{"id":1409,"date":"2026-01-28T13:28:59","date_gmt":"2026-01-28T03:28:59","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1409"},"modified":"2026-01-28T13:34:06","modified_gmt":"2026-01-28T03:34:06","slug":"year12-math-4-4-4-statistical-inference-confidence-intervals","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1409","title":{"rendered":"Year12 MATH 4-4-4 statistical inference (confidence intervals)"},"content":{"rendered":"\n

In QLD Year 12 Specialist Mathematics (Unit 4, Topic 3: Statistical Inference), confidence intervals (CIs)<\/strong> are used to estimate an unknown population parameter (such as the population mean, \u03bc<\/mi>mu<\/annotation><\/semantics><\/math>) by creating a range of plausible values based on sample data. <\/p>\n\n\n\n

Unlike point estimates (a single value), confidence intervals provide a range that accounts for sampling variability, offering a specific level of confidence (e.g., 95% or 99%) that the true population parameter lies within the bounds. <\/p>\n\n\n\n

Key Concepts in Unit 4 <\/strong><\/p>\n\n\n\n

Purpose:<\/strong> To infer population characteristics from a sample, specifically addressing uncertainty due to sampling.<\/p>\n\n\n\n

Key Formula (Population Mean):<\/strong> The approximate confidence interval for a population mean \u03bc<\/mi>mu<\/annotation><\/semantics><\/math> is:
x<\/mi>\u0304<\/mo><\/mover>\u00b1<\/mo>z<\/mi>s<\/mi>n<\/mi><\/msqrt><\/mfrac><\/mrow>x bar plus or minus z the fraction with numerator s and denominator the square root of n end-root end-fraction<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

Where:<\/p>\n\n\n\n

x<\/mi>\u0304<\/mo><\/mover>x bar<\/annotation><\/semantics><\/math> is the sample mean.<\/p>\n\n\n\n

z<\/mi>z<\/annotation><\/semantics><\/math> is the z<\/mi>z<\/annotation><\/semantics><\/math>-score corresponding to the confidence level (e.g., 1.96<\/mn>1.96<\/annotation><\/semantics><\/math> for 95%).<\/p>\n\n\n\n

s<\/mi>s<\/annotation><\/semantics><\/math> is the sample standard deviation.<\/p>\n\n\n\n

n<\/mi>n<\/annotation><\/semantics><\/math> is the sample size.<\/p>\n\n\n\n

Standard Error <\/strong>(<\/mo>s<\/mi>\/<\/mi>n<\/mi><\/msqrt><\/mrow>(s\/\\sqrt{n}<\/annotation><\/semantics><\/math>):<\/strong> This measures the precision of the sample mean as an estimate of the population mean.<\/p>\n\n\n\n

Margin of Error (\ud835\udc67\u00d7Standard Error):<\/strong> This represents the distance from the sample mean to each endpoint of the interval.<\/p>\n\n\n\n

Confidence Levels:<\/strong><\/p>\n\n\n\n

90% CI:<\/strong> z<\/mi>\u2248<\/mo>1.645<\/mn><\/mrow>z is approximately equal to 1.645<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n

95% CI:<\/strong> z<\/mi>\u2248<\/mo>1.96<\/mn><\/mrow>z is approximately equal to 1.96<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n

99% CI:<\/strong> z<\/mi>\u2248<\/mo>2.576<\/mn><\/mrow>z is approximately equal to 2.576<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n

Interpretation:<\/strong> A 95% confidence interval means that if you took many random samples and built a 95% CI from each, approximately 95% of those intervals would contain the true population mean. <\/p>\n\n\n\n

Key Skills Required <\/p>\n\n\n\n

    \n
  1. Constructing Confidence Intervals:<\/strong> Calculating the upper and lower bounds for the population mean.<\/li>\n\n\n\n
  2. Determining Sample Size:<\/strong> Calculating the necessary sample size (n<\/mi>n<\/annotation><\/semantics><\/math>) to achieve a specific margin of error.<\/li>\n\n\n\n
  3. Understanding Width:<\/strong> Recognizing that higher confidence levels (e.g., 99%) result in wider intervals (less precision), while larger sample sizes (n<\/mi>n<\/annotation><\/semantics><\/math>) result in narrower intervals (more precision).<\/li>\n\n\n\n
  4. Assumptions:<\/strong> These intervals generally assume random sampling and a normal distribution of the population, or a large enough sample size (n<\/mi>n<\/annotation><\/semantics><\/math>) for the central limit theorem to apply.\u00a0<\/li>\n<\/ol>\n\n\n\n

    This topic is essential for interpreting statistical results and is assessed in the external examinations for Specialist Mathematics. <\/p>\n\n\n\n

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

    QLD 12\u5e74\u751f\u5c02\u9580\u6570\u5b66\uff08\u30e6\u30cb\u30c3\u30c84\u3001\u30c8\u30d4\u30c3\u30af3\uff1a\u7d71\u8a08\u7684\u63a8\u8ad6\uff09\u3067\u306f\u3001\u4fe1\u983c\u533a\u9593\uff08CI\uff09\u3092\u7528\u3044\u3066\u3001\u6a19\u672c\u30c7\u30fc\u30bf\u306b\u57fa\u3065\u3044\u3066\u59a5\u5f53\u306a\u5024\u306e\u7bc4\u56f2\u3092\u4f5c\u6210\u3057\u3001\u672a\u77e5\u306e\u6bcd\u6570\u30d1\u30e9\u30e1\u30fc\u30bf\uff08\u6bcd\u6570\u5e73\u5747\u5024 (\u03bc<\/mi>\\mu<\/annotation><\/semantics><\/math> ) \u306a\u3069\uff09\u3092\u63a8\u5b9a\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

    \u70b9\u63a8\u5b9a\u5024\uff08\u5358\u4e00\u306e\u5024\uff09\u3068\u306f\u7570\u306a\u308a\u3001\u4fe1\u983c\u533a\u9593\u306f\u6a19\u672c\u5909\u52d5\u3092\u8003\u616e\u3057\u305f\u7bc4\u56f2\u3092\u63d0\u4f9b\u3057\u3001\u771f\u306e\u6bcd\u6570\u30d1\u30e9\u30e1\u30fc\u30bf\u304c\u305d\u306e\u7bc4\u56f2\u5185\u306b\u3042\u308b\u3068\u3044\u3046\u7279\u5b9a\u306e\u4fe1\u983c\u5ea6\uff08\u4f8b\uff1a95%\u307e\u305f\u306f99%\uff09\u3092\u793a\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

    \u30e6\u30cb\u30c3\u30c84\u306e\u4e3b\u8981\u6982\u5ff5<\/p>\n\n\n\n

    \u76ee\u7684\uff1a\u6a19\u672c\u304b\u3089\u6bcd\u6570\u7279\u6027\u3092\u63a8\u8ad6\u3059\u308b\u3053\u3068\u3002\u7279\u306b\u3001\u6a19\u672c\u62bd\u51fa\u306b\u8d77\u56e0\u3059\u308b\u4e0d\u78ba\u5b9f\u6027\u306b\u5bfe\u51e6\u3059\u308b\u3053\u3068\u3002<\/p>\n\n\n\n

    \u91cd\u8981\u306a\u516c\u5f0f\uff08\u6bcd\u5e73\u5747\uff09\uff1a\u6bcd\u5e73\u5747 (\u03bc<\/mi>\\mu<\/annotation><\/semantics><\/math> ) \u306e\u304a\u304a\u3088\u305d\u306e\u4fe1\u983c\u533a\u9593\u306f\u3001\u6b21\u306e\u3068\u304a\u308a\u3067\u3059\u3002
    x<\/mi>\u02c9<\/mo><\/mover>\u00b1<\/mo>z<\/mi>s<\/mi>n<\/mi><\/msqrt><\/mfrac><\/mrow>\\={x}\\pm z\\frac{s}{\\sqrt{n}}<\/annotation><\/semantics><\/math> \u3053\u3053\u3067\uff1a
    (x<\/mi>\u02c9<\/mo><\/mover>\\={x} <\/annotation><\/semantics><\/math>) \u306f\u6a19\u672c\u5e73\u5747\u3067\u3059\u3002
    (z<\/mi>z<\/annotation><\/semantics><\/math>) \u306f\u4fe1\u983c\u6c34\u6e96\u306b\u5bfe\u5fdc\u3059\u308b (z<\/mi>z<\/annotation><\/semantics><\/math>) \u30b9\u30b3\u30a2\u3067\u3059\uff08\u4f8b\uff1a95% \u306e\u5834\u5408\u306f (1.96)\uff09\u3002
    (s<\/mi>s<\/annotation><\/semantics><\/math>) \u306f\u6a19\u672c\u6a19\u6e96\u504f\u5dee\u3067\u3059\u3002
    (n<\/mi>n<\/annotation><\/semantics><\/math>) \u306f\u6a19\u672c\u30b5\u30a4\u30ba\u3067\u3059\u3002<\/p>\n\n\n\n

    \u6a19\u6e96\u8aa4\u5dee (s<\/mi>\/<\/mi>n<\/mi><\/msqrt><\/mrow>s\/\\sqrt{n}<\/annotation><\/semantics><\/math>)\uff1a\u3053\u308c\u306f\u3001\u6bcd\u5e73\u5747\u306e\u63a8\u5b9a\u5024\u3068\u3057\u3066\u306e\u6a19\u672c\u5e73\u5747\u306e\u7cbe\u5ea6\u3092\u6e2c\u5b9a\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

    \u8aa4\u5dee\u5e45 (z<\/mi>\u00d7<\/mo>\u6a19\u6e96\u8aa4\u5dee<\/mtext><\/mrow>z\\times \\text{\u6a19\u6e96\u8aa4\u5dee}<\/annotation><\/semantics><\/math>)\uff1a\u3053\u308c\u306f\u3001\u6a19\u672c\u5e73\u5747\u304b\u3089\u533a\u9593\u306e\u5404\u7aef\u70b9\u307e\u3067\u306e\u8ddd\u96e2\u3092\u8868\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

    \u4fe1\u983c\u6c34\u6e96\uff1a
    90%\u4fe1\u983c\u533a\u9593\uff1a(z<\/mi>\u2248<\/mo>1.645<\/mn><\/mrow>z\\approx 1.645<\/annotation><\/semantics><\/math>).
    95%\u4fe1\u983c\u533a\u9593\uff1a(z<\/mi>\u2248<\/mo>1.96<\/mn><\/mrow>z\\approx 1.96<\/annotation><\/semantics><\/math>).
    99%\u4fe1\u983c\u533a\u9593\uff1a(z<\/mi>\u2248<\/mo>2.576<\/mn><\/mrow>z\\approx 2.576<\/annotation><\/semantics><\/math>).<\/p>\n\n\n\n

    \u89e3\u91c8\uff1a95%\u4fe1\u983c\u533a\u9593\u3068\u306f\u3001\u591a\u6570\u306e\u7121\u4f5c\u70ba\u6a19\u672c\u3092\u63a1\u53d6\u3057\u3001\u305d\u308c\u305e\u308c\u304b\u308995%\u4fe1\u983c\u533a\u9593\u3092\u69cb\u7bc9\u3057\u305f\u5834\u5408\u3001\u305d\u308c\u3089\u306e\u533a\u9593\u306e\u7d0495%\u306b\u771f\u306e\u6bcd\u5e73\u5747\u304c\u542b\u307e\u308c\u308b\u3053\u3068\u3092\u610f\u5473\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

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

    \u4fe1\u983c\u533a\u9593\u306e\u69cb\u7bc9\uff1a\u6bcd\u5e73\u5747\u306e\u4e0a\u9650\u3068\u4e0b\u9650\u3092\u8a08\u7b97\u3059\u308b\u3002<\/p>\n\n\n\n

    \u6a19\u672c\u30b5\u30a4\u30ba\u306e\u6c7a\u5b9a\uff1a\u7279\u5b9a\u306e\u8aa4\u5dee\u5e45\u3092\u9054\u6210\u3059\u308b\u305f\u3081\u306b\u5fc5\u8981\u306a\u6a19\u672c\u30b5\u30a4\u30ba (n<\/mi>n<\/annotation><\/semantics><\/math>) \u3092\u8a08\u7b97\u3059\u308b\u3002<\/p>\n\n\n\n

    \u5e45\u306e\u7406\u89e3\uff1a\u4fe1\u983c\u6c34\u6e96\uff08\u4f8b\uff1a99%\uff09\u304c\u9ad8\u3044\u307b\u3069\u533a\u9593\u306f\u5e83\u304f\u306a\u308a\uff08\u7cbe\u5ea6\u306f\u4f4e\u4e0b\u3059\u308b\uff09\u3001\u6a19\u672c\u30b5\u30a4\u30ba (n<\/mi>n<\/annotation><\/semantics><\/math>) \u304c\u5927\u304d\u3044\u307b\u3069\u533a\u9593\u306f\u72ed\u304f\u306a\u308b\uff08\u7cbe\u5ea6\u306f\u5411\u4e0a\u3059\u308b\uff09\u3053\u3068\u3092\u8a8d\u8b58\u3059\u308b\u3002<\/p>\n\n\n\n

    \u4eee\u5b9a\uff1a\u3053\u308c\u3089\u306e\u533a\u9593\u306f\u3001\u4e00\u822c\u7684\u306b\u7121\u4f5c\u70ba\u6a19\u672c\u62bd\u51fa\u3068\u6bcd\u96c6\u56e3\u306e\u6b63\u898f\u5206\u5e03\u3001\u307e\u305f\u306f\u4e2d\u5fc3\u6975\u9650\u5b9a\u7406\u3092\u9069\u7528\u3059\u308b\u306e\u306b\u5341\u5206\u306a\u5927\u304d\u3055\u306e\u6a19\u672c\u30b5\u30a4\u30ba (n<\/mi>n<\/annotation><\/semantics><\/math>) \u3092\u524d\u63d0\u3068\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

    \u3053\u306e\u30c8\u30d4\u30c3\u30af\u306f\u7d71\u8a08\u7d50\u679c\u3092\u89e3\u91c8\u3059\u308b\u4e0a\u3067\u4e0d\u53ef\u6b20\u3067\u3042\u308a\u3001\u5c02\u9580\u6570\u5b66\u306e\u5916\u90e8\u8a66\u9a13\u3067\u8a55\u4fa1\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n

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

    In QLD Year 12 Specialist Mathematics (Unit 4, Topic 3: Statistical Inference), confidence intervals (CIs) are used to estimate an unknown population parameter (such as the population mean, \u03bcmu) by creating a range of plausible values based on sample data.  Unlike point estimates (a single value), confidence intervals provide a range that accounts for sampling […]<\/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-1409","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1409","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=1409"}],"version-history":[{"count":2,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1409\/revisions"}],"predecessor-version":[{"id":1414,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1409\/revisions\/1414"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1409"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1409"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1409"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}