第 9 章 簡單隨機抽樣與抽樣分配

1-1 學習目的 林惠玲 陳正倉著 雙葉書廊發行 2000

1-1 本章結構

4. 無法全部觀察，樣本較母體小，在資料搜集與整 理時較容易且較精確。 1-1 抽樣的重要性 人們在研究某些問題或現象時，有時並不直接探討母體而係經由對樣本的探討，以獲致某些樣本統計量，然後再利用這些樣本統計量去推測母體的參數，主要是因為： 1. 有限的資源， 2. 毀壞性的測驗， 3. 概念性的母體， 4. 無法全部觀察，樣本較母體小，在資料搜集與整 理時較容易且較精確。 林惠玲 陳正倉著 雙葉書廊發行 2000

Why Sample the Population? The physical impossibility of checking all items in the population. The cost of studying all the items in a population. The sample results are usually adequate. Contacting the whole population would often be time-consuming. The destructive nature of certain tests.

圖9.1 等待看牙時間（母體)

圖9.2 等待看牙時間(樣本1)

圖9.3 等待看牙時間(樣本2)

1-1 估計誤差

圖9.4 抽樣誤差與非抽樣誤差

圖9.5 資料蒐集成本與抽樣誤差的關係

抽樣單位與抽樣底冊 抽樣單位 : 抽樣母體中的一個母體元素或一組母體元素。 抽樣底冊 : 抽樣單位的名冊或一覽表。 抽樣底冊最大的問題是可能漏掉部份元素，而發生涵蓋不全的 現象。例如以電話簿做為底冊，而部份住戶沒登錄或沒裝電話 ，因而發生涵蓋不全。再者有些住宿學生或國防役軍人沒有電 話，因此這些人都會被漏掉。政府所做的調查有的以戶籍 （戶口名簿）為底冊，有的以普查結果為底冊，有的以電話簿 為底冊。

Probability Sampling A probability sample is a sample selected such that each item or person in the population being studied has a known likelihood of being included in the sample.

Methods of Probability Sampling Simple Random Sample: A sample formulated so that each item or person in the population has the same chance of being included. Systematic Random Sampling: The items or individuals of the population are arranged in some order. A random starting point is selected and then every kth member of the population is selected for the sample.

Methods of Probability Sampling Stratified Random Sampling: A population is first divided into subgroups, called strata, and a sample is selected from each stratum. Cluster Sampling: A population is first divided into primary units then samples are selected from the primary units.

Methods of Probability Sampling In nonprobability sample inclusion in the sample is based on the judgment of the person selecting the sample. The sampling error is the difference between a sample statistic and its corresponding population parameter.

1-1 簡單隨機抽樣

簡單隨機抽樣的方法

1-1 簡單隨機抽樣

表9.1 亂 數 表

圖9.6 數列對話方塊

圖9.7 隨機抽樣對話方塊

1-1 母體參數與樣本統計量

1-1 樣本平均數的抽樣分配

表9.2 展示小姐的月薪的次數分配

表9.3 展示小姐月薪的母體機率分配

圖9.8 展示小姐月薪的母體機率分配

圖9.10 展示小姐月薪的抽樣

表9.5 展示小姐月薪的樣本平均數

表9.6 展示小姐月薪的抽樣分配

圖9.11 展示小姐月薪的抽樣分配圖

圖9.9 樣本平均數的抽樣分配

表9.4 樣本平均數的機率分配

圖9.12 秘書小姐年資的機率分配圖

表9.7 秘書小姐年資的樣本平均數

表9.8 秘書小姐的年資的抽樣分配

圖9.13 秘書小姐的年資抽樣分配圖

圖9.14 擲骰子出現點數的機率分配圖

表9.9 擲骰子兩次的樣本平均數的抽樣分配

圖9.15 擲骰子兩次樣本平均數的抽樣分配圖

表9.10 三個抽樣分配的比較

1-1 樣本平均數的期望值與變異數

1-1 抽樣分配的變異數與標準差

大數法則

圖9.16 大數法則

圖9.17 X 的機率分配與 的抽樣分配

表9.11 樣本數不同的抽樣分配

圖9.18 樣本大小不同的抽樣分配

1-1 樣本平均數抽樣分配的形狀

1-1 樣本平均數抽樣分配的形狀 應用中央極限定理的注意事項

圖9.19 中央極限定理 母體分配 母體分配

圖9.19 中央極限定理（續） 抽樣分配 抽樣分配

圖9.19 中央極限定理（續） 抽樣分配 抽樣分配

表9.12 的抽樣分配

圖9.20 紡織業平均銷售額的抽樣分配

樣本和的抽樣分配

圖9.21 台鐵乘客平均乘車公里數的機率

圖9.22 品質管制圖

Using the Sampling Distribution of the Sample Mean (Sigma Known) - Example The Quality Assurance Department for Cola, Inc., maintains records regarding the amount of cola in its Jumbo bottle. The actual amount of cola in each bottle is critical, but varies a small amount from one bottle to the next. Cola, Inc., does not wish to underfill the bottles. On the other hand, it cannot overfill each bottle. Its records indicate that the amount of cola follows the normal probability distribution. The mean amount per bottle is 31.2 ounces and the population standard deviation is 0.4 ounces. At 8 A.M. today the quality technician randomly selected 16 bottles from the filling line. The mean amount of cola contained in the bottles is 31.38 ounces. Is this an unlikely result? Is it likely the process is putting too much soda in the bottles? To put it another way, is the sampling error of 0.18 ounces unusual?

Using the Sampling Distribution of the Sample Mean (Sigma Known) - Example Step 1: Find the z-values corresponding to the sample mean of 31.38

Using the Sampling Distribution of the Sample Mean (Sigma Known) - Example Step 2: Find the probability of observing a Z equal to or greater than 1.80

We conclude the process is putting too much cola in the bottles. Using the Sampling Distribution of the Sample Mean (Sigma Known) - Example What do we conclude? It is unlikely, less than a 4 percent chance, we could select a sample of 16 observations from a normal population with a mean of 31.2 ounces and a population standard deviation of 0.4 ounces and find the sample mean equal to or greater than 31.38 ounces. We conclude the process is putting too much cola in the bottles.

1-1 樣本比例的抽樣分配

圖9.23 點二項分配

表9.13 母體比例

表9.14 樣本比例的抽樣分配

圖9.24 樣本比例的抽樣分配

1-1 樣本比例的抽樣分配

樣本比例抽樣分配的應用

圖9.25 樣本平均數的抽樣

表9.15 100 組樣本平均數

表9.16 樣本平均數的抽樣分配

圖9.26 直方圖