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  • 2022-06-26
    从一批滚珠中随机抽取 50 个，测得它们的直径 (单位: [tex=1.714x1.286]iyMAMKMBnJEIhXjMfm0lzg==[/tex] ） 为[img=610x298]1781c7da9c435ac.png[/img]是否可以认为这批钢珠的直径服从正态分布? [tex=4.0x1.357]RmH8DdE3ny+Ivnl3HG6EvQ==[/tex]
    • 数据中最小的为 14.2，最大者为 15.9，设 [tex=7.571x1.214]Q8folL/tTOEo1xQJtjfO+TE8q03VKN1A7ZkEvyz45ho=[/tex] ，欲把 [tex=1.929x1.286]0UMnlwcnmtQAgoeNciVtQA==[/tex] 分成七个（相等的）区间，则区间长度(组距 ) 为 [tex=7.857x2.357]SMqvbTeB6i31o/00I/9xQyrz0cHZs0YSirbhatZ/btE=[/tex] 得分点 [tex=25.929x1.214]WVPXRnEC8uQrtzRCmXrBN85hxc0x43U/OWOl4o9BIVNJ0zxGLPeRSeqC059HAHtY0ceI4kWlBDTA6QBOmeDmHk+cRm5h3sL1Tols2+abIlk=[/tex]它们把实数轴分成七个不相交的区间，样本值分成了七组:[tex=10.5x9.357]OOdTrLGt+hva56tTPivt0/s5MgEuFwvY8zPDUB+7Udi3/Y1uFSoHhA93jzAGRVBUopxppZ8BG0mMiOh2zXE8ao4WXwgxkPujJszhE5sLQzgxoL8g3ZWIUdKCV2sdNuTVAq7fs4w1wjkl7K1IzEO2j3awr46eZZBjT6SN0tNemij+ed+ifU753aKIJyoWkMW+bsGeoEs7kZDD5eOaUb6kL8/GtFwhDaVYQa+0qybrJIfqFhDn4hOnevn7DJQRbpFHjrErrMW60ku72u2IDKgyq/BHA/pFMmGArja+ozpB1dS9rA0yV6h3wE9mxnPMZEaM2AjQh7OyzIxqN/kL6svcj7VAz7yoP3L7DW71yo5Wjto=[/tex]设钢珠的直径为 [tex=1.143x1.214]5tXpix/jb7tkCHi6+JV4sA==[/tex] 其分布函数为 [tex=2.0x1.357]6D04mYW2ivsCmiBu0E4w8w==[/tex] ，我们的问题是检验假设:[tex=9.286x2.786]1FUgc2WIGYLRgHbkgvSOHLB8uK8MAr0EJk++YzNb7svKZW/b6tOEzDsl+0WeOFdYSO9r7dLvZ8fQvhrwd9RGQg==[/tex]其中 [tex=2.071x1.429]4vDI+uNqsdUlkzr52dAX9fixKGQiBuCHYJ19k8ndfBA=[/tex] 未知.在 [tex=1.214x1.214]LiqE1PJ3GxIgL1RVr3tmDA==[/tex] 成立之下， [tex=0.643x1.0]hK6dRoCn+OGpoJ7dSqNW4g==[/tex] 和 [tex=1.0x1.214]+33urkkz3/Nyr4sGrqM3/w==[/tex] 的极大似然估计为 [tex=22.071x3.286]bjzFNLIrA+jatC9MeBlfqkPMzx5RtvxD3TNf/NPYEfVcPIcavGyMtgtb4s3d264arhEf4xXHNxiKyp/FpoxTtivakohGB10oNhzmDPSadG/V1cyZlwp1kUgTDoZxihBE9xBpahIc8B2GhBtFOUVOaGb3lwJNTu6WahZ0qee6ldo=[/tex]在上面的表中第 1 组和第 7 组的频数过小，把它们并入相邻的组内，即分成 5 组，分点为 [tex=19.714x1.214]BFF2ar4pPaqbvexLqTbPrP6aBNuOs2NH4oQEUuIyo30ElY7zJkTZzOEITwu/ehY9zSU3eHgXGni6/Rve3rRj35aNtfMMOxZWxp1E/3bYhF4=[/tex][tex=21.429x2.786]NpTpfyeTX1u1u/KAlYsr1h9u40U4NWfQtMde2I8eqh4APkvM5lV0YHvkC9yFSA+oDW53B/FIhq/5iV7kcAPFeyQ8okTgKLm0VnJTfMzNzEbN2Tdstvp2z6mzpbY8LlGL[/tex][tex=22.571x15.143]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[/tex]统计量[tex=12.357x2.857]UF8FB+x3BIWTPSiOk1sRv03J49iyn/0jgMiJRfZJYvoiGYwMi7l0JwiIqguLtMLcG1/ZbfT1u8zHw/cLRhGdnICuJ0utR6lL0RVm6myI5saGI1BOmu8vCv1jBt3ZhHuJ[/tex]的值计算如下表：[tex=27.286x9.357]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[/tex]即 [tex=5.357x1.429]bfUeEeMpbmz35EKyWDqwkn1PTtQlE1qSBekWd4mcXMM=[/tex] 对于 [tex=3.214x1.0]CigpJ31b6Mhsw1i1RjLe8A==[/tex] 查 [tex=1.071x1.429]637LVdgs6x2/Us8WxEQwHA==[/tex] 分布表得临界值 [tex=10.714x1.286]ozCbwo//3HwuZtJmmvqc42NUZgVTpyKHrIV6H8V/V4TDJKcIi5mKjBNLLJPiK5LLalKI2Tx6jBQ1Tqcya1eOGA==[/tex]因 [tex=13.857x1.286]6aJMELSme18QU5/3T8nMTxazFICZDKd9UfJHTFB6953Pt8+y8lQppbS3NFYJejti[/tex] 故接受 [tex=1.5x1.214]J9POlIUThUeKa4CuSsEvGw==[/tex] 即认为钢珠直径服从正态分布 [tex=6.929x1.357]TSAZ0QW3Qi5Fv/cUABlg3fYpT6VtbwmkxoV39T1pMiU=[/tex]
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    内容

    • 0

      某车床生产滚珠，随机抽取了 50 个产品，测得它们的直径为（单位：mm）：[p=align:center][tex=27.786x5.643]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[/tex]经过计算知道，样本均值 [tex=3.143x1.0]Nu7kFCv6RursUxjtMmCS1w==[/tex], 样本方差 [tex=6.357x1.286]8TimK0xTOp3oyhKZmq45QKSdUt9yVE8N0yb0MpN0DGM=[/tex] 问滚珠直径是否服从正态分布 [tex=7.143x1.571]OeqL4YDs+qPjTtxDniVek4d5G+ktC/sxmPsOzW+p53QG0I2eNEyfp6UsTAfcQ8gA[/tex]？

    • 1

      某车床生产滚珠，随机抽取了 50 个产品，测得它们的直径为（单位：mm）：[p=align:center][tex=27.786x5.643]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[/tex]经过计算知道，样本均值 [tex=3.143x1.0]Nu7kFCv6RursUxjtMmCS1w==[/tex], 样本方差 [tex=6.357x1.286]8TimK0xTOp3oyhKZmq45QKSdUt9yVE8N0yb0MpN0DGM=[/tex] 问滚珠直径是否服从正态分布 [tex=7.143x1.571]OeqL4YDs+qPjTtxDniVek4d5G+ktC/sxmPsOzW+p53QG0I2eNEyfp6UsTAfcQ8gA[/tex]？

    • 2

      某车床生产滚珠，随机抽取了 50 个产品，测得它们的直径为（单位：mm）：[p=align:center][tex=27.786x5.643]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[/tex]经过计算知道，样本均值 [tex=3.143x1.0]Nu7kFCv6RursUxjtMmCS1w==[/tex], 样本方差 [tex=6.357x1.286]8TimK0xTOp3oyhKZmq45QKSdUt9yVE8N0yb0MpN0DGM=[/tex] 问滚珠直径是否服从正态分布 [tex=7.143x1.571]OeqL4YDs+qPjTtxDniVek4d5G+ktC/sxmPsOzW+p53QG0I2eNEyfp6UsTAfcQ8gA[/tex]？

    • 3

      某车床生产滚珠，随机抽取了 50 个产品，测得它们的直径为（单位：mm）：[p=align:center][tex=27.786x5.643]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[/tex]经过计算知道，样本均值 [tex=3.143x1.0]Nu7kFCv6RursUxjtMmCS1w==[/tex], 样本方差 [tex=6.357x1.286]8TimK0xTOp3oyhKZmq45QKSdUt9yVE8N0yb0MpN0DGM=[/tex] 问滚珠直径是否服从正态分布 [tex=7.143x1.571]OeqL4YDs+qPjTtxDniVek4d5G+ktC/sxmPsOzW+p53QG0I2eNEyfp6UsTAfcQ8gA[/tex]？

    • 4

      某车床生产滚珠，随机抽取了 50 个产品，测得它们的直径为（单位：mm）：[p=align:center][tex=27.786x5.643]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[/tex]经过计算知道，样本均值 [tex=3.143x1.0]Nu7kFCv6RursUxjtMmCS1w==[/tex], 样本方差 [tex=6.357x1.286]8TimK0xTOp3oyhKZmq45QKSdUt9yVE8N0yb0MpN0DGM=[/tex] 问滚珠直径是否服从正态分布 [tex=7.143x1.571]OeqL4YDs+qPjTtxDniVek4d5G+ktC/sxmPsOzW+p53QG0I2eNEyfp6UsTAfcQ8gA[/tex]？