QM Spring 2002 Business Statistics 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:  and  – Qualitative data:  (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 Parameter Statistic

Sampling Distributions  Quantitative data – Expected value for x-bar is the population or process average (i.e.,  ) – Expected variation in x-bar from one sample average to another is Known as the standard error of the mean Equal to  /√n – Distribution of x-bar is approx normal (CLT)  Qualitative data – E(p) is  – Standard error is √  (1-  )/n – Distribution of p is approx normal (CLT)

A Review Example from the Homework  Supposedly, WNB executive salaries equal industry on average (  = 80,000)  But sample results were – x-bar = $68,270 – s = $18,599  If truly  = 80,000 – Assume for now that  = s = – What is P(x-bar < 68270)? – What is P(x-bar 91730) ?

Some Answers  Given assumptions about  and  – Standard error:  /√n = 18599/√15 = 4800 – An x-bar value of is standard errors from the supposed population average Table probability = Thus P(x-bar < 68270) = – = 0.7% And P(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 Parameter Statistic

Two Types of Statistical Inference  Hypothesis testing – Starts with a hypothesis (i.e., claim, assumption, standard, etc.) about a population parameter ( , , ,  , 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 (H 0 ): What we’ll believe until proven otherwise We state this first if we’re seeing if something’s changed – Alternate (H A ): Opposite of H 0 If we’re trying to prove something, we state it as H A and start with this, not the null  Then state willingness to make wrong conclusion (  )  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 H 0  Type II: accepting a false H 0  Probabilities  = P(Type I)  = P(Type II) Power = P(Rejecting false H 0 ) = P(No error)  Controlling risks – Decision rule controls  – Sample size controls   Worst error: Type III (solving the wrong problem)!  Hence, be sure H 0 and H A 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 p- values 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