4: \u5e30\u7121\u4eee\u8aac\\(H_0\\)\u304c\u6b63\u3057\u3044\u3068\u4eee\u5b9a\u3057\u3066\u691c\u5b9a\u7d71\u8a08\u91cf\u3092\u8a08\u7b97<\/strong><\/span><\/p>\r\n\u5e30\u7121\u4eee\u8aac\\(H_0:p=1\/2\\)\u306e\u3082\u3068\u3067\u306e\u691c\u5b9a\u7d71\u8a08\u91cf\\(Z_0\\)\u306f<\/p>\r\n
$$ Z_0=\\frac{\\bar{x}-\\mu_0}{\\sigma_0\/\\sqrt{N}} $$<\/p>\r\n
\u3068\u8868\u3055\u308c\u307e\u3059\u3002\u306a\u304a\u3001\\(\\mu = 1\/(1-p),\\ \\sigma^2 = p\/(1-p)^2\\)\u3060\u3063\u305f\u306e\u3067<\/p>\r\n
$$ \\begin{align*} \\mu_0=\\frac{1}{1-1\/2}=2,\\ \\sigma_0=\\frac{\\sqrt{1\/2}}{1-1\/2}=\\sqrt{2} \\end{align*} $$<\/p>\r\n
\u3068\u306a\u308a\u307e\u3059\u3002\u305d\u3053\u3067\u3001\u691c\u5b9a\u7d71\u8a08\u91cf\\(Z_0\\)\u306b<\/p>\r\n
$$ \\mu_0=2, \\sigma_0=\\sqrt{2}, \\bar{x}=1.89, N=100 $$<\/p>\r\n
\u3092\u4ee3\u5165\u3057\u3066\u5f97\u3089\u308c\u308b\u5024\\(z\\)\u3092\u6c42\u3081\u307e\u3059\u3002<\/p>\r\n
$$ \\begin{align*} z =\\frac{1.89-2}{\\sqrt{2}\/\\sqrt{100}} \\approx-0.7778 \\end{align*} $$<\/p>\r\n
<\/p>\r\n
5: \u68c4\u5374\u306e\u53ef\u5426\u3092\u5224\u65ad<\/strong><\/span><\/p>\r\n\\(z > -1.645\\)\u3067\u3042\u3063\u305f\u306e\u3067\u3001\u5e30\u7121\u4eee\u8aac\\(H_0: p=1\/2\\)\u306f\u68c4\u5374\u3055\u308c\u307e\u305b\u3093<\/strong>\u3002<\/p>\r\n\u3059\u306a\u308f\u3061\u3001\u4eca\u56de\u53ce\u96c6\u3057\u305f\u30c7\u30fc\u30bf\u304b\u3089\u306f\u300c\u30ab\u30b9\u30df\u306e\u30b3\u30a4\u30f3\u3067\u8868\u304c\u51fa\u308b\u78ba\u7387\u306f1\/2\u3088\u308a\u7d71\u8a08\u7684\u306b\u6709\u610f\u306b\u4f4e\u3044\u300d\u3053\u3068\u3092\u793a\u3059\u3053\u3068\u306f\u3067\u304d\u307e\u305b\u3093\u3067\u3057\u305f<\/strong>\u3002<\/p>\r\n<\/p>\n\n
\u307e\u3068\u3081<\/h2>\n
\u672c\u30d6\u30ed\u30b0\u8a18\u4e8b\u3067\u306f\u3001\u30ab\u30b9\u30df\u306e\u30b3\u30a4\u30f3\u30c8\u30b9\u304c\u5e7e\u4f55\u5206\u5e03\u3092\u7528\u3044\u305f\u78ba\u7387\u30e2\u30c7\u30eb\u3067\u8868\u73fe\u3067\u304d\u308b\u3053\u3068\u3092\u5229\u7528\u3057\u3066\u3001\u7d71\u8a08\u7684\u4eee\u8aac\u691c\u5b9a\uff08\u7279\u306b\u4e2d\u5fc3\u6975\u9650\u5b9a\u7406\u3092\u7528\u3044\u305fZ\u691c\u5b9a\uff09\u3092\u884c\u3044\u307e\u3057\u305f<\/strong>\u3002<\/p>\r\n\u4eca\u56de\u96c6\u3081\u305f\u30c7\u30fc\u30bf\u304b\u3089\u306f\u3001\u300c\u30ab\u30b9\u30df\u306e\u30b3\u30a4\u30f3\u3067\u8868\u304c\u51fa\u308b\u78ba\u7387\u306f1\/2\u3088\u308a\u7d71\u8a08\u7684\u306b\u6709\u610f\u306b\u4f4e\u3044\u300d\u3068\u3044\u3046\u3053\u3068\u306f\u793a\u3055\u308c\u307e\u305b\u3093\u3067\u3057\u305f<\/strong>\u3002<\/p>\r\n\u3053\u3053\u304b\u3089\u306f\u500b\u4eba\u7684\u306a\u611f\u60f3\u306b\u306a\u308a\u307e\u3059\u304c\u3001\u5b9f\u969b\u306b\u30ab\u30b9\u30df\u3092\u4f7f\u3063\u3066\u30c7\u30fc\u30bf\u53ce\u96c6\u3057\u306a\u304c\u3089\u300c\u88cf\u3070\u304b\u308a\u51fa\u308b\u304b\u3089\u7d76\u5bfe1\/2\u3088\u308a\u4f4e\u304f\u8a2d\u5b9a\u3055\u308c\u3066\u3044\u308b\u306a\u300d\u3068\u611f\u3058\u3066\u3044\u305f\u3082\u306e\u306e\u3001\u88cf\u304c\u51fa\u305f\u5834\u9762\u304c\u8a18\u61b6\u306b\u6b8b\u308a\u3084\u3059\u304b\u3063\u305f\u3060\u3051\u306a\u306e\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u3002\u306a\u306e\u3067\u3001\u304a\u305d\u3089\u304f\u30aa\u30f3\u30e9\u30a4\u30f3\u5bfe\u6226\u3067\u3082\u30ab\u30b9\u30df\u304c\u30b3\u30a4\u30f3\u3067\u8868\u3092\u51fa\u3059\u78ba\u7387\u306f1\/2\u306b\u8a2d\u5b9a\u3055\u308c\u3066\u3044\u308b\u306e\u3067\u306f\u306a\u3044\u304b\u306a<\/strong>\u3068\u8003\u3048\u3066\u3044\u307e\u3059\u3002<\/p>\r\n\u30b5\u30f3\u30d7\u30eb\u30b5\u30a4\u30ba\\(N=100\\)\u304c\u5341\u5206\u3067\u306f\u306a\u3044\u3068\u3044\u3046\u61f8\u5ff5\u304c\u3042\u308b\u305f\u3081\u3001\u4eca\u56de\u7d39\u4ecb\u3057\u305f\u624b\u9806\u3092\u7528\u3044\u3066\u66f4\u306b\u591a\u304f\u306e\u30b5\u30f3\u30d7\u30eb\u306b\u5bfe\u3057\u3066\u4eee\u8aac\u691c\u5b9a\u3092\u5b9f\u65bd\u3067\u304d\u308c\u3070\u3001\u3088\u308a\u4fe1\u983c\u3067\u304d\u308b\u7d50\u679c\u304c\u5f97\u3089\u308c\u308b\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\r\n
<\/p>\n\n
\u4ed8\u9332<\/h2>\n
1\u679a\u306e\u30ab\u30b9\u30df\u304c\u30b3\u30a4\u30f3\u30c8\u30b9\u3092\u884c\u3046\u56de\u6570\\(X\\)\u306e\u671f\u5f85\u5024\\(\\mu\\)\u3068\u5206\u6563\\(\\sigma^2\\)\u306e\u5c0e\u51fa<\/span><\/strong><\/p>\r\n\u671f\u5f85\u5024\\(\\mu\\)\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\r\n
$$ \\begin{split} \\mu &= E[X]\\\\ &= \\sum_{k=1}^{\\infty} k \\cdot P(X=k)\\\\ &= \\sum_{k=1}^{\\infty} k \\cdot p^{k-1}(1-p)\\\\ &= (1-p) \\sum_{k=1}^{\\infty} k \\cdot p^{k-1}\\\\ &= \\sum_{k=1}^{\\infty} k \\cdot p^{k-1} – p \\sum_{k=1}^{\\infty} k \\cdot p^{k-1}\\\\ &= \\sum_{k=1}^{\\infty} k \\cdot p^{k-1} – \\sum_{k=1}^{\\infty} k \\cdot p^{k}\\\\ &= \\sum_{k=1}^{\\infty} p^{k-1}\\\\ &= \\frac{1}{1-p} \\end{split} $$<\/p>\r\n
\u5206\u6563\\(\\sigma^2\\)\u306e\u8a08\u7b97\u306e\u305f\u3081\u306b\u3001\\(X^2\\)\u306e\u671f\u5f85\u5024\u3092\u6c42\u3081\u3066\u304a\u304d\u307e\u3059\u3002<\/p>\r\n
$$ \\begin{split} E[X^2] &= \\sum_{k=1}^{\\infty} k^2 \\cdot P(X=k)\\\\ &= \\sum_{k=1}^{\\infty} k^2 \\cdot p^{k-1}(1-p)\\\\ &= (1-p) \\sum_{k=1}^{\\infty} k^2 \\cdot p^{k-1}\\\\ &= \\sum_{k=1}^{\\infty} k^2 \\cdot p^{k-1} – p \\sum_{k=1}^{\\infty} k^2 \\cdot p^{k-1}\\\\ &= \\sum_{k=1}^{\\infty} k^2 \\cdot p^{k-1} – \\sum_{k=1}^{\\infty} k^2 \\cdot p^{k}\\\\ &= \\sum_{k=1}^{\\infty} (2k-1)p^{k-1}\\\\ &= 2 \\sum_{k=1}^{\\infty} k \\cdot p^{k-1} – \\sum_{k=1}^{\\infty} p^{k-1}\\\\ &= \\frac{2}{(1-p)^2} – \\frac{1}{1-p}\\\\ &=\\frac{1+p}{(1-p)^2} \\end{split} $$<\/p>\r\n
\u4ee5\u4e0a\u304b\u3089\u3001\u5206\u6563\\(\\sigma^2\\)\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\r\n
$$ \\begin{split} \\sigma^2 &= E[X^2] – E[X]^2\\\\ &= \\frac{1+p}{(1-p)^2} – \\bigg\\{ \\frac{1}{1-p} \\bigg\\}^2\\\\ &= \\frac{p}{(1-p)^2} \\end{split} $$<\/p>\r\n
<\/p>","protected":false},"excerpt":{"rendered":"
\u6700\u8fd1\u30ea\u30ea\u30fc\u30b9\u3055\u308c\u305f\u30dd\u30b1\u30ab\u30a2\u30d7\u30ea\u300c\u30dd\u30b1\u30dd\u30b1\u300d\u306e\u30ab\u30b9\u30df\u3067\u300c\u30b3\u30a4\u30f3\u306e\u8868\u304c\u51fa\u308b\u78ba\u7387\u304c\u4e0b\u65b9\u4fee\u6b63\u3055\u308c\u305f\u306e\u3067\u306f\u306a\u3044\u304b\u300d\u3068\u3044\u3046\u5642\u304c\u805e\u304b\u308c\u307e\u3059\u3002\u672c\u30d6\u30ed\u30b0\u8a18\u4e8b\u3067\u306f\u3001\u5b9f\u969b\u306b\u30ab\u30b9\u30df\u3092\u4f7f\u3063\u3066\u8868\u304c\u51fa\u7d9a\u3051\u308b\u56de\u6570\u3092\u96c6\u8a08\u3057\u300c\u30b3\u30a4\u30f3\u306e\u8868\u304c\u51fa\u308b\u78ba\u7387\u304c1\/2\u3067 […]<\/p>\n","protected":false},"author":46,"featured_media":5928,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"content-type":"","swell_btn_cv_data":"","footnotes":"","_wp_rev_ctl_limit":""},"categories":[1246],"tags":[39,801,57],"class_list":["post-5899","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-data-infrastructure","tag-39","tag-801","tag-57"],"_links":{"self":[{"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/posts\/5899","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/users\/46"}],"replies":[{"embeddable":true,"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/comments?post=5899"}],"version-history":[{"count":0,"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/posts\/5899\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/media\/5928"}],"wp:attachment":[{"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/media?parent=5899"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/categories?post=5899"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/tags?post=5899"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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