\n"
+     ]
+    }
+   ],
+   "source": [
+    "significance = 0.01\n",
+    "\n",
+    "sample_success_a, sample_size_a = (data[data['label']==1]['use_sensors'].value_counts().values[1], \n",
+    "                                   data[data['label']==1]['use_sensors'].shape[0])\n",
+    "sample_success_b, sample_size_b = (data[data['label']==0]['use_sensors'].value_counts().values[1], \n",
+    "                                   data[data['label']==0]['use_sensors'].shape[0])\n",
+    "\n",
+    "successes = np.array([sample_success_a, sample_success_b])\n",
+    "samples = np.array([sample_size_a, sample_size_b])\n",
+    "\n",
+    "stat, p_value = proportions_ztest(count = successes, nobs = samples, \n",
+    "                                  alternative = 'larger')\n",
+    "\n",
+    "print('z_stat: %0.3f, p_value: %0.3f' % (stat, p_value))\n",
+    "if p_value > significance:\n",
+    "    print(\"Fail to reject the null hypothesis\")\n",
+    "else:\n",
+    "    print(\"Reject the null hypothesis. Proportion of malicious apps using sensors is higher.\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 19,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "p1: 0.187\n",
+      "p2: 0.137\n"
+     ]
+    }
+   ],
+   "source": [
+    "print('p1: %0.3f' % (sample_success_a / sample_size_a))\n",
+    "print('p2: %0.3f' % (sample_success_b / sample_size_b))"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.7.6"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}