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Estimation and Hypothesis (Hadyana Sukandar) Lesson Objectives 1. Know what is parameter estimation 2. Understand hypothesis testing & the “types of errors” in decision making. 3. Learn how to use test statistics to examine hypothesis about population mean, proportion Statistic Parameter Mean: X estimates ____ Standard deviation: s estimates ____ Proportion: p estimates ____ from entire from sample population Population Point estimate Interval estimate I am 95% Mean confident that Mean, , is is between 40 & X = 50 unknown 60 Sample CONFIDENCE INTERVAL Definitions : 1. Estimator : a formula or process for using sample data to estimate a population parameter 2. Estimate : a specific value or range of values used to approximate some population parameter 3. Point Estimate : a single value (or point) used to approximate a population parameter The sample mean x is the best point estimate of the population mean µ 4. Confidence Interval (or Interval Estimate) : a range (or an interval) of values used to estimate the true value of the population parameter. Lower # < population parameter < Upper # As an example : Lower # < µ< Upper # 5. Degree of confidence (level of confidence or confidence coefficient) : The probability 1 - (often expressed as the equivalent percentage value) that is the relative frequency of times the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times. Usually 95 %, (= 5%) or 99 % (=1 %). Interpreting a Confidence Interval for (for example : Systolic Blood pressure of medical student) : 100 < < 140 We are 95 % confident that the interval from 100 to 140 actually does contain the true value of . This means that if we were to select many different samples of size 100 and construct the confidence interval, 95 % of them would actually contain the value of the population mean . PARAMETER = STATISTIC ITS ERROR Standard Error S Quantitative Variable SE (Mean) = n p(1-p) Qualitative Variable SE (p) = n Confidence Interval α/2 α/2 1-α _ X SE SE Z-axis 95% Samples X - 1.96 SE X + 1.96 SE Confidence Interval α/2 α/2 1-α SE SE Z-axis p 95% Samples p - 1.96 SE p + 1.96 SE Interpretation of CI Probabilistic Practical In repeated sampling 100(1-)% We are 100(1-)% confident that of all intervals around sample the single computed CI contains means will in the long run include Example (Sample size≥30) An epidemiologist studied the blood glucose level of a random sample of 100 patients. The mean was 170, with a SD of 10. = X + Z SE SE = s/√n = 10/10 = 1 95 Then CI: % = 170 + 1.96 1 168.04 ≤ 171.96 Example (Proportion) In a survey of 140 asthmatics, 35% had allergy to house dust. Construct the 95% CI for the population proportion. = p + Z √ p(1-p)/n SE= √ 0.35(1-0.35)/140 = 0.04 0.35 – 1.96 0.04 0.35 + 1.96 0.04 0.27 ≤ 0.43 27% 43% HYPOTHESIS What is Hypothesis ? Definition : Hypothesis in statistics, is a claim or statement about a property of population. Components of formal hypothesis test : - Null Hypothesis : H0 - Statement about value of population parameter - Must contain condition of equality : = ; ; or - Test the null Hypothesis directly - Recect H0 or fail to reject H0 Alternative Hypothesis - Must be true if H0 is false - ≠; < ; > - Opposite of null Note : If you are conducting a study and want to use a hypothesis test to support your claim, the claim must be worded so that it becomes the alternative hypothesis. LEVEL OF SIGNIFICANCE, The probability that the test statistic will fall in the critical region when the null hypothesis is actually true. Common choices are 0.05; 0.01 and 0.10. TYPE ERROR : TYPE I AND TYPE I Type I error : - the mistake of rejecting the null hypothesis when it is true. - (alpha) is used to represent the probability of a type I error. Example : Rejecting a claim that the mean systolic blood pressure is 110 mmHg when the mean really does equal 110 mmHg. Type II error : - the mistake of failing to reject the null hypothesis when it is false. - (beta) is used to represent the probability of a type II error. Example : Failling to reject the claim that the mean systolic blood pressure is 110 mmHg when the mean really different from 110 mmHg. Table Type I and Type II Errors True State of Nature The null hypothesis is The null hypothesis is true false Decision : Type I error Correct decision (rejecting a true null hypothesis) Correct decision Type II error (rejecting a falls null hypothesis) p value Test - Probability of Obtaining a Test Statistic More Extreme ( or ) than Actual Sample Value Given H0 Is True . - Called Observed Level of Significance - Used to Make Rejection Decision * If p value , Do Not Reject H0 * If p value < , Reject H0 Hypothesis Testing: Steps Test the Assumption that the true mean SBP of participants is 120 mmHg. * State H0 H0 : µ = 120 * State H1 H1 : µ 120 * Choose = 0.05 * Choose n n = 100 * Choose Test: Z, t, X2 Test (or p Value) * Compute Test Statistic (or compute P value) * Search for Critical Value * Make Statistical Decision rule * Express Decision Example : One sample-mean Test - Assumptions - Population is normally distributed - State the null and alternative hypotheses H0: µ = µ0 H1: µ µ0 - t test statistic : sample mean null value x 0 t standard error s n