233 Where You’re Going In Chapter 5, you will learn how to recognize normal (bell-shaped) distributions and how to use their properties in real-life applications. Suppose that you are a farmer planning to buy 20 Holstein cows and 10 Jersey cows from a breeder.You want to know the probabilities that the groups of cows will produce certain average daily amounts of milk. You will learn how to calculate this type of probability using a sampling distribution of sample means and the Central Limit Theorem in Section 5.4. The graphs below show the distributions of sample means of milk produced daily by the two breeds of cows. The table shows the information given to you by the breeder. Assume that the amounts of milk produced are normally distributed. Amount of milk produced per day (in pounds) Breed Mean Standard deviation Holstein 69.3 11.7 Jersey 49.7 10.1 You can use this information to make calculations about average amounts of milk produced daily by the cows. For instance, the probability that the 20 Holstein cows will produce an average of at least 65 pounds of milk per day is about 94.95%, and the probability that the 10 Jersey cows will produce an average of between 50 and 60 pounds of milk per day is about 46.35%. Where You’ve Been In Chapters 1 through 4, you learned how to collect and describe data, find the probability of an event, and analyze discrete probability distributions. You also learned that when a sample is used to make inferences about a population, it is critical that the sample not be biased. For instance, how would you organize a study to determine which breed of dairy cow is the most profitable? When the U.S. Department of Agriculture performs this study, it uses random sampling and then records the measures of various milk production and physical traits such as pounds produced, fat percentage, protein percentage, productive life, somatic cell count, and calving ability. The studies have repeatedly shown Holstein cows to be the most profitable breed of dairy cow. Other top breeds are Jersey, Brown Swiss, and Ayrshire cows. 50 55 60 65 70 75 80 85 90 3 6 9 12 15 Milk produced (in pounds) Percent Average Daily Milk Production by Holstein Cows Milk produced (in pounds) Percent Average Daily Milk Production by Jersey Cows 35 40 45 50 55 60 65 70 2 4 6 8 10 12 14
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