224 CHAPTER 5 Discrete Probability Distributions As in earlier sections, finding values for m and s can be great fun, but it is especially important to interpret and understand those values, so the range rule of thumb and the rare event rule for inferential statistics can be very helpful. Here is a brief summary of the range rule of thumb: Values are significantly low or high if they differ from the mean by more than 2 standard deviations, as described by the following: Range Rule of Thumb Significantly lowvalues … 1m - 2s2 Significantly highvalues Ú 1m + 2s2 Values not significant: Between 1m - 2s2 and 1m + 2s2 b. The phrase “fewer than three” vegetarians means that the number of vegetarians is 0 or 1 or 2. (See the bottom of the preceding page for the probability values.) P1fewer than 3 vegetarians2 = P10or 1or 22 = P102 + P112 + P122 = 0.774 + 0.204 + 0.021 = 0.999 YOUR TURN. Do Exercise 15 “Cashless Society” using Table A-1. Proportions of Males>Females It is well known that when a baby is born, boys and girls are not equally likely. It is currently believed that 105 boys are born for every 100 girls, so the probability of a boy is 0.512. Kristen Navara of the University of Georgia conducted a study showing that around the world, more boys are born than girls, but the difference becomes smaller as people are located closer to the equator. She used latitudes, temperatures, unemployment rates, gross and national products from 200 countries and conducted a statistical analysis showing that the proportions of boys appear to be affected only by latitude and its related weather. So far, no one has identified a reasonable explanation for this phenomenon. It k w is a n li c PART 1 Using Mean and Standard Deviation for Critical Thinking Section 5-1 included formulas for finding the mean, variance, and standard deviation from any discrete probability distribution. A binomial distribution is a particular type of discrete probability distribution, so we could use those same formulas, but if we know the values of n and p, it is much easier to use the following: PART 2 For Binomial Distributions FORMULA 5-6 Mean: m = np FORMULA 5-7 Variance: s 2 = npq FORMULA 5-8 Standard Deviation: s = 1npq The Chapter Problem and Example 3 involve n = 460 overtime wins in NFL football games. We get p = 0.5 and q = 0.5 by assuming that winning the overtime coin toss does not provide an advantage, so both teams have the same 0.5 chance of winning the game in overtime. a. Find the mean and standard deviation for the number of wins in groups of 460 games. b. Use the range rule of thumb to find the values separating the numbers of wins that are significantly low or significantly high. c. Is the result of 252 overtime wins in 460 games significantly high? CP EXAMPLE 5 Using Parameters to Determine Significance
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