Note 9 of 5E Statistics with Economics and Business Applications Chapter 7 Estimation of Means and Proportions Point Estimation, Interval Estimation/Confidence Interval

Note 9 of 5E Review Review I. What’s in last lecture? Random Sample, Central Limit Theorem Chapter 6 II. What's in this lecture? Point Estimation Interval Estimation/Confidence Interval Read Chapter 7

Note 9 of 5E Probability vs Statistics Reasoning In the last Chapter we looked at sampling: staring with a population, we imagined taking many samples and investigated how sample statistics were distributed (sampling distribution) In this Chapter, we do the reverse: given one sample, we ask what was the random system that generated its statistics This shift our mode of thinking from deductive reasoning to induction Population Sample Probability/deduction Statistics/induction

Note 9 of 5E Probability vs Statistics Reasoning Deductive reasoning: from a hypothesis to a conclusion Inductive reasoning argues backward: from a set of observations to a reasonable hypothesis In many ways, science, including statistics, is like detective work. Beginning with a set of observations, we ask what can be said about the system that generated them “Data! Data! Data! I can't make bricks without clay.” Sherlock Holmes.

Note 9 of 5EParameters Populations are described by their probability distributions and/or parameters. –For quantitative populations, the location and shape are described by  and  –Binomial populations are determined by a single parameter, p. If the values of parameters are unknown, we make inferences about them using sample information.

Note 9 of 5E Types of Inference Estimation (Chapter 7):Estimation (Chapter 7): –Estimating or predicting the value of the parameter – – “What is (are) the most likely values of  or p?” Hypothesis Testing (Chapter 8):Hypothesis Testing (Chapter 8): –Deciding about the value of a parameter based on some preconceived idea. –“Did the sample come from a population with  or p =.2?”

Note 9 of 5E Types of Inference Examples:Examples: –A consumer wants to estimate the average price of similar homes in her city before putting her home on the market. Estimation: Estimation: Estimate , the average home price. Hypothesis test Hypothesis test: Is the new average resistance,   greater than the old average resistance,    – A manufacturer wants to know if a new type of steel is more resistant to high temperatures than an old type was.

Note 9 of 5E Types of Inference Whether you are estimating parameters or testing hypotheses, statistical methods are important because they provide: –Methods for making the inference –A numerical measure of the goodness or reliability of the inference

Note 9 of 5E Definitions estimator An estimator is a rule, usually a formula, that tells you how to calculate the estimate based on the sample. – Point estimation: – Point estimation: A single number is calculated to estimate the parameter. – Interval estimation/Confidence Interval: – Interval estimation/Confidence Interval: Two numbers are calculated to create an interval within which the parameter is expected to lie. It is constructed so that, with a chosen degree of confidence, the true unknown parameter will be captured inside the interval.

Note 9 of 5E Point Estimator of Population Mean A sample of weights of 34 male freshman students was obtained If one wanted to estimate the true mean of all male freshman students, you might use the sample mean as a point estimate for the true mean. An point estimate of population mean,, is the sample mean

Note 9 of 5E Point Estimation of Population Proportion A sample of 200 students at a large university is selected to estimate the proportion of students that wear contact lens. In this sample 47 wear contact lens. An point estimate of population mean, p, is the sample proportion, where x is the number of successes in the sample.

Note 9 of 5E Properties of Point Estimators sampling distribution.Since an estimator is calculated from sample values, it varies from sample to sample according to its sampling distribution. estimatorunbiasedAn estimator is unbiased if the mean of its sampling distribution equals the parameter of interest. It does not systematically overestimate or underestimate the target parameter. Both sample mean and sample proportion are unbiased estimators of population mean and proportion. The following sample variance is an unbiase estimator of population variance.

Note 9 of 5E Properties of Point Estimators unbiased smallest spreadvariability Of all the unbiased estimators, we prefer the estimator whose sampling distribution has the smallest spread or variability.

Note 9 of 5E Interval Estimators/Confidence Intervals Confidence intervals depend on sampling distributions The shape of sampling distributions depend on sample sizes We will learn different methods for large and small sample sizes For large sample sizes, central limit theorem applies which allow us to use normal distributions For small sample sizes, we need to learn a new distribution

Note 9 of 5E Key Concepts I. Types of Estimators 1. Point estimator: a single number is calculated to estimate the population parameter. Interval estimator/confidence interval 2. Interval estimator/confidence interval: range of values which is likely to include an unknown population parameter. II. Properties of Good Estimators 1. Unbiased: the average value of the estimator equals the parameter to be estimated. 2. Minimum variance: of all the unbiased estimators, the best estimator has a sampling distribution with the smallest standard error.