UJI HIPOTESISISI:

The equality part of the hypotheses always appears in the null hypothesis. In general, a hypothesis test about the value of a population proportion p must take one of the following three forms (where p0 is the hypothesized value of the population proportion).A Summary of Forms for Null and Alternative Hypotheses About a Population ProportionOne-tailed(lower tail)One-tailed(upper tail)Two-tailed

Test StatisticTests About a Population Proportionwhere:assuming np > 5 and n(1 p) > 5

Rejection Rule: p Value ApproachH0: ppReject H0 if z > zReject H0 if z < -zReject H0 if z < -z or z > z H0: ppH0: ppTests About a Population ProportionReject H0 if p value < aRejection Rule: Critical Value Approach

Example: National Safety Council For a Christmas and New Years week, theNational Safety Council estimated that500 people would be killed and 25,000injured on the nations roads. TheNSC claimed that 50% of theaccidents would be caused bydrunk driving.Two-Tailed Test About aPopulation Proportion

A sample of 120 accidents showed that67 were caused by drunk driving. Usethese data to test the NSCs claim witha = .05.Two-Tailed Test About aPopulation ProportionExample: National Safety Council

Two-Tailed Test About aPopulation Proportion1. Determine the hypotheses.2. Specify the level of significance.3. Compute the value of the test statistic.a = .05 p Value and Critical Value Approaches

p-Value Approach4. Compute the p -value.5. Determine whether to reject H0.Because pvalue = .2006 > a = .05, we cannot reject H0.Two-Tailed Test About aPopulation ProportionFor z = 1.28, cumulative probability = .8997

pvalue = 2(1 - .8997) = .2006

Two-Tailed Test About aPopulation Proportion Critical Value Approach5. Determine whether to reject H0.For a/2 = .05/2 = .025, z.025 = 1.964. Determine the critical value and rejection rule.Reject H0 if z < -1.96 or z > 1.96Because 1.278 > -1.96 and < 1.96, we cannot reject H0.

Hypothesis Testing and Decision Making In many decision-making situations the decision maker may want, and in some cases may be forced, to take action with both the conclusion do not reject H0 and the conclusion reject H0. In such situations, it is recommended that the hypothesis-testing procedure be extended to include consideration of making a Type II error.

Calculating the Probability of a Type II Error in Hypothesis Tests About a Population Mean1. Formulate the null and alternative hypotheses.3. Using the rejection rule, solve for the value of the sample mean corresponding to the critical value of the test statistic.2. Using the critical value approach, use the level of significance to determine the critical value and the rejection rule for the test.

Calculating the Probability of a Type II Error in Hypothesis Tests About a Population Mean4. Use the results from step 3 to state the values of the sample mean that lead to the acceptance of H0; this defines the acceptance region.

Example: Metro EMS (revisited) The EMS director wants toperform a hypothesis test, with a.05 level of significance, to determinewhether or not the service goal of 12 minutes or less isbeing achieved. Recall that the response times fora random sample of 40 medicalemergencies were tabulated. Thesample mean is 13.25 minutes.The population standard deviationis believed to be 3.2 minutes.Calculating the Probability of a Type II Error

3. Value of the sample mean that identifies the rejection region:2. Rejection rule is: Reject H0 if z > 1.6451. Hypotheses are: H0: and Ha:Calculating the Probability of a Type II Error

12.0001 1.645 .9500 .050012.4 0.85 .8023 .197712.8 0.06 .5239 .476112.8323 0.00 .5000 .500013.2 -0.73 .2327 .767313.6 -1.52 .0643 .935714.0 -2.31 .0104 .98965. Probabilities that the sample mean will be in the acceptance region:Calculating the Probability of a Type II Error

Calculating the Probability of a Type II ErrorCalculating the Probability of a Type II Error When the true population mean m is far above the null hypothesis value of 12, there is a low probability that we will make a Type II error. When the true population mean m is close to the null hypothesis value of 12, there is a high probability that we will make a Type II error.Observations about the preceding table:Example: m = 12.0001, b = .9500Example: m = 14.0, b = .0104

Power of the Test The probability of correctly rejecting H0 when it is false is called the power of the test. For any particular value of m, the power is 1 b. We can show graphically the power associated with each value of m; such a graph is called a power curve. (See next slide.)

Power CurvemH0 False

Chart4

0.9894968367

0.9825806757

0.9720994708

0.9568221687

0.9354045425

0.9065253214

0.8690720589

0.8223540257

0.7663046745

0.7016280202

0.6298462098

0.553221201

0.5000000002

0.4745493487

0.3968609203

0.3230728692

0.2556657735

0.1964394617

0.1463882732

0.1057059957

0.0739016081

0.0500076127

1 - beta

Probability of CorrectlyRejecting Null Hypothesis

Sheet1

muzbeta1 - beta

14.0-2.30786976490.01050316330.9894968367

13.9-2.11022741110.01741932430.9825806757

13.8-1.91258505730.02790052920.9720994708

13.7-1.71494270360.04317783130.9568221687

13.6-1.51730034980.06459545750.9354045425

13.5-1.31965799610.09347467860.9065253214

13.4-1.12201564230.13092794110.8690720589

13.3-0.92437328850.17764597430.8223540257

13.2-0.72673093480.23369532550.7663046745

13.1-0.5290885810.29837197980.7016280202

13.0-0.33144622730.37015379020.6298462098

12.9-0.13380387350.4467787990.553221201

12.832300.49999999980.5000000002

12.80.06383848030.52545065130.4745493487

12.70.2614808340.60313907970.3968609203

12.60.45912318780.67692713080.3230728692

12.50.65676554150.74433422650.2556657735

12.40.85440789530.80356053830.1964394617

12.31.05205024910.85361172680.1463882732

12.21.24969260280.89429400430.1057059957

12.11.44733495660.92609839190.0739016081

12.00011.6447796680.94999238730.0500076127

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0

0

0

0

0

0

0

0

0

0

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0

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1 - beta

Probability of CorrectlyRejecting Null Hypothesis

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The specified level of significance determines the probability of making a Type I error. By controlling the sample size, the probability of making a Type II error is controlled.Determining the Sample Size for a Hypothesis Test About a Population Mean

m0a maReject H0b Determining the Sample Size for a Hypothesis Test About a Population Mean

wherez = z value providing an area of in the tailz = z value providing an area of in the tail = population standard deviation0 = value of the population mean in H0a = value of the population mean used for the Type II errorDetermining the Sample Size for a Hypothesis Test About a Population MeanNote: In a two-tailed hypothesis test, use z /2 not z

Relationship Among a, b, and nOnce two of the three values are known, the other can be computed.For a given level of significance a, increasing the sample size n will reduce b.For a given sample size n, decreasing a will increase b, whereas increasing a will decrease b.

Determining the Sample Size for a Hypothesis Test About a Population Mean Lets assume that the director of medical services makes the following statements about the allowable probabilities for the Type I and Type II errors:If the mean response time is m = 12 minutes, I am willing to risk an a = .05 probability of rejecting H0.If the mean response time is 0.75 minutes over the specification (m = 12.75), I am willing to risk a b = .10 probability of not rejecting H0.

Determining the Sample Size for a Hypothesis Test About a Population Mean

a = .05, b = .10z = 1.645, z = 1.280 = 12, a = 12.75 = 3.2