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10.1: Multinomial Experiments Multinomial experiment A probability experiment consisting of a fixed number of trials in which there are more than two possible outcomes for each independent trial. A binomial experiment had only two possible outcomes. The probability for each outcome is fixed and each outcome is classified into categories. 1 Larson/Farber Example: A radio station claims that the distribution of music preferences for listeners in the broadcast region is as shown below. Distribution of music Preferences Classical4%Oldies2% Country36%Pop18% Gospel11%Rock29% Each outcome is classified into categories. The probability for each possible outcome is fixed.

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Chi-Square Goodness-of-Fit Test Used to test whether a frequency distribution fits an expected distribution. H 0 : The frequency distribution FITS the specified distribution. H a : The frequency distribution DOES NOT FIT the specified distribution. 2 Larson/Farber Example: To test the radio station’s claim, the executive can perform a chi-square goodness-of-fit test using the following hypotheses. H 0 : The distribution of music preferences in the broadcast region is 4% classical, 36% country, 11% gospel, 2% oldies, 18% pop, and 29% rock. (claim) H a : The distribution of music preferences differs from the claimed or expected distribution.

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Chi-Square Goodness-of-Fit Test Observed frequency O - frequency for the category observed in the sample. 3 Larson/Farber 4th ed Expected frequency E - calculated frequency for the category. Expected frequencies are obtained assuming the specified (or hypothesized) distribution. The expected frequency for the i th category is: E i = np i n = number of trials (sample size) p i = assumed probability of i th category. Example: A marketing executive randomly selects 500 radio music listeners from the broadcast region and asks each whether he or she prefers classical, country, gospel, oldies, pop, or rock music. The results are shown at the right. Find the observed frequencies and the expected frequencies for each type of music. Survey results (n = 500) Classical8 Country210 Gospel72 Oldies10 Pop75 Rock125 Type of music % of listeners Observed frequency Expected frequency Classical 4%8 Country36%210 Gospel11%72 Oldies 2%10 Pop18%75 Rock29%125 500(0.04) = 20 500(0.36) = 180 500(0.11) = 55 500(0.02) = 10 500(0.18) = 90 500(0.29) = 145 n = 500

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Chi-Square Goodness-of-Fit Test 1.The observed frequencies must be obtained by using a random sample. 2.Each expected frequency must be greater than or equal to 5. If these two conditions are satisfied, then the sampling distribution for the goodness-of-fit test is approximated by a chi-square distribution with k – 1 degrees of freedom, where k is the number of categories and test statistic is: You may perform a hypothesis test using Table 6 Appendix B to find critical values 4 Larson/Farber The test is always a right-tailed test. O = Observed frequency in each category E = Expected frequency of each category

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Example1: Goodness of Fit Test Use the music preference data to perform a chi-square goodness-of-fit test to test whether the distributions are different. Use α = 0.01. Survey results (n = 500) Classical8 Country210 Gospel72 Oldies10 Pop75 Rock125 Distribution of music preferences Classical4% Country36% Gospel11% Oldies2% Pop18% Rock29% 5 Larson/Farber 4th ed H 0 : music preference is 4% classical, 36% country, 11% gospel, 2% oldies, 18% pop, and 29% rock H a :music preference differs from the claimed or expected distribution =.01 d.f. = n –1 = 6 –1 = 5 01 χ2χ2 0 15.086 Type of music Observed frequency Expected frequency Classical820 Country210180 Gospel7255 Oldies10 Pop7590 Rock125145 Decision: Reject H 0 There is enough evidence to conclude that the distribution of music preferences differs from the claimed distribution.

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Example2: Goodness of Fit Test The manufacturer of M&M’s candies claims that the number of different- colored candies in bags of dark chocolate M&M’s is uniformly distributed. To test this claim, you randomly select a bag that contains 500 dark chocolate M&M’s. The results are shown in the table. Using α = 0.10, perform a chi- square goodness-of-fit test to test the claimed or expected distribution. What can you conclude? (Adapted from Mars Incorporated) 6 Larson/Farber 4th ed ColorFrequency Brown80 Yellow95 Red88 Blue83 Orange76 Green78 d.f. = 6 –1 = 5 n = 500 The claim is that the distribution is uniform, so the expected frequencies of the colors are equal. To find each expected frequency, divide the sample size by the number of colors. E = 500/6 ≈ 83.3 H 0 : distribution of different-colored candies in bags of dark chocolate M & Ms is uniform. H a :distribution of different-colored candies in bags of dark chocolate M & Ms is not uniform. 0.10 χ2χ2 09.236 Expected Frequency 83.3 Decision: Fail to Reject H 0 – there is not enough evidence to dispute the claim that the distribution is uniform.

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