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Pengujian Hipotesis Proporsi dan Beda Proporsi Pertemuan 21 Matakuliah: I0134/Metode Statistika Tahun: 2007

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Bina Nusantara A Summary of Forms for Null and Alternative Hypotheses about a Population Proportion 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 p 0 is the hypothesized value of the population proportion). H 0 : p > p 0 H 0 : p < p 0 H 0 : p = p 0 H a : p p 0 H a : p p 0

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Bina Nusantara Tests about a Population Proportion: Large-Sample Case (np > 5 and n(1 - p) > 5) Test Statistic where: Rejection Rule One-Tailed Two-Tailed H 0 : p p Reject H 0 if z > z H 0 : p p Reject H 0 if z < -z H 0 : p p Reject H 0 if |z| > z

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Bina Nusantara Example: NSC Two-Tailed Test about a Population Proportion: Large n For a Christmas and New Year’s week, the National Safety Council estimated that 500 people would be killed and 25,000 injured on the nation’s roads. The NSC claimed that 50% of the accidents would be caused by drunk driving. A sample of 120 accidents showed that 67 were caused by drunk driving. Use these data to test the NSC’s claim with = 0.05.

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Bina Nusantara Example: NSC Two-Tailed Test about a Population Proportion: Large n – Hypothesis H 0 : p =.5 H a : p.5 – Test Statistic

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Bina Nusantara Example: NSC Two-Tailed Test about a Population Proportion: Large n – Rejection Rule Reject H 0 if z 1.96 – Conclusion Do not reject H 0. For z = 1.278, the p-value is.201. If we reject H 0, we exceed the maximum allowed risk of committing a Type I error (p-value >.050).

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Bina Nusantara 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 H 0 and the conclusion reject H 0. In such situations, it is recommended that the hypothesis-testing procedure be extended to include consideration of making a Type II error.

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Bina Nusantara Calculating the Probability of a Type II Error in Hypothesis Tests about a Population Mean 1. Formulate the null and alternative hypotheses. 2. Use the level of significance to establish a rejection rule based on the test statistic. 3. Using the rejection rule, solve for the value of the sample mean that identifies the rejection region. 4. Use the results from step 3 to state the values of the sample mean that lead to the acceptance of H 0 ; this defines the acceptance region. 5. Using the sampling distribution of for any value of from the alternative hypothesis, and the acceptance region from step 4, compute the probability that the sample mean will be in the acceptance region.

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Bina Nusantara Inferences About the Difference Between the Proportions of Two Populations Sampling Distribution of Interval Estimation of p 1 - p 2 Hypothesis Tests about p 1 - p 2

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Bina Nusantara Expected Value Standard Deviation Distribution Form If the sample sizes are large (n 1 p 1, n 1 (1 - p 1 ), n 2 p 2, and n 2 (1 - p 2 ) are all greater than or equal to 5), the sampling distribution of can be approximated by a normal probability distribution. Sampling Distribution of

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Bina Nusantara Interval Estimation of p 1 - p 2 Interval Estimate Point Estimator of

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Bina Nusantara Example: MRA MRA (Market Research Associates) is conducting research to evaluate the effectiveness of a client’s new advertising campaign. Before the new campaign began, a telephone survey of 150 households in the test market area showed 60 households “aware” of the client’s product. The new campaign has been initiated with TV and newspaper advertisements running for three weeks. A survey conducted immediately after the new campaign showed 120 of 250 households “aware” of the client’s product. Does the data support the position that the advertising campaign has provided an increased awareness of the client’s product?

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Bina Nusantara Example: MRA Point Estimator of the Difference Between the Proportions of Two Populations p 1 = proportion of the population of households “aware” of the product after the new campaign p 2 = proportion of the population of households “aware” of the product before the new campaign = sample proportion of households “aware” of the product after the new campaign = sample proportion of households “aware” of the product before the new campaign

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Bina Nusantara Example: MRA Interval Estimate of p 1 - p 2 : Large-Sample Case For =.05, z.025 = 1.96:.08 + 1.96(.0510).08 +.10 or -.02 to +.18 – Conclusion At a 95% confidence level, the interval estimate of the difference between the proportion of households aware of the client’s product before and after the new advertising campaign is -.02 to +.18.

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Bina Nusantara Hypothesis Tests about p 1 - p 2 Hypotheses H 0 : p 1 - p 2 < 0 H a : p 1 - p 2 > 0 Test statistic Point Estimator of where p 1 = p 2 where:

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Bina Nusantara Example: MRA Hypothesis Tests about p 1 - p 2 Can we conclude, using a.05 level of significance, that the proportion of households aware of the client’s product increased after the new advertising campaign? p 1 = proportion of the population of households “aware” of the product after the new campaign p 2 = proportion of the population of households “aware” of the product before the new campaign – Hypotheses H 0 : p 1 - p 2 < 0 H a : p 1 - p 2 > 0

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Bina Nusantara Example: MRA Hypothesis Tests about p 1 - p 2 – Rejection Rule Reject H 0 if z > 1.645 – Test Statistic – Conclusion Do not reject H 0.

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