1. © 2010 Pearson Prentice Hall. All rights reserved 7-1

2. © 2010 Pearson Prentice Hall. All rights reserved The area under the graph of the density function over an interval represents the probability of observing a value of the random variable in that interval. 7-2

3. © 2010 Pearson Prentice Hall. All rights reserved Relative frequency histograms that are symmetric and bell-shaped are said to have the shape of a normal curve. 7-3

4. © 2010 Pearson Prentice Hall. All rights reserved If a continuous random variable is normally distributed, or has a normal probability distribution, then a relative frequency histogram of the random variable has the shape of a normal curve (bell-shaped and symmetric). 7-4

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13. © 2010 Pearson Prentice Hall. All rights reserved The table gives the area under the standard normal curve for values to the left of a specified Z-score, z o, as shown in the figure. 7-13

14. © 2010 Pearson Prentice Hall. All rights reserved Area under the normal curve to the right of z o = 1 – Area to the left of z o 7-14

15. © 2010 Pearson Prentice Hall. All rights reserved 7-15

16. © 2010 Pearson Prentice Hall. All rights reserved The notation z α (prounounced “z sub alpha”) is the z-score such that the area under the standard normal curve to the right of z α is α. 7-16

17. © 2010 Pearson Prentice Hall. All rights reserved Notation for the Probability of a Standard Normal Random Variable P(a < Z < b) represents the probability a standard normal random variable is between a and b P(Z > a)represents the probability a standard normal random variable is greater than a. P(Z < a) represents the probability a standard normal random variable is less than a. 7-17

18. © 2010 Pearson Prentice Hall. All rights reserved For any continuous random variable, the probability of observing a specific value of the random variable is 0. For example, for a standard normal random variable, P(a) = 0 for any value of a. This is because there is no area under the standard normal curve associated with a single value, so the probability must be 0. Therefore, the following probabilities are equivalent: P(a < Z < b) = P(a < Z < b) = P(a < Z < b) = P(a < Z < b) 7-18

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21. © 2010 Pearson Prentice Hall. All rights reserved Suppose that we obtain a simple random sample from a population whose distribution is unknown. Many of the statistical tests that we perform on small data sets (sample size less than 30) require that the population from which the sample is drawn be normally distributed. Up to this point, we have said that a random variable X is normally distributed, or at least approximately normal, provided the histogram of the data is symmetric and bell-shaped. This method works well for large data sets, but the shape of a histogram drawn from a small sample of observations does not always accurately represent the shape of the population. For this reason, we need additional methods for assessing the normality of a random variable X when we are looking at sample data. 7-21

22. © 2010 Pearson Prentice Hall. All rights reserved A normal probability plot plots observed data versus normal scores. A normal score is the expected Z-score of the data value if the distribution of the random variable is normal. The expected Z-score of an observed value will depend upon the number of observations in the data set. 7-22

23. © 2010 Pearson Prentice Hall. All rights reserved 7-23

24. © 2010 Pearson Prentice Hall. All rights reserved The idea behind finding the expected Z-score is that if the data comes from a population that is normally distributed, we should be able to predict the area left of each of the data values. The value of f i represents the expected area left of the i th data value assuming the data comes from a population that is normally distributed. For example, f 1 is the expected area left of the smallest data value, f 2 is the expected area left of the second smallest data value, and so on. 7-24

25. © 2010 Pearson Prentice Hall. All rights reserved If sample data is taken from a population that is normally distributed, a normal probability plot of the actual values versus the expected Z-scores will be approximately linear. 7-25

26. © 2010 Pearson Prentice Hall. All rights reserved Criteria for a Binomial Probability Experiment An experiment is said to be a binomial experiment provided: 1. The experiment is performed n independent times. Each repetition of the experiment is called a trial. Independence means that the outcome of one trial will not affect the outcome of the other trials. 2. For each trial, there are two mutually exclusive outcomes - success or failure. 3. The probability of success, p, is the same for each trial of the experiment. 7-26

27. © 2010 Pearson Prentice Hall. All rights reserved For a fixed p, as the number of trials n in a binomial experiment increases, the probability distribution of the random variable X becomes more nearly symmetric and bell-shaped. As a general rule of thumb, if np(1 – p) > 10, the probability distribution will be approximately symmetric and bell-shaped. 7-27

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