Chapter 6 Probability & The Normal Distribution

機率在統計裏扮演的角色 Probability vs. inferential statistics Different sample, different variability, different outcome The importance of random sample

Random sample 隨機樣本 Equal chance of being selected Constant probability for each selection Sampling with replacement Simple random sample Convenience samples

Random sample  normal distribution

The Normal Distribution Bell shaped, symmetric, & unimodal Notation: X~N(，2) 學生身高(X)  X~(135, 102) Characteristics: Symmetrical Mean=median 大部分分數落在mean，少部分分數落在兩尾 兩尾向兩端無限延伸 常態分配曲線下的面積總合=1

常態分配機率範圍 隨機變數的值落在平均數1個標準差的範圍的機率為 68.26% 隨機變數的值落在平均數2個標準差的範圍的機率為 95.44% 隨機變數的值落在平均數3個標準差的範圍的機率為 99.74%

68.26% 95.44% 9974% -3 -2 -  +  +2   +3 99.7%

Why do we care about the normal distribution? Many human characteristics fall into an approximately normal distribution Normal distribution of scores is assumed when running most statistical analysis

The concept of probability or chance occurrence is the foundation of hypothesis testing in statistics 機率的觀念是利用統計方法來驗證假設的基礎!!

The Standard Normal Distribution 標準常態分配 Notation: Z~N(0, 1) Characteristics: The standard normal distribution has a mean of 0 and standard deviation of 1 The original scores need to convert to z score! Areas under the curve has fixed probabilities associated with z-scores These areas are presented in normal curve table or z-table.

Z score 相對應的probability P(-1 Z  1)= P(-1  Z  0) + P(0  Z  1) = .6826 P(-2 Z  2)= P(-2  Z  0) + P(0  Z  2) = .9544 P(-3 Z  3)= P(-3  Z  0) + P(0  Z  3) = .9974

68.2% 95.4% 99.7% 1-3s 1-2s 1+3s 1+2s 1+s 1-s

Other common z scores Probability Z score 68% 80% 90% 95% 95.4% 99% 99.7% 1.0 1.26 1.65 1.96 2.0 2.56 3.0