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我觉得你问的问题很好，体现了统计学的思想方法。
这题本身没有问题，而且题目非常严密，说的是"is nearest to"，有没有想过为什么？
Analogy:
For a normal distribution with mean=100 and S.D.=5
The probability that you get an observation of 50 is very closed to 0.
Right?
In fact, you can calculate (assuming normal distribution and random sampling) the probability using the Z-table , but here you just need a statistical intuition to see that the probability is closed to 0, because 50 looks way too far from 100.
那么回到这道题，题目特意强调了，40%是个估计("Obviously that doesn’t mean that if we invited 1,000 we’d necessarily get at least 400, so we need to think in terms of the risks of getting too few returns or exceeding the budget.")。事实上这个数存在一个概率分布，40%是expectation（数学期望），题目中没有告诉我们S.D.是多少，甚至都没有告诉我们是个什么分布（比如是否normal distribution），缺少这些条件，我们无法精确计算概率。
the number of the 2nd group responders的这个概率分布，根据题目给的条件，the expectation is 560*0.4=224, the range is [0, 560]
那么我们可以算得总的expenses: the expectation is (350+224)*$50=$28,700.
发现this is way too far from $45,000. 所以真实数据超过$45,000的概率当然是closed to 0，就像我前面那个例子一样。
这样看，这题已经解决了。如果你感兴趣的话，可以算下the range of the expenses，新发出去的560份，理论上最多收到560个回复，最少收到0个回复，于是发现expenses是[$17,500 , $45,500]。就说理论上可能会超过预算$45,000。所以说概率不为0，但是如前所述，近似于0。
Email #1 Email from administrator to research staff January 15, 10:46 a.m. Yesterday was the deadline for our receipt of completed surveys from doctors who were invited to participate in the Medical Practice Priorities Survey. Did we get enough returns from this original group of invitees to get reliable statistics? Do we need to invite additional participants? Email #2 January 15. 11:12 a.m. Altogether we got exactly 350 actual completions. We need at least 700 and were hoping for even more, so we plan to invite a second group to participate. Both the results from this first group and other research indicates that with this type of survey and this type of participants there is about a 40 percent probability that any given invitee will submit the completed survey in the time we’ll allow. (Obviously that doesn’t mean that if we invited 1,000 we’d necessarily get at least 400, so we need to think in terms of the risks of getting too few returns or exceeding the budget.) All of the participants who submitted their surveys by the deadline will get the $50 payment we promised. What is our total budget for compensation to participants? Email #3 January 15, 1:54 p.m. The budget we allocated for compensation to those who complete and submit the Medical Practice Priorities Survey is $45,000. We will honor our commitment to pay $50 to each participant—in the second group as well as the first--who completes the survey and submits it by the deadline we specify when we invite them to participate. However, we will need to try not to exceed the total amount that is budgeted for this purpose.
Suppose that the total number of invitees in the second group is 560. Than, if all of the information in the three emails is accurate, the probability that the budget for compensating participants will be exceed is nearest to:
a)1.0
b)0.8
c)0.5
d)0.2
e)0.0
我一开始的想法是：560×0.4=224 .... 224*50=11200 小于预算 所以可能性为0。答案也是0
但是仔细想想，按这种做法，题目的答案只可能是0或者1
算出来小于预算就是0，算出来大于预算就是1。这又有什么意义？ 既然是问可能性，则应该是调查人数在551-560时的可能性（超出预算的可能人数）。然后每个人回复的可能性为0.4……这样来算才是可能性。
不过样题中没有N次方的计算器……所以可能这道题做事这么做，但是题目本身是有问题的？希望大家讨论讨论
-- by 会员 zenglishiwo (2012/5/21 10:11:31)
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发表于 2012-5-21 10:11:31
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发表于 2012-5-24 03:23:06
