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H-SC MATH 121 - Lecture 33 - Hypothesis Testing Mean

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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 ofxSlide 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, 2007IntroductionIn 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 HypothesesThe null and alternative hypotheses will be statements concerning .Null hypothesis.H0:  = 0.Alternative hypothesis (choose one).H1:  < 0.H1:  > 0.H1:   0.The HypothesesSee Example 10.1, p. 616.The hypotheses areH0:  = 15 mg.H1:  < 15 mg.Level of SignificanceThe level of significance is the same as before.If the value is not given, assume that  = 0.05.The Test StatisticThe 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 eitherThe sample size n is at least 30, orThe population is normal,Then the Central Limit Theorem for Means will apply. (See p. 615.)The Sampling Distribution ofxIf the population is normal, then the distribution ofx 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 0nNxThe Sampling Distribution ofxTherefore, the test statistic isIt is exactly standard normal.nxZ/0The Sampling Distribution ofxOn the other hand, if The population is not normal, But the sample size is at least 30, then the distribution ofx is approximately normal, with mean 0 and standard deviation /n.We are still assuming that  is known..,ely approximat is 0nNxThe Sampling Distribution ofxTherefore, the test statistic is It is approximately standard normal.The approximation is good enough that we can use normalcdf.nxZ/0The 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 StatisticIn our sample, we find thatx = 12.528.We are assuming that  = 4.8.Therefore,.575.296.0472.2258.415528.12ZCompute the p-ValueThe p-value is P(x < -2.575).Use normalcdf(-E99, -2.575) = 0.005012.Therefore, p-value = 0.005012.DecisionBecause the p-value is less than, we will reject the null hypothesis.ConclusionWe conclude that the carbon monoxide content of cigarettes is lower today than it was in the past.Hypothesis Testing on the TI-83Press 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-83Assuming you selected Stats,Enter 0, the hypothetical mean.Enter . (Remember,  is known.)Enterx.Enter n, the sample size.Select the type of alternative hypothesis.Select Calculate and press ENTER.Hypothesis Testing on the TI-83A 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.ExampleRe-do Example 10.1 on the TI-83 (using Stats).The TI-83 reports thatz = -2.575.p-value = 0.005012.Hypothesis Testing on the TI-83Suppose 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.ExampleRe-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 thatz = -2.575.p-value = 0.005012.x = 12.528.s = 4.740 (


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H-SC MATH 121 - Lecture 33 - Hypothesis Testing Mean

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