Expected value of a function of X Let

Expected value of a function of X 複 習 • Let X be a discrete random variable, and let Y be any function of X such that Y = g(X). Then the expected value of Y, or the expected value of g(X), is ©蘇國賢 2001 2

Properties of Expectations 複 習 • Theorem 1: if Y=a．X+b, where a and b are constant, then E(Y) = a．E(X) + b =1 E(X) ©蘇國賢 2001 3

Properties of Expectations 複 習 • Theorem 2: if X 1, X 2, X 3…Xn are n random variable such that each expectation E(Xi) exists (i = 1, 2, …n), then E(X 1+X 2…+Xn) = E(X 1) +E(X 2) +… E(Xn) ©蘇國賢 2001 4

Properties of Expectations 複 習 • It follows from theorem 1 and theorem 2 that for any constant a 1, a 2, …an and b, E(a 1 X 1+a 2 X 2…+an. Xn) = a 1 E(X 1) +a 2 E(X 2) +… +an. E(Xn) • 同理： • E(a 1 X 1 -a 2 X 2…-an. Xn) = a 1 E(X 1) -a 2 E(X 2) -… -an. E(Xn) ©蘇國賢 2001 5

Properties of Expectations 複 習 • Theorem 3: • If X 1, …Xn are n independent variables such that each expectation E(Xi) exists, then • E(X 1．X 2．X 3…Xn)=E(X 1)E(X 2)…E(Xn) ©蘇國賢 2001 8

Properties of Expectations 複 習 • Proof • X 1, …Xn are n independent variables, P(X 1．X 2 ．X 3…Xn)= P(X 1)P(X 2)…P(Xn) ©蘇國賢 2001 9

Variance of Discrete Random Variable 複 習 • 非連續隨機變數的變異數 ©蘇國賢 2001 10

Properties of the variance 複 習 • Var(c)=0, c 為常數項 • 更正式的陳述：Var(X)=0 if and only if there exists a constant c such that P(X=c)=1. ©蘇國賢 2001 11

Properties of the variance • • 複 習 Theorem 4: Var(a. X+b)=a 2 Var(X) Proof. 因為E(a. X+b)=a. E(X)+b=au+b Var(a. X+b)=E[((a. X+b) – E(a. X+b))2] =E[(a. X+b – au –b)2]=E[(a. X-au)2] =a 2 E[(X-u)2] ©蘇國賢 2001 12

Properties of the variance 複 習 • Theorem 5: • If X 1, …Xn are independent random variables, then Var(X 1+…+Xn) = Var(X 1)+ …+Var(Xn) ©蘇國賢 2001 13

Properties of the variance • • 複 習 Proof. 以n=2為例，If E(X 1) = u 1 and E(X 2) = u 2 E(X 1+X 2)=u 1+u 2 Var(X 1+X 2)=E[(X 1+X 2 -u 1 -u 2)2] =E[(X 1 -u 1)2+(X 2 -u 2)2+2(X 1 -u 1)(X 2 -u 2)] =Var(X 1) + Var(X 2) +2 E[(X 1 -u 1)(X 2 -u 2)] X 1, X 2 are independent, 根據theorem 3 E[(X 1 -u 1)(X 2 -u 2)]=E(X 1 -u 1)E(X 2 -u 2)=0 ©蘇國賢 2001 14

Linear combinations of normally distributed variables • Theorem 6. If the random variables X 1, . . Xk are independent and if Xi has a normal distribution with mean ui and variance σ2, then the sum X 1+X 2+…Xk has a normal distribution with mean u 1+…uk and variance σ12 +…+σk 2 • 兩常態獨立樣本相加減所得到新的分 配仍為常態分配。 ©蘇國賢 2001 15

Page 477, Figure 10. 1 ©蘇國賢 2001 16

Sampling distribution of the difference between two sample mean 定 義 假設有兩獨立母體 1 1 如果樣本數 夠大，其樣 本平均數的 抽樣分配必 21/n 1 為常態，且 1 ©蘇國賢 2001 2 2 22/n 2 2 17

Sampling distribution of the difference between two sample mean 定 義 • 當樣本n很大時，樣本平均數差之抽樣分配為近 似常態分配（theorem 6) 22/n 2 21/n 1 1 ©蘇國賢 2001 2 1 - 2 18

Sampling distribution of the difference between two sample mean 根據theorem 2 根據theorem 5 為一平均值， 變異數已知的 常態分配 ©蘇國賢 2001 1 - 2 19

例題 • A financial loan officer claims that the mean monthly payment for credit cards is $80 and the variance is 1, 400 for female. For males, the mean is $80 and the variance is 1, 320. You take a random sample of 100 females and an independent random sample of 120 males. What is the probability that the sample mean for females will be at least $5 higher than the sample mean for males? ©蘇國賢 2001 20

例題 • The television picture tubes of manufacturer A have a mean lifetime of 6. 5 years and a standard deviation of 0. 9 year, while those of manufacturer B have a mean lifetime of 6 years and a standard deviation of 0. 8 year. What is the probability that a random sample of 36 tubes from manufacturer A will have a mean lifetime that is at least 1 year more than the mean lifetime of a sample of 49 tubes from manufacturer B? ©蘇國賢 2001 25

Sampling distribution of the difference between two sample mean兩樣本平均數差的 抽樣分配 • 假設有兩獨立分配母體 1 1 2 其樣本平均數的 抽樣分配分別為 21/n 1 ©蘇國賢 2001 1 2 22/n 2 2 30

Sampling distribution of the difference between two sample mean ©蘇國賢 2001 1 - 2 32

Sampling distribution of the difference between two sample mean 22/n 2 21/n 1 1 2 • 當樣本n>30時，兩樣本平均數差之抽樣分配為近似 常態分配 ©蘇國賢 2001 1 - 2 33

Sampling distribution of the difference between two sample mean, When population variances unknown 如果樣本數夠大(n>30)，則我們可以用樣本變 異數s 2來取代未知的母體變異數 2。 如果我們假設D 0=0, 則 ©蘇國賢 2001 35

Confidence intervals for the difference of Two means Confidence interval for (u 1 -u 2) when variance are known— independent sample Suppose we have independent random samples of size n 1 and n 2 from two normal populations having unknown means u 1 and u 2 and known variance 12 and 22. If the observed sample means are x 1 and x 2, a 100(1 - )% confidence interval for (u 1 – u 2) is given by ©蘇國賢 2001 36

Confidence intervals for the difference of Two means If sample sizes are large n>30, and the population variances are unknown, then ©蘇國賢 2001 37

例題 Reject H 0 ©蘇國賢 2001 41

Confidence intervals for the difference of Two means Confidence interval for (u 1 -u 2) when variance are UNKNOWN and sample sizes are small, population variances are assumed equal Suppose we have independent random samples of size n 1 and n 2 from normal populations having unknown means u 1 and u 2 and a common unknown variance 2. If the observed sample means are x 1 and x 2, a 100(1 - )% confidence interval for (u 1 – u 2) is given by ©蘇國賢 2001 48

Confidence intervals for the difference of Two means Pooled estimate of the common variance ©蘇國賢 2001 Degree of freedom 49

例題 因為 =5%，d. f. =(n 1 -1)+(n 2 -1) =30，查表求雙尾 的critical value，得 t. 025, 30= 2. 042 Failed to reject ©蘇國賢 2001 51

Page 484, Procedure 10. 1 A ©蘇國賢 2001 53

Page 484, Procedure 10. 1 A (cont. ) ©蘇國賢 2001 54

Page 485, Procedure 10. 1 B ©蘇國賢 2001 55

Page 485, Procedure 10. 1 B (cont. ) ©蘇國賢 2001 56

Page 488, Procedure 10. 2 ©蘇國賢 2001 57

Nonpooled t-Test for Two population means (unequal variances and small samples) 如果兩母體的變異數為未知且不同，且 樣本數很小，則: ©蘇國賢 2001 59

Page 497, Procedure 10. 3 A ©蘇國賢 2001 60

Page 497, Procedure 10. 3 A (cont. ) ©蘇國賢 2001 61

Page 498, Procedure 10. 3 B ©蘇國賢 2001 62

Page 498, Procedure 10. 3 B (cont. ) ©蘇國賢 2001 63

Page 501, Procedure 10. 4 ©蘇國賢 2001 64

例題10. 6 499頁 • Step 1: state the null and alternative hypotheses • Step 2: Decide on the significant level α • α=. 01 • Step 3: compute the value of test statistic ©蘇國賢 2001 66

例題10. 6 499頁 ©蘇國賢 2001 Reject null hypothesis 67