# Where the probability of a type ii error is β or

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where the probability of a Type II error is β or less if the true difference μ d Δ. Note : If σ d is unknown, substitute an estimated value to obtain approximate sample size. / £ ¡ ¢ EXAMPLE 6.9 An experiment was done to determine the effect on dairy cattle of a diet supplemented with liquid whey. While no differences were noted in milk pro- duction measurements among cattle given a standard diet (7.5 kg of grain plus hay by choice) with water and those on the standard diet and liquid whey only, a considerable difference between the groups was noted in the amount of hay ingested. Suppose that one tests the null hypothesis of no difference in mean hay consumptiuon for the two diet groups of dairy cattle. For a two-tailed test with α = . 05, determine the approximate number of dairy cattle that should be included in each group if we want β . 10 for | μ 1 - μ 2 | ≥ . 5. Previous experimentation has shown σ to be approximately .8. SOLUTION From the description of the problem, we have α = . 05 , β . 10 for Δ = | μ 1 - μ 2 | ≥ . 5 and σ = . 8. Table 2 in the Appendix gives us z . 025 = 1 . 96 and z . 010 = 1 . 28. Substituting into the formula we have n 2( . 8) 2 (1 . 96 + 1 . 28) 2 ( . 5) 2 = 53 . 75 , or 54 . That is, we need 54 cattle per group to run the desired test. 14

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5. A Little Bit of History / £ ¡ ¢ Carl Friedrich Gauss The normal probability distribution is often referred to as the Gaussian distribution in honor of Carl Gauss, the individual thought to have dis- covered the idea. However, it was actually Abraham de Moivre who first wrote down the equation of the normal distribution. Gauss was born in Brunswick, Germany, on April 30, 1777. Gauss’ mathematical prowess was evident early in his life. At age eight he was able to instantly add the first 100 integers. In 1792, Gauss entered the Brunswick Collegium Corolinum and remained there for three years. In 1795, Gauss entered the University of G¨ottingen. In 1799, Gauss earned his doctorate. The subject of his dissertation was the Fundamental Theorem of Algebra. In 1809, Gauss published a book on the mathematics of planetary orbits. In this book, he further developed the theory of least-squares regression by analyzing the errors. The analysis of these errors led to the discovery that errors follow a normal distribution. Gauss was considered to be “glacially cold” as a person and had troubled relationships with his family. Gauss died February 23, 1855. / £ ¡ ¢ Abraham de Moivre Abraham de Moivre was born in France on May 26, 1667. He is known as a great contributor to the areas of probability and trigonometry. De Moivre studied for five years at the Protestant academy at Sedan. From 1682 to 1684, he studied logic at Saumur. In 1685, he moved to Eng- land. De Moivre was elected a fellow of the Royal Society in 1697. He was part of the commission to settle the dispute between Newton and Leibniz regarding the discoverer of calculus. He published The Doctrine of Chance in 1718. In 1733, he developed the equation that describes the normal curve. Unfortunately, de Moivre had a difficult time being accepted in English society (perhaps due to his accent) and was able to make only a meager living tutor- ing mathematics. An interesting piece of information regarding de Moivre; he correctly predicted the day of his death, Nov. 27, 1754.
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