4.3 The Normal Distribution
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1 4.3 The Normal Distribution Objectives. Definition of normal distribution. Standard normal distribution. Specialties of the graph of the standard normal distribution. Percentiles of the standard normal distribution. Z-values. Nonstandard normal distribution. Normal approximation to a discrete population. Normal approximation to a binomial distribution Definition. A continuous rv X is said to have a normal distribution with parameters and (or and ), where and 0, if the pdf of X is f x;, Abbreviation: X N, 1 x e x Meaning of the parameters and., V X E X Interactive website with normal curves: 1
2 4.3. Graphs of Normal Curves. Mean of, N determines the axis of symmetry x. Since Normal Probability Density Function is symmetric and bell-shaped, its mean is also its mode (the highest point) and its median (the point that splits the area under the curve in half). (!) Therefore, the area to each side of the is 0.5. Standard Deviation is a measure of spread (variability). When is large the normal curve is wide and, therefore, low. When is small, the normal curve is narrow and high. 1 Inflection point on the graph,.
3 The Empirical Rule for N, If the population distribution of a variable is (approximately) normal, then 1. Roughly 68% of the values are within 1 SD of the mean.. Roughly 95% of the values are within SDs of the mean. 3. Roughly 99.7% of the values are within 3 SDs of the mean. Remark When dealing with, N we need to compute x P a X b e dx b a 1 There is no antiderivative as an elementary function, so standard techniques of computing such integrals do not work. These integrals are computed by using numerical techniques and tabulations. 3
4 Standard Normal Distribution. Definitions. The normal distribution with parameters 0 and 1 is called the standard normal distribution. A random variable having a standard normal distribution is called a standard normal random variable and will be denoted by Z. The pdf of Z is The graph of ; 0,1 z 1 f z; 0,1 e z The standard normal distribution Z N0,1 z 1 f z e is called standard normal curve, or z curve. Cumulative function of a standard normal distribution. z ;0,1 z P Z z f t dt is used as a reference distribution from which information about other normal distributions can be obtained. N 0,1 Center at 0. Inflection point 1 1, 4
5 Standartization. Process of transforming, X N in Z N 0,1 Computing probabilities for Z N 0,1 Example (4.13 p. 154 textbook.) Appendix Table A3 Determine the following standard normal probabilities: a) PZ 1.5 ; b) PZ 1.5 ; c) PZ 1.5, and d) P 0.5 Z 1.5 PZ PZ
6 PZ P 0.5 Z P 0.5 Z
7 Percentiles of the Standard Normal Distribution. Z-table. For any p between 0 and 1, Appendix Table A.3 can be used to obtain the (100p) th percentile of the standard normal distribution. The 99th percentile of the standard normal distribution is the value on the horizontal axis such that the area under the z curve to the left of the value is Appendix Table A.3 gives for fixed z the area under the standard normal curve to the left of z. So, we are given the area and need to determine the value of z. This is the inverse problem. Thus, the table is used in an inverse fashion: Find in the middle of the table.9900; the row and column in which it lies identify the 99 th z percentile. Here the intersection of the row marked.3 and column marked.03, so the 99th percentile is (approximately) z
8 Example Find the 95 th percentile of N 0;1. 95 th percentile Z Values for Commonly Used Percentiles Percentile 1st Z th 5th 10th 5th 50th 75th 90th 95th 97.5th 99th The most important for theory and practice Notation for z Critical Values. will denote the value on the z-axis for which of the area under the z curve lies to the right of is equal to. Remark. The area under the curve to the left of is. This means that is the percentile of the standard normal distribution 8
9 Illustration for critical value z. Area 1 Area Example Find critical value z for Solution. 1) Critical value z ) In the table body find the value 0.85(0.8508) and consider the row and column that contain this value. 3) Read critical value z 1.04 Area Area 0.15 Critical value z
10 Approximating the Binomial Distribution. The probability histogram of Binomial Distribution is a bit skewed and the normal curve gives a very good approximation for it. Let X be a binomial rv based on n trials with success probability p. Then if the binomial probability histogram is not too skewed, X has approximately a normal distribution with np and npq. area under the normal curve P X x B x, n p to the left of x 0.5 x 0.5 np npq The good approximation is provided when np 10 and nq 10. Example (Example 4.0 p.161). Suppose that 5% of all students at a large public university receive financial aid. Let X be the number of students in a random sample of size 50 who receive financial aid, so that p 0.5. Then 1.5 and Since np nq , the approximation can safely be applied. 10
11 For example, the probability that at most 10 students receive aid is P X 10 B10; 50,
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