10_Hypothesis_Testing.ipynb

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      "Problems 1, 2, 6, 7, 8, 9, 10, 12\n"
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    "%matplotlib inline\n",
    "print \"Problems 1, 2, 6, 7, 8, 9, 10, 12\""
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    "<b>1.</b> Not enough information. Deaths could be attributed to many things. "
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    "<b>(a)</b> Observational study <p></p>\n",
    "<b>(b)</b> The study was trying to limit confounding factors <p></p>\n",
    "<b>(c)</b> That conclusion ignores other confounding factors (people being forced to stop smoking etc) <p></p>"
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    "<b>(a)</b> Null hypothesis: coin is not biased. <p></p>\n",
    "Alternative hypothesis: Coin is biased if I find with 95 % confidence that it is over the upward bound (2 standard deviations). <p></p>\n",
    "<b>(b)</b> z = <sup>((Observed) - (expected))</sup>&frasl;<sub>standard deviation</sub> => z = <sup>((5400) - (5000))</sup>&frasl;<sub>50</sub> = 8  <p></p>\n",
    "<i>p-</i>value: way less than 1% <p></p>\n",
    "<b>(c)</b> The coined is obviously biased towards heads (with 95% + confidence). Reject the null hypothesis. "
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   "source": [
    "<b>approach:</b> find z statistic <p></p>\n",
    "z = <sup>((Observed) - (expected))</sup>&frasl;<sub>standard deviation</sub> => z = <sup>((368) - (350))</sup>&frasl;<sub>var<sup>1/2</sup></sub><p></p>\n",
    "var = n * (E(X<sup>2</sup>) - &micro;<sup>2</sup>) =>\n"
   ]
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   "source": [
    "var = <sup>1</sup>&frasl;<sub>6</sub>(1<sup>2</sup> + 2<sup>2</sup> + 3<sup>2</sup> + 4<sup>2</sup> + 5<sup>2</sup> + 6<sup>2</sup>)  - 3.5<sup>2</sup><p></p>"
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     "text": [
      "The z statistic is only:  1.05397207878  and so it is unlikely the dice are loaded\n"
     ]
    }
   ],
   "source": [
    "var = 100 *  ((1**2 + 2**2 + 3**2 + 4**2 + 5**2 + 6**2)*(1.0/6) - (3.5**2))\n",
    "z = (368 - 350) / (var**(1.0/2))\n",
    "print \"The z statistic is only: \", z, \" and so it is unlikely the dice are loaded\""
   ]
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   "source": [
    "<img src = \"images/Worksheet10/8.png\">"
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   "execution_count": 7,
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     "text": [
      "Higher p-value\n"
     ]
    }
   ],
   "source": [
    "print \"Higher p-value\""
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    "<img src = \"images/Worksheet10/9.png\">"
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     "text": [
      "(a) std =  0.0196009839403\n",
      "z value:  5.56094532458\n",
      "difference is real and definitely not chance \n"
     ]
    }
   ],
   "source": [
    "\n",
    "\n",
    "var = ((.219)*(1-.219))/700\n",
    "var2 = ((.11)*(1-.11))/700\n",
    "std = (var + var2)**(.5)\n",
    "print '(a) std = ', std\n",
    "z = (.219 - .110 - 0 ) / std\n",
    "print 'z value: ', z\n",
    "print 'difference is real and definitely not chance '"
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     "text": [
      "(b) std =  0.0253568249477\n"
     ]
    }
   ],
   "source": [
    "var = ((.319)*(1-.319))/700\n",
    "var2 = ((.369)*(1-.369))/700\n",
    "std = (var + var2)**(.5)\n",
    "print '(b) std = ', std"
   ]
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     "name": "stdout",
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     "text": [
      "z value:  1.97185570761\n",
      "difference is real\n"
     ]
    }
   ],
   "source": [
    "z = (.369 - .319 ) / std\n",
    "print 'z value: ', z\n",
    "print 'difference is real'"
   ]
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    "<img src = \"images/Worksheet10/10.png\">"
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   "source": [
    "<b>approach:</b> find z statistic <p></p>\n",
    "z = <sup>((Observed) - (expected))</sup>&frasl;<sub>standard deviation</sub> => z = <sup>((12.2) - (9.2))</sup>&frasl;<sub>var<sup>1/2</sup></sub> =><p></p>"
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   "source": [
    "&#963; = (&#963;<sub>private</sub><sup>2</sup> + &#963;<sub>public</sub><sup>2</sup>)<sup>1/2</sup> = 14.43 (this works because var can be added)"
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   "execution_count": 16,
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     "name": "stdout",
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     "text": [
      "z:  6.20863905406 \n",
      "This value suggests this difference in averages is probably not due to chance\n"
     ]
    }
   ],
   "source": [
    "(10.5**2 + 9.9**2)**(.5)\n",
    "sig1 = 12.2/ (1000**0.5)\n",
    "sig2 = 9.2 / (1000**0.5)\n",
    "std = (sig1**2 + sig2**2)**.5\n",
    "z = (12.2 - 9.2)/std\n",
    "z\n",
    "print \"z: \", z, \"\\nThis value suggests this difference in averages is probably not due to chance\""
   ]
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