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# ch10 - AMS 315/576 Lecture Notes Chapter 10 CATEGORICAL...

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¤ § ¥ ƒ AMS 315/576 Lecture Notes Chapter 10. CATEGORICAL DATA 10.1 Inferences About one population Proportion π The public opinion polls are becoming increasingly popular and vital in our societies nowadays. Almost daily, the news media report the results of some poll. There are polls aimed to determine the percentage of people in favor of the President’s various policies, the fraction of voters in favor of a certain candidate, the percentage of customers who plan to buy a certain product within the next year, and the proportion of college students who smoke cigarettes. In each case, we are interested in estimating the percentage (or proportion) of a population with a certain characteristic. In this section we consider methods for making inferences about one population proportion when the sample is considered large. 1. Sample. The sample is summarized entirely by 2 statistics: (1). the sample size n , and (2). the total number of subjects (or objects) in the sample that possess the characteristic of interest, x . 2. Point estimator. The population proportion π is estimated by the sample proportion ˆ π = x/n . When the sample is considered large, namely 5 and n (1 - π ) 5, the distribution of ˆ π can be approximated by a normal distribution with mean π and variance π (1 - π ) /n . 3. Large sample confidence interval. 1

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¤ § ¥ ƒ Confidence Interval for π , with Confidence Coefficient of (1 - α ) ˆ π ± z α/ 2 r b π (1 - b π ) n , where ˆ π = x n . Note: the large sample confidence interval is valid when n b π = x 5 and n (1 - b π ) = n - x 5. 4. Sample size calculation. ¤ § ¥ ƒ Sample Size Required for a 100(1 - α )% Confidence Interval for π of the Form ˆ π ± E n = z 2 α/ 2 π (1 - π ) E 2 π is known n = z 2 α/ 2 4 E 2 π is unknown Note. The above sample size will ensure a 100(1 - α )% Confidence Interval for π to be of the required length 2E, or equivalnetly, a maximum error of E with probability 100(1 - α )%. 2
5. Hypothesis testing. ¤ § ¥ ƒ Summary of a Statistical Test for π H 0 : π = π 0 ( π 0 is specified ) H a : 1. π > π 0 2. π < π 0 3. π 6 = π 0 T.S.: z 0 = ˆ π - π 0 q π 0 (1 - π 0 ) n R.R.: For a probability α of a Type I error 1. reject H 0 if z 0 > z α 2. reject H 0 if z 0 < - z α 3. reject H 0 if | z 0 | > z α/ 2 Note: the large sample test is valid when 0 5 and n (1 - π 0 ) 5. / £ ¡ ¢ EXAMPLE 10.1 Sports car owners in a town complain that their cars are judged differently from family-style cars at the state vehicle inspection station. Previous records indicate that 30% of all passenger cars fail the inspection on the first time through. In a random sample of 150 sports cars, 60 failed the inspection on the first time through. Is there sufficient evidence to indicate that the percentage of first failures for sports cars is higher than the percentage for all passenger cars?

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ch10 - AMS 315/576 Lecture Notes Chapter 10 CATEGORICAL...

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