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Business Statistics QM 2113 - Spring 2002 Introduction to Inference: Hypothesis Testing Student Objectives Review concepts of sampling distributions List and distinguish between the two types of inference Summarize hypothesis testing procedures Conduct hypothesis tests concerning population/process averages Understand how to use tables for the t distribution Recall: Parameters versus Statistics Descriptive numerical measures calculated from the entire population are called parameters. – Quantitative data: m and s – Qualitative data: p (proportion) Corresponding measures for a sample are called statistics. – Quantitative data: x-bar and s – Qualitative data: p The Sampling Process Population or Process Sample Statistic Parameter Sampling Distributions Quantitative data – Expected value for x-bar is the population or process average (i.e., m) – Expected variation in x-bar from one sample average to another is • Known as the standard error of the mean • Equal to s/√n – Distribution of x-bar is approx normal (CLT) Qualitative data – E(p) is p – Standard error is √p(1-p)/n – Distribution of p is approx normal (CLT) A Review Example from the Homework Supposedly, WNB executive salaries equal industry on average (m = 80,000) But sample results were – x-bar = $68,270 – s = $18,599 If truly m = 80,000 – Assume for now that s = s = 18599 – What is P(x-bar < 68270)? – What is P(x-bar < 68270 or x-bar > 91730) ? Some Answers Given assumptions about m and s – Standard error: s/√n = 18599/√15 = 4800 – An x-bar value of 68270 is -2.44 standard errors from the supposed population average • Table probability = 0.4927 • Thus P(x-bar < 68270) = 0.5000 – 0.4927 = 0.7% • And P(x-bar < 68270 or x-bar > 91730) = 1.4% Now, consider how this might be put to use in addressing the claim – Bring action against WNB (false claim?) – What’s the probability of doing so in error? Putting Sampling Theory to Work We need to make decisions based on characteristics of a process or population But it’s not feasible to measure the entire population or process; instead we do sampling Therefore, we need to make conclusions about those characteristics based upon limited sets of observations (samples) These conclusions are inferences applying knowledge of sampling theory The Sampling Process Population or Process Sample Statistic Parameter Two Types of Statistical Inference Hypothesis testing – Starts with a hypothesis (i.e., claim, assumption, standard, etc.) about a population parameter (m, p, s, b1, distribution, . . . ) – Sample results are compared with the hypothesis – Based upon how likely the observed results are, given the hypothesis, a conclusion is made Estimation: a population parameter is concluded to be equal to a sample result, give or take a margin of error, which is based upon a desired level of confidence Hypothesis Testing Start by defining hypotheses – Null (H0): • What we’ll believe until proven otherwise • We state this first if we’re seeing if something’s changed – Alternate (HA): • Opposite of H0 • If we’re trying to prove something, we state it as HA and start with this, not the null Then state willingness to make wrong conclusion (a) Determine the decision rule (DR) Gather data and compare results to DR The Logic Involved Suppose someone makes a statement and you wonder about whether or not it’s true You typically do some research and get some evidence If the evidence contradicts the statement but not by much, you typically let it slide (but you’re not necessarily convinced) However, if the evidence is overwhelming, you’re convinced and you take action This is hypothesis testing! Statistics helps us to determine what is “overwhelming” Errors in Hypothesis Testing Type I: rejecting a true H0 Type II: accepting a false H0 Probabilities a = P(Type I) b = P(Type II) Power = P(Rejecting false H0) = P(No error) Controlling risks – Decision rule controls a – Sample size controls b Worst error: Type III (solving the wrong problem)! Hence, be sure H0 and HA are correct Stating the Decision Rule First, note that no analysis should take place before DR is in place! Can state any of three ways – Critical value of observed statistic (x-bar) – Critical value of test statistic (z) – Critical value of likelihood of observed result (p-value) Generally, test statistics are used when results are generated manually and pvalues are used when results are determined via computer Always indicate on sketch of distribution Some Exercises Addressing the Mean Don’t forget to sketch distributions! Large sample (CLT applies) – One tail hypothesis (#8-3) – Two tail hypothesis (#8-8) Small sample (introducing the t distribution) – One tail hypothesis (#8-5) – Note: we’re really always using the t distribution • Applies whenever s is used to estimate the standard error • It just becomes obvious when sample sizes are small Homework Section 8-1: – Reread – Rework exercises Read Section 8-3 Work exercises: 28, 29, 30, 34