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Chapter 5 Notes

Chapter 5 Notes - 5.1 5.2 xx Standard Normal Curve x NOT...

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5.1 & 5.2 Let’s suppose we have the numbers, 1, 2, 3, 4, and 5. What is the probability that 1 is chosen? P(1 is chosen) = 1/5. We can also say that P(4 is chosen) = 1/5. So, the probability that any one number is chosen is 1/5. Hence, we can make the following probability distribution: x P( x ) 1 1/5 2 1/5 3 1/5 4 1/5 5 1/5 This gives rise to the following histogram: P(x) 1/5 x 0 1 2 3 4 5 Note that we get a uniform distribution since each number has an equal chance of getting picked. Since the histogram promotes continuity, we can consider all the real numbers from 0 to 5. That is, let’s consider numbers such as .1, 4.5, and 3.666667. We saw from previous sections that probability and percentage are related. For example, the probability of choosing a 1 is 1/5 is due to the fact that 1 makes up 1/5 or 0.2 or 20% of all the numbers 1 through 5. The probability of choosing a 2 or 3 is 2/5 which also means that 2 and 3 make up 2/5 or .4 or 40% of all the numbers 1 through 5. So, from the histogram above, let’s talk about the probability of choosing a number that is between 0 and 1. Since all the numbers from 0 to 1 make up 20% of all the numbers from 0 to 5. We can see that by the picture: P(x) 1/5

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x 0 1 2 3 4 5 So, we can say P(0 < x < 1) = 0.2. P(0 < x < 1) = 0.2 means the probability that a number between 0 and 1 is chosen is 0.2. Now, let’s consider the shaded rectangle above. Finding the area of the shaded rectangle, we have A = lw Þ A = (1)(1/5) Þ A = 1/5 Now, let’s consider P(2 < x < 5). That is, P(x) 1/5 x 0 1 2 3 4 5 The shaded rectangle makes up .6 or 60% of the whole rectangle But area = (5-2)(1/5) = 3(1/5) = 3/5 or .6. It makes sense to say P(0 < x < 5) = 1 because since the numbers being considered are between 0 and 5. So, the probability that we can pick a number and that number is between 0 and 5 is 1. But, note that area of total rectangle = (5 – 0)(1/5) = 5(1/5) = 1. Thus, probability and “area under the P(x) line” are related. Now, the above pertains only to distributions that are uniform. Distributions for the most part are normal. That is: z
So, to calculate a probability associated with a normal distribution, we must be able to find the area under the normal curve which is more complicated than finding the area under a line. In fact, finding the area under the normal curve would require methods of calculus. Since calculus is not a prerequisite for this course, we will use a normal curve called the standard normal curve. Standard Normal Curve – Normal curve with 0 x = and s = 1. Using calculus, one can find that the total area under the standard normal curve is 1. We do p. 240 #6. To start, we draw a picture: ------------------- -------------------- -------------------- 0 1.96 z The “---------“ marks are being used to indicate a shading under the standard normal curve between 0 and 1.96. We want to find this shaded area. Also, NOTE that z = 0 is always the center point on the z-axis just like 0 is usually the center point on a number line. To find the area, we use Table A-2 at the very back cover of the textbook. Above Table A-2, there is a picture of the normal curve with shaded area between 0 and z. Since z is to the right of 0, z is a positive number.

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Chapter 5 Notes - 5.1 5.2 xx Standard Normal Curve x NOT...

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