Stats 11 (Fall 2004) Lecture Note Instructor: Hongquan XuIntroduction to Statistical Methods for Business and EconomicsChapter 9 Significance Testing: Using Data to Test HypothesesIn Chapter 8, we learned confidence intervals for an unknown parameter. Chapter 9 deals with significancetest. These are different methods for coping with the uncertainty about the true value of a parameter causedby the sampling variation in estimates.• Confidence interval: A fixed level of confidence is chosen. We determine a range of possible valuesfor the parameter that are consistent with the data (at the chosen confidence level).• Significance test: Only one possible value for the parameter, called the hypothesized value, is tested.We determine the strength of the evidence provided by the data against the proposition that thehypothesized value is the true value.Note: This is a combination of Sections 9.1–9.5.Hypothesis Testing:Techniques for comparing two competing theories or hyp othes es about the population.Key Components:• Population(s) of interest• Research hypothesis• Two competing hypothesesH0: null hyp othesisH1: alternative hypothesis• Data:• Test Statistic:• Decision Rule:Setting up the HypothesesThe null and alternative hypotheses are statements about a population parameter (not about the results inthe sample). There will be a ”direction of extreme” that is indicated by the alternative hypothesis. Tosee these ideas, let’s try writing out some hypotheses to be put to the test.1Example 1. Mean area of apartments in a new development is advertised to be 1250 square feet. A tenantgroup thinks the apartments are smaller than advertised. They hire an engineer to measure a sample ofapartments to test their suspicion.H0:H1:Example 2. A so c iologist asks a large sample of HS students which subject they like best. She suspectsthat a higher pe rcent of males than of females will name math as favorite.H0:H1:Example 3. Last year, service technicians took an average of 2.6 hours to resp ond to trouble calls. Managerwishes to assess if this year’s data show a different average response time.H0:H1:Let’s Do a Test:To learn how to do a test about the population mean (in the case when we assume the data are a randomsample from a normal population with a known population standard deviation), we will just dive right inand use our knowledge from Section 7.2 to work it through.Example: SSHA is a psychological test that measures motivation, attitude toward school, study habits.Scores: 0 to 200. The model for SSHA scores for US college students is normal with a mean of 115 and astandard deviation of 30.A teacher thinks that older students have better attitudes (a higher score implies a better attitude). Shewill give the SSHA test to a random sample of 20 older students (at least 30 years old).What is the population under study?State the hypotheses to be tested regarding the mean of the population under study:H0:H1:What will we base our decision on? We will look at the sample mean X com puted for our random sampleof test scores.2Draw a picture of the sampling distribution for X if the null hypothesis H0were true.What kind of values of x will lead us to reject H0?Values that are too small? too large? both directions?The 20 test scores have been obtained and suppose the sample mean was 127.2.Note that 127.2 is larger than 115, but is it large enough? Let’s compute its standard score.The standard score is called the z-test statistic and is given by:z =x − µ0σ√n=How likely is it to get a z-test statistic of 1.82 or even larger if the H0is true?So if H0were true, we would expect to get a test statistic of 1.82 or larger only % of the time (inrepeated samples). This probability you computed is called the p-value. It measures the compatibility ofthe data with the null hypothesis.P-value = the probability of getting a test statistic as extreme or more extreme than the actualobserved test statistic, assuming H0is true. (note that the direction of extreme matches H1.)Our text (p. 381) says:The P-value is the probability that, if the hypothesis was true, sampling variation wouldproduce an estimate that is further away from the hypothesized value than the estimate wegot from our data.The P-value measures the strength of the evidence against H0.The smaller the p-value, the stronger the evidence against H0.In our example, p-value = is somewhat small. How small is small enough?We compare the p-value to the fixed amount of evidence required to reject H0. This fixed level, decided inadvance, is our significance level, represented by α.1. Suppose the significance level α = 5%.What is the decision? Accept H0(i.e. fail to Reject H0) or Reject H0What is the conclusion?32. Suppose the significance level α = 1%.What is the decision? Accept H0(i.e. fail to Reject H0) or Reject H0What is the conclusion?Decision Rule: Reject H0ifand conclude that the data areApproach to hypothesis testing:1. State the null and alternative hypotheses.2. Calculate the test statistic.3. Find the p-value (assuming is true).4. State the decision (by comparing to a significance level) and conclusion in terms of the scenario.Summary: We have just conducted our first One-Sample Z-test for the Population Mean µ. It is a One-Sidedto the Right (Upp er level) Z-test.In the above example, we assume we know the population s.d. σ, so the statistic Z =X−µ0σ /√nhas a standardnormal N(0, 1) distribution.If we do NOT know σ, we would estimate σ by sample s.d. S and use statistic T =X−µ0S/√n. What is thedistribution of T under the null hypothesis H0?Note that N(0, 1) is a special T distribution with d.f.=∞.Next we will summarize the T-test for population mean µ in general and do examples of One-Sided to theLeft, and Two-sided testing problems.4One-sample T test for H0: µ = µ0Assume:Test Statistic: has a distribution under H0The observed value (computed from data) isIf H1: µ > µ0then the p-value is:If H1: µ < µ0then the p-value is:If H1: µ 6= µ0then the p-value is:The test requires we have a random sample from a normal population. (If the sample size is large, theassumption of normality is not so crucial and the result is approximate.)5Example – Using the t table to find a p-valueH0: µ = 0 versus H1: µ > 0 with n = 15 observations and the observe d t statistic is t = 1.97(a) What are the degrees of freedom?(b) Give the 2 critical values t∗from the t table which bound t = 1.97.(c) What are the right-tail