Making Decisions about a Population Mean with ConfidenceIntroductionThe Steps of Testing a Hypothesis (p-Value Approach)The HypothesesSlide 5Level of SignificanceThe Test StatisticThe Sampling Distribution ofxSlide 9Slide 10Slide 11The Decision TreeSlide 13Slide 14Slide 15Slide 16Slide 17Slide 18Calculate the Value of the Test StatisticCompute the p-ValueDecisionConclusionHypothesis Testing on the TI-83Slide 24Slide 25ExampleSlide 27Slide 28Making Decisions about a Population Mean with ConfidenceLecture 33Sections 10.1 – 10.2Tue, Oct 30, 2007IntroductionIn Chapter 10 we will ask the same basic questions as in Chapter 9, except they will concern the mean.Find an estimate for the mean.Test a hypothesis about the mean.The Steps of Testing a Hypothesis (p -Value Approach)1. State the null and alternative hypotheses.2. State the significance level.3. Give the test statistic, including the formula.4. Compute the value of the test statistic.5. Compute the p-value.6. State the decision. 7. State the conclusion.The HypothesesThe null and alternative hypotheses will be statements concerning .Null hypothesis.H0: = 0.Alternative hypothesis (choose one).H1: < 0.H1: > 0.H1: 0.The HypothesesSee Example 10.1, p. 616.The hypotheses areH0: = 15 mg.H1: < 15 mg.Level of SignificanceThe level of significance is the same as before.If the value is not given, assume that = 0.05.The Test StatisticThe choice of test statistic will depend on the sample size and what is known about the population. (Details to follow.)If we assume that is known and that eitherThe sample size n is at least 30, orThe population is normal,Then the Central Limit Theorem for Means will apply. (See p. 615.)The Sampling Distribution ofxIf the population is normal, then the distribution ofx is also normal, with mean 0 and standard deviation /n.for any sample size (no matter how small).This assumes that is known..,exactly is 0nNxThe Sampling Distribution ofxTherefore, the test statistic isIt is exactly standard normal.nxZ/0The Sampling Distribution ofxOn the other hand, if The population is not normal, But the sample size is at least 30, then the distribution ofx is approximately normal, with mean 0 and standard deviation /n.We are still assuming that is known..,ely approximat is 0nNxThe Sampling Distribution ofxTherefore, the test statistic is It is approximately standard normal.The approximation is good enough that we can use normalcdf.nxZ/0The Decision TreeIs known?yes noThe Decision TreeIs known?yes noIs the population normal?yes noThe Decision TreenXZ/Is known?yes noIs the population normal?yes noThe Decision TreenXZ/Is known?yes noIs the population normal?yes noIs n 30?yes noThe Decision TreenXZ/Is known?yes noIs the population normal?yes noIs n 30?yes nonXZ/The Decision TreenXZ/Is known?yes noIs the population normal?yes noIs n 30?yes nonXZ/Give upThe Decision TreenXZ/Is known?yes noIs the population normal?yes noIs n 30?yes nonXZ/Give upCome backtomorrowCalculate the Value of the Test StatisticIn our sample, we find thatx = 12.528.We are assuming that = 4.8.Therefore,.575.296.0472.2258.415528.12ZCompute the p-ValueThe p-value is P(x < -2.575).Use normalcdf(-E99, -2.575) = 0.005012.Therefore, p-value = 0.005012.DecisionBecause the p-value is less than, we will reject the null hypothesis.ConclusionWe conclude that the carbon monoxide content of cigarettes is lower today than it was in the past.Hypothesis Testing on the TI-83Press STAT.Select TESTS. Select Z-Test. Press ENTER.A window appears requesting information.Select Data if you have the sample data entered into a list.Otherwise, select Stats.Hypothesis Testing on the TI-83Assuming you selected Stats,Enter 0, the hypothetical mean.Enter . (Remember, is known.)Enterx.Enter n, the sample size.Select the type of alternative hypothesis.Select Calculate and press ENTER.Hypothesis Testing on the TI-83A window appears with the following information.The title “Z-Test.”The alternative hypothesis.The value of the test statistic Z.The p-value of the test.The sample mean.The sample size.ExampleRe-do Example 10.1 on the TI-83 (using Stats).The TI-83 reports thatz = -2.575.p-value = 0.005012.Hypothesis Testing on the TI-83Suppose we had selected Data instead of Stats.Then somewhat different information is requested.Enter the hypothetical mean.Enter . (Why?)Identify the list that contains the data.Skip Freq (it should be 1).Select the alternative hypothesis.Select Calculate and press ENTER.ExampleRe-do Example 10.1 on the TI-83 (using Data).Enter the data in the chart on page 616 into list L1.The TI-83 reports thatz = -2.575.p-value = 0.005012.x = 12.528.s = 4.740 (
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