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When we free ourselves of desire, we will know serenity and freedom. 1 Inferences about One Population Mean Estimation: z interval and t interval Statistical tests: z test and t test 2 Estimation To estimate a numerical summary in population (parameter): 3 Point estimator — A (same) numerical summary in sample; a statistic Interval estimator — a “random” interval which includes the parameter most of time Example: Serum Cholesterol Level 4 Population: all 20- to 74-year-old male smokers in the U.S. X: serum cholesterol level Distribution: normal with unknown mean m Sample: 12 randomly selected male smokers with mean x 217 mg / 100ml Question: what is the mean serum cholesterol level of adults U.S. male smokers? Confidence Interval ˆL ˆL (Y1 ,..., Yn ) ˆL ˆL (Y1 ,..., Yn ) ˆL ˆL ( X1 ,..., X n ) ˆU ˆU ( X1 ,..., X n ) 5 Ideally, a short interval with high confidence interval is preferred. 6 Estimation for m Point estimator: x Confidence interval – Normality with known s or large sample (n > 30) : Z interval x z / 2 – n Normality with unknown s: t interval x t / 2 7 s s n The t distribution The distribution of t, the standardized sample mean with estimated standard deviation for a normal population xm t s/ n 8 9 One Population 10 Sample Size for Estimating m ( z / 2 ) s n 2 E 2 2 if s is unknown, use s from prior data or the upper bound of s. Where E is the largest tolerable error. 11 Hypothesis Test The null hypothesis: H 0 : m m0 211mg / 100ml The alternative hypothesis: H a : m 211mg / 100ml Question: Should we reject or not reject Ho? 12 The Logic of Hypothesis Test “Assume the null hypothesis Ho is a (possible) truth until proven false” Analogical to “Presumed innocent until proven guilty” The logic of the US judicial system 13 Rejection Region 14 When the observed test statistic is too “extreme” under Ho (i.e. in the opposite direction of Ho), the Ho should be rejected Rejection Region is the region when the observed test statistic falls in, we will reject Ho A test result is determined by its rejection region, while rejection region is determined by the significance level Types of Errors H0 true we accept H0 we reject H0 Good! (Correct!) Type I Error, or “ Error” H0 false Type II Error, or “ Error” Good! (Correct) 15 Terms 16 Type I error rate = probability of incorrectly rejecting Ho Type II error rate = probability of incorrectly accepting Ho Significance level is the pre-set largest tolerable level Power = probability of correctly rejecting Ho = 1- Z Test Statistic For normal populations or large samples (n > 30): Z x m0 sx x m0 s/ n And the computed value of Z is denoted by Z*. 17 Types of Tests 18 Types of Tests 19 Types of Tests 20 Example: Serum Cholesterol Level 21 Assuming the population S.D. is 46 mg/100ml, conduct the test at 5 % level What is the power of the test at the true m221 mg/100ml? 22 Steps in Hypothesis Test 1. 2. 3. 4. 5. 23 Set up the null (Ho) and alternative (Ha) hypotheses Find an appropriate test statistic (T.S.) Find the rejection region (R.R.) Reject Ho if the observed test statistic falls in R.R. Report the result in the context of the situation t Test Statistic For normal populations with unknown s x m0 t s/ n the same formula for Z but replacing s by s 24 One Population 25 Sample Size for Testing m The type I, II error rates are controlled at , respectively and the maximum tolerable error is : ( z z ) s 2 One-tailed tests: n 2 ( z / 2 z ) s 2 Two-tailed tests: 26 n 2 2 2 P-value 27 p-value is the probability of seeing what we observe as far as (or further) from Ho (in the direction of Ha) given Ho is true; the smallest level to reject Ho p-value is computed by assuming Ho is true and then determining the probability of a result as extreme (or more extreme) as the observed test statistic in the direction of the Ha. The smaller p-value is, the less likely that what we observe will occur given Ho is true. Smaller p-value means stronger evidence against Ho. Computing the p-Value for the Z-Test 28 Computing the p-Value for the Z-Test 29 Computing the p-Value for the Z-Test P-value = P(|Z| > |z*| )= 2 x P(Z > |z*|) 30 Hypothesis Test using p-Value 1. 2. 3. 4. 5. 31 Set up the null (Ho) and alternative (Ha) hypotheses Find an appropriate test statistic (T.S.) Find the p-value Reject Ho if the p-value < Report the result in the context of the situation Example: Serum Cholesterol Level 32 Assuming the population S.D. is 46 mg/100ml, conduct the test at 5 % level

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