英文文章與技術文件翻譯至正體中文 on the GPT Store

Introduction to 英文文章與技術文件翻譯至正體中文

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• Bobby calculates the true mean and variance for the pizza poll and finds � = 3 5 , � 2 = 3 16 . μ= 5 3 ​ , σ 2 = 16 3 ​ . The whole point of this mini-poll is to test out the estimator 1 � ∑ � = 1 � ( � � − � ˉ ) 2 , n 1 ​ i=1 ∑ n ​ (X i ​ − X ˉ ) 2 , so he draws 5 5 votes at random and plugs the results into this expression. Because independent and identically distributed random variables � 1 , … , � � X 1 ​ ,…,X n ​ obey � [ ∑ � = 1 � ( � � − � ˉ ) 2 ] = ( � − 1 ) � 2 ⇒ � [ 1 � − 1 ∑ � = 1 � ( � � − � ˉ ) 2 ] = � 2 , E[ i=1 ∑ n ​ (X i ​ − X ˉ ) 2 ] ⇒E[ n−1 1 ​ i=1 ∑ n ​ (X i ​ − X ˉ ) 2 ] ​ =(n−1)σ 2 =σ 2 , ​ the sample variance � � 2 = 1 � − 1 ∑ � = 1 � ( � � − � ˉ ) 2 s n 2 ​ = n−1 1 ​ i=1 ∑ n ​ (X i ​ − X ˉ ) 2 is an unbiased estimator of the variance. Remember, a statistic is unbiased if the average of the statistic over many, many samples equals the true population parameter. In other words, its expected value is the population parameter. This is the mathematically precise argument for the ( � − 1 ) (n−1) in the denominator of � � 2 . s n 2 ​ . You may wonder why we fussed over the ( � − 1 ) (n−1) in this lesson — after all, isn't the relative difference between it and � n negligible for really large sample sizes? Well, it's precisely when � n isn't large that we need to worry about this factor. We'll see specific instances of this later in the chapter on Student's � t-tests. Review and reflect