问一道题：NO.PZ2017092702000113

问题如下：

For a sample size of 17, with a mean of 116.23 and a variance of 245.55, the width of a 90% confidence interval using the appropriate t-distribution is closest to:

选项：

13.23.

13.27.

13.68.

解释：

B is correct.

The confidence interval is calculated using the following equation: $\overline X\pm t_{\alpha/2}\frac s{\sqrt n}$ Sample standard deviation (s) = $\sqrt{245.55}$ = 15.670. For a sample size of 17, degrees of freedom equal 16, so t_0.05= 1.746. The confidence interval is calculated as

$116.23\text{ }\pm\text{ }1.746\frac{15.67}{\sqrt{17}}\text{ }=\text{ }116.23\text{ }\pm\text{ }6.6357$ Therefore, the interval spans 109.5943 to 122.8656, meaning its width is equal to approximately 13.271. (This interval can be alternatively calculated as 6.6357 × 2).

什么情况下z a/2, T a/2 的置信区间直接用，比如99%=2.58，什么时候要查表？

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星星_品职助教 · 2020年04月14日

同学你好，

上课要求背诵的1.65,1.96,2.58这些值也都是查表得到的。适用于正态分布。

例如这道题，如果题干说求90%正态分布的置信区间，则直接应用1.65即可。但这道题说的是t分布，且题干没有说可以用正态分布的临界值替代t分布的临界值，这个时候就要根据自由度重新查表了。这道题N=17，属于小样本，所以查表值1.746和1.65有一定差距，如果是N很大，t分布是趋近于正态分布的，最后的结果和1.65也不会差很远的。

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NO.PZ2017092702000113 问题如下 For a sample size of 37, with a meof 116.23 ana varianof 245.55, the wih of a 90% confinintervusing the appropriate t-stribution is closest to: A.8.5480. B.8.6970 C.8.8456. B is correct. The confinintervis calculateusing the following equation:X‾±tα/2sn\overline X\pm t_{\alpha/2}\frs{\sqrt n}X±tα/2ns Sample stanrviation (s) = 245.55\sqrt{245.55}245.55 = 15.670. For a sample size of 37, grees of freem equ36, so t0.05 = 1.688. The confinintervis calculateasTherefore, the intervspans 120.5785 to 111.8815, meaning its wih is equto approximately 8.6970. (This intervcalternatively calculate4.3485 × 2). 样本标准差的计算如下245.55\sqrt{245.55}245.55 = 15.670. 当样本=37，自由度=36，那么 t0.05 = 1.688. 置信区间计算如下因此，置信区间为111.8815 - 120.5785，这意味着其宽度大约等于 8.6970。（这个区间也可以计算为 4.3485 × 2）。为什么解答过程中提到了自由度36，是用来做什么的呢

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问一道题：NO.PZ2017092702000113

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