100Bnote1

# 100Bnote1 - STAT 100B Note for Midterm Note Please try to...

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Unformatted text preview: STAT 100B Note for Midterm Note: Please try to understand the logic flow, instead of just memorizing the formulas. 1 Topics (1) Point estimate: bias and variance. (2) Hypothesis testing: null hypothesis, alternative hypothesis, test statistic, reference distribution, p-value, decision rule, type I error, type II error. (3) Confidence interval. (4) Inference of population proportion (or probability) (5) Inference of population average (or expectation). (6) Linear regression, best linear unbiased estimator, least squares principle. 2 Probability preparation (1) For a random variable X , E( aX + b ) = a E( X )+ b , Var( aX + b ) = a 2 Var( X ), where a and b are two constants. (2) For a sequence of random variables, X 1 , X 2 , ..., X n , E( ∑ n i =1 X i ) = ∑ n i =1 E( X i ). If these random variables are independent, then Var( ∑ n i =1 X i ) = ∑ n i =1 Var( X i ). (3) We can apply (1) and (2) together. For instance, if E( X i ) = μ and Var( X i ) = σ 2 , for i = 1 ,...,n , then E( ¯ X ) = μ and Var( ¯ X ) = σ 2 /n . (4) Normal distribution. If a random variable X ∼ N ( μ,σ 2 ), then Z = ( X- μ ) /σ ∼ N (0 , 1), and the statement X ∈ ( a,b ) is equivalent to Z ∈ ( a ,b ), where a = ( a- μ ) /σ , and b = ( b- μ ) /σ , so Pr( X ∈ ( a,b )) = Pr( Z ∈ ( a ,b )). We can look up the normal table to get Pr( Z ∈ ( a ,b )). 3 Inference of population proportion or probability The data and the model: X 1 ,X 2 ,...,X n are independent binary random variables from Bernoulli( p ). Example 1: p is the probability of getting a head in coin flipping. X i = 1 if the i-th flip is a head, and X i = 0 otherwise. Example 2: p is the population proportion of those who are going to vote for a candidate. X i = 1 if the i-th person in the random sample says that he or she is going to vote for this candidate, and X i = 0 otherwise....
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100Bnote1 - STAT 100B Note for Midterm Note Please try to...

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