{"id":1381,"date":"2026-01-27T16:12:07","date_gmt":"2026-01-27T06:12:07","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1381"},"modified":"2026-01-27T16:14:47","modified_gmt":"2026-01-27T06:14:47","slug":"year12-math-3-4-3-statistical-inference-sample-proportions","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1381","title":{"rendered":"Year12 MATH 3-4-3 statistical inference (sample proportions)"},"content":{"rendered":"\n

In QLD Year 12 Mathematical Methods (General Subject) Unit 4, statistical inference (sample proportions)<\/strong> is the process of using data from a sample to estimate an unknown population proportion<\/mark> (p<\/mi>p<\/annotation><\/semantics><\/math>). It is the culmination of probability studies, where students move from understanding theoretical distributions to analyzing real-world data to make predictions. <\/p>\n\n\n\n

In this unit, statistical inference is specifically focused on: <\/p>\n\n\n\n

    \n
  • Two-outcome populations:<\/strong> Situations where outcomes are binary (e.g., yes\/no, success\/failure, vote Liberal\/Labor).<\/li>\n\n\n\n
  • Estimating proportions:<\/strong> Using sample proportions (p<\/mi>\u0302<\/mo><\/mover>p hat<\/annotation><\/semantics><\/math>) to infer the true, unknown population proportion (p<\/mi>p<\/annotation><\/semantics><\/math>).\u00a0<\/li>\n<\/ul>\n\n\n\n

    Key Concepts and Components <\/p>\n\n\n\n

      \n
    1. Sample Proportion (\ud835\udc5d\u0302):<\/strong> Defined as p<\/mi>\u0302<\/mo><\/mover>=<\/mo>X<\/mi>n<\/mi><\/mfrac><\/mrow>p hat equals the fraction with numerator cap X and denominator n end-fraction<\/annotation><\/semantics><\/math>, where X<\/mi>cap X<\/annotation><\/semantics><\/math> is the number of successes in a sample of size n<\/mi>n<\/annotation><\/semantics><\/math>. The sample proportion is a random variable, meaning its value varies from sample to sample.<\/li>\n\n\n\n
    2. Sampling Distribution of \ud835\udc5d\u0302:<\/strong> For large sample sizes, the distribution of the sample proportion p<\/mi>\u0302<\/mo><\/mover>p hat<\/annotation><\/semantics><\/math> is approximately normal, with:\n
        \n
      • Mean:<\/strong> \u03bc<\/mi>p<\/mi>\u0302<\/mo><\/mover><\/msub>=<\/mo>p<\/mi><\/mrow>mu sub p hat end-sub equals p<\/annotation><\/semantics><\/math><\/li>\n\n\n\n
      • Standard Deviation:<\/strong> \u03c3<\/mi>p<\/mi>\u0302<\/mo><\/mover><\/msub>=<\/mo>p<\/mi>(<\/mo>1<\/mn>\u2212<\/mo>p<\/mi>)<\/mo><\/mrow>n<\/mi><\/mfrac><\/msqrt><\/mrow>sigma sub p hat end-sub equals the square root of the fraction with numerator p open paren 1 minus p close paren and denominator n end-fraction end-root<\/annotation><\/semantics><\/math><\/li>\n<\/ul>\n<\/li>\n\n\n\n
      • Confidence Intervals for Proportions:<\/strong> This is the core application, providing a range of values within which the true population proportion (p<\/mi>p<\/annotation><\/semantics><\/math>) is likely to lie, based on a given level of confidence.\n
          \n
        • Formula:<\/strong> p<\/mi>\u0302<\/mo><\/mover>\u00b1<\/mo>z<\/mi>p<\/mi>\u0302<\/mo><\/mover>(<\/mo>1<\/mn>\u2212<\/mo>p<\/mi>\u0302<\/mo><\/mover>)<\/mo><\/mrow>n<\/mi><\/mfrac><\/msqrt><\/mrow>p hat plus or minus z the square root of the fraction with numerator p hat open paren 1 minus p hat close paren and denominator n end-fraction end-root<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n
        • Common Levels:<\/strong> 95% confidence intervals are standard (using z<\/mi>=<\/mo>1.96<\/mn><\/mrow>z equals 1.96<\/annotation><\/semantics><\/math>).<\/li>\n<\/ul>\n<\/li>\n\n\n\n
        • Margin of Error:<\/strong> The distance from the sample proportion (\ud835\udc5d\u0302) to the endpoints of the confidence interval, reflecting the uncertainty of the estimate. It is calculated as z<\/mi>p<\/mi>\u0302<\/mo><\/mover>(<\/mo>1<\/mn>\u2212<\/mo>p<\/mi>\u0302<\/mo><\/mover>)<\/mo><\/mrow>n<\/mi><\/mfrac><\/msqrt><\/mrow>z the square root of the fraction with numerator p hat open paren 1 minus p hat close paren and denominator n end-fraction end-root<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n
        • Simulation:<\/strong> Using technology to simulate repeated sampling to understand how \ud835\udc5d\u0302 varies and to observe the, approximately, normal distribution of p<\/mi>\u0302<\/mo><\/mover>p hat<\/annotation><\/semantics><\/math>.\u00a0<\/li>\n<\/ol>\n\n\n\n

          Key Skills Required <\/strong><\/p>\n\n\n\n

            \n
          • Calculating the point estimate (p<\/mi>\u0302<\/mo><\/mover>p hat<\/annotation><\/semantics><\/math>) and the confidence interval for a given sample.<\/li>\n\n\n\n
          • Interpreting what a confidence interval means in context.<\/li>\n\n\n\n
          • Using technology (e.g., calculators) to determine interval estimates for proportions.<\/li>\n\n\n\n
          • Determining the necessary sample size (n<\/mi>n<\/annotation><\/semantics><\/math>) to achieve a specific margin of error.\u00a0<\/li>\n<\/ul>\n\n\n\n

            This topic often utilizes simulation and technology to visualize how the sample proportion approaches the population proportion as the sample size increases. <\/p>\n\n\n\n

            ***********************<\/p>\n\n\n\n

            QLD 12\u5e74\u751f\u6570\u5b66\u6cd5\uff08\u4e00\u822c\u79d1\u76ee\uff09\u30e6\u30cb\u30c3\u30c84\u3067\u306f\u3001\u7d71\u8a08\u7684\u63a8\u8ad6\uff08\u6a19\u672c\u5272\u5408\uff09\u3068\u306f\u3001\u6a19\u672c\u30c7\u30fc\u30bf\u3092\u7528\u3044\u3066\u672a\u77e5\u306e\u6bcd\u96c6\u56e3\u5272\u5408\uff08p\uff09\u3092\u63a8\u5b9a\u3059\u308b\u30d7\u30ed\u30bb\u30b9\u3067\u3059\u3002\u3053\u308c\u306f\u78ba\u7387\u8ad6\u306e\u96c6\u5927\u6210\u3067\u3042\u308a\u3001\u751f\u5f92\u306f\u7406\u8ad6\u7684\u306a\u5206\u5e03\u306e\u7406\u89e3\u304b\u3089\u73fe\u5b9f\u4e16\u754c\u306e\u30c7\u30fc\u30bf\u306e\u5206\u6790\u3078\u3068\u9032\u307f\u3001\u4e88\u6e2c\u3092\u7acb\u3066\u307e\u3059\u3002<\/p>\n\n\n\n

            \u3053\u306e\u30e6\u30cb\u30c3\u30c8\u3067\u306f\u3001\u7d71\u8a08\u7684\u63a8\u8ad6\u306f\u7279\u306b\u4ee5\u4e0b\u306e\u70b9\u306b\u7126\u70b9\u3092\u5f53\u3066\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

            2\u3064\u306e\u7d50\u679c\u3092\u6301\u3064\u6bcd\u96c6\u56e3\uff1a\u7d50\u679c\u304c2\u5024\u3067\u3042\u308b\u72b6\u6cc1\uff08\u4f8b\uff1a\u8cdb\u6210\/\u53cd\u5bfe\u3001\u6210\u529f\/\u5931\u6557\u3001\u81ea\u7531\u515a\/\u52b4\u50cd\u515a\u3078\u306e\u6295\u7968\uff09\u3002
            \u5272\u5408\u306e\u63a8\u5b9a\uff1a\u6a19\u672c\u5272\u5408\uff08p<\/mi>\u02c6<\/mo><\/mover>\\^{p}<\/annotation><\/semantics><\/math>\uff09\u3092\u7528\u3044\u3066\u3001\u771f\u306e\u672a\u77e5\u306e\u6bcd\u96c6\u56e3\u5272\u5408\uff08p<\/mi>p<\/annotation><\/semantics><\/math>\uff09\u3092\u63a8\u5b9a\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

            \u4e3b\u8981\u306a\u6982\u5ff5\u3068\u69cb\u6210\u8981\u7d20<\/p>\n\n\n\n

            \u6a19\u672c\u5272\u5408\uff08p<\/mi>\u02c6<\/mo><\/mover>\\^{p}<\/annotation><\/semantics><\/math>\uff09\uff1a(p<\/mi>\u02c6<\/mo><\/mover>=<\/mo>X<\/mi>n<\/mi><\/mfrac>)<\/mo><\/mrow> \\^{p}=\\frac{X}{n})<\/annotation><\/semantics><\/math>\u3068\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002\u3053\u3053\u3067\u3001(X)\u306f\u6a19\u672c\u30b5\u30a4\u30ba(n)\u306b\u304a\u3051\u308b\u6210\u529f\u6570\u3067\u3059\u3002\u6a19\u672c\u5272\u5408\u306f\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u3067\u3042\u308a\u3001\u305d\u306e\u5024\u306f\u6a19\u672c\u3054\u3068\u306b\u7570\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

            (p<\/mi>\u02c6<\/mo><\/mover>\\^{p}<\/annotation><\/semantics><\/math>) \u306e\u6a19\u672c\u5206\u5e03\uff1a\u6a19\u672c\u30b5\u30a4\u30ba\u304c\u5927\u304d\u3044\u5834\u5408\u3001\u6a19\u672c\u5272\u5408 (p<\/mi>\u02c6<\/mo><\/mover>\\^{p}<\/annotation><\/semantics><\/math>) \u306e\u5206\u5e03\u306f\u307b\u307c\u6b63\u898f\u5206\u5e03\u3068\u306a\u308a\u3001\u5e73\u5747\uff1a(\u03bc<\/mi>p<\/mi>\u02c6<\/mo><\/mover><\/msub>=<\/mo>p<\/mi><\/mrow>\\mu_ {\\^{p}}=p<\/annotation><\/semantics><\/math>)\u3001\u6a19\u6e96\u504f\u5dee\uff1a(<\/em>\u03c3<\/mi>p<\/mi>\u02c6<\/mo><\/mover><\/msub>=<\/mo>p<\/mi>(<\/mo>1<\/mn>\u2212<\/mo>p<\/mi>)<\/mo><\/mrow>n<\/mi><\/mfrac><\/msqrt><\/mrow>\\sigma_ {\\^{p}}=\\sqrt{\\frac{p(1-p)}{n}}<\/annotation><\/semantics><\/math>) \u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

            \u5272\u5408\u306e\u4fe1\u983c\u533a\u9593\uff1a\u3053\u308c\u306f\u4e2d\u6838\u7684\u306a\u5fdc\u7528\u3067\u3042\u308a\u3001\u4e0e\u3048\u3089\u308c\u305f\u4fe1\u983c\u6c34\u6e96\u306b\u57fa\u3065\u3044\u3066\u3001\u771f\u306e\u6bcd\u96c6\u56e3\u5272\u5408 ((p)) \u304c\u542b\u307e\u308c\u308b\u53ef\u80fd\u6027\u306e\u3042\u308b\u5024\u306e\u7bc4\u56f2\u3092\u793a\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

            \u5f0f\uff1a(p<\/mi>\u02c6<\/mo><\/mover>\u00b1<\/mo>z<\/mi>p<\/mi>\u02c6<\/mo><\/mover>(<\/mo>1<\/mn>\u2212<\/mo>p<\/mi>\u02c6<\/mo><\/mover>)<\/mo><\/mrow>n<\/mi><\/mfrac><\/msqrt><\/mrow>\\^{p}\\pm z\\sqrt{\\frac{\\^{p}(1-\\^{p})}{n}}<\/annotation><\/semantics><\/math>)\u3002\u4e00\u822c\u7684\u306a\u6c34\u6e96\uff1a95%\u4fe1\u983c\u533a\u9593\u304c\u6a19\u6e96\u3067\u3059\uff08(z=1.96) \u3092\u4f7f\u7528\uff09\u3002<\/p>\n\n\n\n

            \u8aa4\u5dee\u5e45\uff1a\u6a19\u672c\u5272\u5408\uff08p<\/mi>\u02c6<\/mo><\/mover>\\^{p}<\/annotation><\/semantics><\/math>\uff09\u304b\u3089\u4fe1\u983c\u533a\u9593\u306e\u4e21\u7aef\u307e\u3067\u306e\u8ddd\u96e2\u3067\u3042\u308a\u3001\u63a8\u5b9a\u5024\u306e\u4e0d\u78ba\u5b9f\u6027\u3092\u53cd\u6620\u3057\u307e\u3059\u3002(z<\/mi>p<\/mi>\u02c6<\/mo><\/mover>(<\/mo>1<\/mn>\u2212<\/mo>p<\/mi>\u02c6<\/mo><\/mover>)<\/mo><\/mrow>n<\/mi><\/mfrac><\/msqrt><\/mrow>z\\sqrt{\\frac{\\^{p}(1-\\^{p})}{n}}<\/annotation><\/semantics><\/math>) \u3068\u3057\u3066\u8a08\u7b97\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n

            \u30b7\u30df\u30e5\u30ec\u30fc\u30b7\u30e7\u30f3\uff1a\u6280\u8853\u3092\u7528\u3044\u3066\u7e70\u308a\u8fd4\u3057\u6a19\u672c\u62bd\u51fa\u3092\u30b7\u30df\u30e5\u30ec\u30fc\u30c8\u3057\u3001(p<\/mi>\u02c6<\/mo><\/mover>\\^{p}<\/annotation><\/semantics><\/math>) \u304c\u3069\u306e\u3088\u3046\u306b\u5909\u5316\u3059\u308b\u304b\u3092\u7406\u89e3\u3057\u3001(p<\/mi>\u02c6<\/mo><\/mover>\\^{p}<\/annotation><\/semantics><\/math>) \u304c\u8fd1\u4f3c\u7684\u306b\u6b63\u898f\u5206\u5e03\u3057\u3066\u3044\u308b\u3053\u3068\u3092\u89b3\u5bdf\u3059\u308b\u3002<\/p>\n\n\n\n

            \u5fc5\u9808\u30b9\u30ad\u30eb<\/p>\n\n\n\n

            \u4e0e\u3048\u3089\u308c\u305f\u6a19\u672c\u306b\u3064\u3044\u3066\u70b9\u63a8\u5b9a\u5024\uff08p<\/mi>\u02c6<\/mo><\/mover>\\^{p}<\/annotation><\/semantics><\/math>\uff09\u3068\u4fe1\u983c\u533a\u9593\u3092\u8a08\u7b97\u3059\u308b\u3002
            \u6587\u8108\u306b\u304a\u3044\u3066\u4fe1\u983c\u533a\u9593\u306e\u610f\u5473\u3092\u89e3\u91c8\u3059\u308b\u3002
            \u6280\u8853\uff08\u4f8b\uff1a\u96fb\u5353\uff09\u3092\u7528\u3044\u3066\u5272\u5408\u306e\u533a\u9593\u63a8\u5b9a\u5024\u3092\u6c42\u3081\u308b\u3002
            \u7279\u5b9a\u306e\u8aa4\u5dee\u5e45\u3092\u9054\u6210\u3059\u308b\u305f\u3081\u306b\u5fc5\u8981\u306a\u6a19\u672c\u30b5\u30a4\u30ba\uff08n<\/mi>n<\/annotation><\/semantics><\/math>\uff09\u3092\u6c7a\u5b9a\u3059\u308b\u3002<\/p>\n\n\n\n

            \u3053\u306e\u30c8\u30d4\u30c3\u30af\u3067\u306f\u3001\u591a\u304f\u306e\u5834\u5408\u3001\u30b7\u30df\u30e5\u30ec\u30fc\u30b7\u30e7\u30f3\u3068\u30c6\u30af\u30ce\u30ed\u30b8\u30fc\u3092\u5229\u7528\u3057\u3066\u3001\u30b5\u30f3\u30d7\u30eb \u30b5\u30a4\u30ba\u304c\u5897\u52a0\u3059\u308b\u306b\u3064\u308c\u3066\u30b5\u30f3\u30d7\u30eb\u306e\u5272\u5408\u304c\u6bcd\u96c6\u56e3\u306e\u5272\u5408\u306b\u8fd1\u3065\u304f\u69d8\u5b50\u3092\u8996\u899a\u5316\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

            <\/p>\n","protected":false},"excerpt":{"rendered":"

            In QLD Year 12 Mathematical Methods (General Subject) Unit 4, statistical inference (sample proportions) is the process of using data from a sample to estimate an unknown population proportion (pp). It is the culmination of probability studies, where students move from understanding theoretical distributions to analyzing real-world data to make predictions.  In this unit, statistical […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-1381","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1381","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1381"}],"version-history":[{"count":2,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1381\/revisions"}],"predecessor-version":[{"id":1386,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1381\/revisions\/1386"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1381"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1381"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1381"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}