統(tǒng)計(jì)問(wèn)題（25）：抽樣調(diào)查大樣本會(huì)導(dǎo)致什么情況發(fā)生,？

Question

If a very large random sample is taken in a population survey measuring children’s height, which if any of the following statements are true?

a) The mean is likely to increase

b) The range is likely to increase

c) The standard error of the mean is likely to increase

d) The distribution will become more normal

Answer

Because children’s height is distributed in a nearly normal manner very large samples will probably include a few more very tall children and very short children than smaller samples and these children’s height will be roughly the same distance away from the mean. Consequently, the range (between the tallest and the shortest) will increase but the mean will remain the same.

A very large sample size will reduce the uncertainty in estimating the mean, which is what the standard error of the mean is measuring. Larger sample sizes will more closely reflect the distribution of the underlying population. If this itself is skewed in some way, as income distribution is—for example, a larger sample will still not be more normally distributed.

In some circumstances a larger sample size can lead to an increase in the sample mean and to an increase in the standard deviation—for example, if the distribution has some very large outlying values that might be missed by a small sample but are likely to be captured by a larger one. This phenomenon might be seen for income distribution in some geographical areas if a few millionaires lived in the area but most of the other residents were poor.

中文解釋?zhuān)?/span>

由于兒童的身高分布幾乎是正態(tài)的，因此,，與較小的樣本相比,，非常大的樣本可能會(huì)包含更多的非常高的兒童女和非常矮的兒童，并且這些兒童的身高與均值的距離大致相同,。因此,，范圍（最高和最短之間，即為極差）將增加,，但均值將保持不變,。

很大的樣本量將減少估計(jì)平均值的不確定性，這個(gè)不確定性由平均值的標(biāo)準(zhǔn)誤差來(lái)測(cè)量,。較大的樣本量將更緊密地反映總?cè)丝诘姆植?。如果總?cè)丝诜植既匀皇瞧珣B(tài)的（例如收入分配），較大的樣本仍不會(huì)是正態(tài)分布,。

在某些情況下,，較大的樣本量可能導(dǎo)致樣本均值的增加和標(biāo)準(zhǔn)偏差的增加，例如,，如果分布中的分布值非常大,，則少量樣本可能會(huì)忽略掉這些值，但很可能會(huì)被更大的捕獲,。如果一些百萬(wàn)富翁居住在該地區(qū),，但其他大多數(shù)居民都是窮人，那么在某些地理區(qū)域的收入分配中可能會(huì)看到這種現(xiàn)象,。

所以答案是選擇 B

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