10 1 Multinomial Experiments Multinomial experiment A probability

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. Example: • A radio station claims that the distribution of music preferences for listeners in the broadcast region is as shown below. Each outcome is classified into categories. Distribution of music Preferences Classical Country Gospel Larson/Farber 4% 36% 11% Oldies Pop Rock 2% 18% 29% The probability for each possible outcome is fixed. 1

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. • Ha: The frequency distribution DOES NOT FIT the specified distribution. 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) Ha: The distribution of music preferences differs from the claimed or expected distribution. Larson/Farber 2

Chi-Square Goodness-of-Fit Test • Observed frequency O - frequency for the category observed in the sample. • Expected frequency E - calculated frequency for the category. § Expected frequencies are obtained assuming the specified (or hypothesized) distribution. The expected frequency for the ith category is: Ei = npi n = number of trials (sample size) pi = assumed probability of ith 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. Type of music Classical Country Gospel Oldies Pop Rock % of listeners Observed frequency 4% 36% 11% 2% 18% 29% 8 210 72 10 75 125 Expected frequency 500(0. 04) = 20 500(0. 36) = 180 Survey results (n = 500) Classical 8 Country 210 Gospel 72 Oldies 10 Pop 75 Rock 125 n = 500(0. 11) = 55 500(0. 02) = 10 500(0. 18) = 90 500(0. 29) = 145 Larson/Farber 4 th ed 3

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: The test is always a right-tailed test. O = Observed frequency in each category E = Expected frequency of each category You may perform a hypothesis test using Table 6 Appendix B to find critical values Larson/Farber 4

Example 1: 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. Distribution of music preferences Classical 4% Country 36% Gospel 11% Oldies 2% Pop 18% Rock 29% Type of Observed music frequency Classical 8 Country 210 Gospel 72 Oldies 10 Pop 75 Rock 125 Survey results (n = 500) Classical 8 Country 210 Gospel 72 Oldies 10 Pop 75 Rock 125 Expected frequency 20 180 55 10 90 145 H 0: music preference is 4% classical, 36% country, 11% gospel, 2% oldies, 18% pop, and 29% rock Ha : music preference differs from the claimed or expected distribution =. 01 d. f. = n – 1 = 6 – 1 = 5 01 0 2 15. 086 χ Decision: Reject H 0 There is enough evidence to conclude that the distribution of music preferences differs from 5 Larson/Farber 4 th ed the claimed distribution.

Example 2: Goodness of Fit Test The manufacturer of M&M’s candies claims that the number of differentcolored 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 chisquare goodness-of-fit test to test the claimed or expected distribution. What can you conclude? (Adapted from Mars Incorporated) Color Frequency Brown Yellow Red Blue Orange Green d. f. = 6 – 1 = 5 80 95 88 83 76 78 n = 500 Expected H : distribution of different-colored candies in Frequency 0 bags of dark chocolate M & Ms is uniform. 83. 3 Ha : distribution of different-colored candies in bags of dark chocolate M & Ms is not uniform. Decision: Fail to Reject H 0 – there is • The claim is that the distribution is uniform, so not enough evidence to dispute the expected frequencies of the colors are equal. claim that the distribution is uniform. • To find each expected frequency, divide the sample size by the number of colors. 0. 10 Larson/Farber 4 th ed • E = 500/6 ≈ 83. 3 0 9. 236 χ2 6