Transcript Oct 6
Wed Oct 6 Inference about Statistics to infer about parameter Confidence interval for versus point estimate for What is Z2 Read 5.1, 5.2 Next time: read 5.3 Ch.5 Inferences about 5.1 Introduction – Objective of statistics is to make inferences about a population based on information contained in a sample. Parameters are certain descriptive measure of the population, e.g. , . E.g. Population (1,2,3,4,5) = 3. Sample of size 2 1,1 y =1 1,2 y =1.5 … 3,3 y =3 5,5 y =5 Sample of size 3 1,1,1 y =1 … … … 5,5,5 y =5 One can use Sample mean to estimate population mean. Such is an action of inference. Two types of inferences: 1. Estimation : to estimate the value of the parameter 2. Hypothesis testing : to decide whether the parameter is a certain specific value e.g. 5.1 A researcher is interested in estimating the percentage of registered voters in her state who have voted in at least one election over the past 2 years. – – – a. Population of interest : registered voters b. How to select a sample : simple random sample from a list of registered voters. c. Is it a problem of estimation or testing? Estimating, parameter is “percentage of ……” 5.2 Estimation of Simplest: use a number to estimate . E.g, y , such approach is called point estimate. How good is the estimate?? Don’t know just by looking at the answer. A better answer : give a confidence interval for . E.g. A random sample of 36 measurements is drawn from a population with a mean of 50 and a standard deviation of 3. a. Describe the sampling distribution ofy . b. Within what interval would you expect y to lie approximately 90% of the time? Answer a. n = 36 > 30, by CLT,y has a normal distribution with mean 50, standard deviation n = 336 = 0.5 b. P(? < y < ? ) = 0.9 P(c < y < d ) = 0.9 P( c 0.550 Z d 0.550 ) = 0.9 P(-1.65 < Z < 1.65 ) = 0.9 .45 .45 -1.65 0 1.65 Thus c 50 0.5 = -1.65 c = 50 – 1.65 * 0.5 d 50 = 1.65 0 .5 d = 50 + 1.65 * 0.5 In general P ( 1.65y <y < 1.65y ) = 0.9 Rearranging the terms inside the probability: P(y 1.65y < <y 1.65y ) = 0.9 Even though we don’t know what is, we know that it falls within (y 1.65y ,y 1.65y ) 90% of the time, where y = n