# Z z α 3 reject h if z z α 2 note n ˆ π and n 1 ˆ

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z 0 < - z α 3. reject H 0 if | z 0 | > z α/ 2 Note: n ˆ π and n (1 - ˆ π ) must be greater than or equal to 5 for both populatons in order for the normal approximation (and hence for this test) to hold. That is, the above test is valid when x 1 5, n 1 - x 1 5, x 2 5, and n 2 - x 2 5. / £ ¡ ¢ EXAMPLE 10.4 In a recent survey of county high school students ( n 1 = 100 males and n 2 = 100 females), 58 of the males and 46 of the females sampled said they consume alcohol on a regular basis. Use the sample data to conduct a test H 0 : π 1 - π 2 = 0 against the one-sided alternative H a : π 1 - π 2 > 0, that a higher proportion of males than females consume alcohol on a regular basis. Use α = . 05. SOLUTION The four parts of the statistical test are shown here: H 0 : π 1 - π 2 = 0 H a : π 1 - π 2 > 0 8

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T.S.: z 0 = ˆ π 1 - ˆ π 2 r b π (1 - b π ) 1 n 1 + 1 n 2 · R.R.: For α = . 05, reject H 0 if z > 1 . 645. From the sample data we find ˆ π 1 = 58 100 = . 58 , ˆ π 2 = 46 100 = . 46 , and ˆ π = 58 + 46 100 + 100 = . 52 , Note also that n ˆ π and n (1 - ˆ π ) are 5 or more for both samples, validating the normal approximations to the binomial. Substituting into the test statistic, we obtain z = . 58 - . 46 q . 52( . 48) ( 1 100 + 1 100 ) = . 12 . 071 = 1 . 69 . Conclusion: Since z = 1 . 69 exceeds 1.645, we reject H 0 : π 1 - π 2 = 0; we have shown that a higher proportion of high school males than females in the county studies consumes alcohol on a regular basis. 10.4 The Multinomial Experiment and Chi-Square Goodness- of-fit Test ¤ § ¥ ƒ The Multinomial Experiment 1. The experiment consists of n identical trials. 2. Each trial results in one of k outcomes. 3. The probability that a single trial will result in outcome i is π i , i = 1 , 2 , . . . , k, and remains constant from trial to trial. (Note: i π i = 1. ) 4. The trials are independent. 5. We are interested in n i , the number of trials resulting in outcome i . (Note: i n i = n .) The probability distribution for the number of observations resulting in each of the k outcomes, called the multinomial distribution , is given by the for- mula P ( n 1 , n 2 , . . . , n k ) = n ! n 1 ! n 2 ! . . . n k ! π n 1 1 π n 2 2 · · · π n k k 9
Recall from Chapter 4, where we discussed the binomial probability distri- bution, that n ! = n ( n - 1) · · · 1 and 0! = 1 . We can use the formula for the multinomial distribution to compute the probability of particular events. Note, when k = 2, the multinomial experiment reduces to the binomial experiment. Equivalently, the multinomial distribution reduces to the binomial distribution. / £ ¡ ¢ EXAMPLE 10.5 (only required for AMS graduate students) Previous experience with the breeding of a particular herd of cattle suggests that the probability of obtaining one healthy calf from a mating is . 83. Similarly, the probabilities of obtaining zero or two healthy calves are, respectively, .15 and .02. If a farmer breeds three dams from the herd, find the probability of obtaining exactly three healthy calves.

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