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The Mean of a Discrete Probability Distribution The mean of a probability distribution for a discrete random variable is x p(x) where the sum is taken over all possible values of x. Agresti/Franklin Statistics, 1e, 1 of 139 Which Wager do You Prefer? You are given $100 and told that you must pick one of two wagers, for an outcome based on flipping a coin: A. You win $200 if it comes up heads and lose $50 if it comes up tails. B. You win $350 if it comes up head and lose your original $100 if it comes up tails. Without doing any calculation, which wager would you prefer? Agresti/Franklin Statistics, 1e, 2 of 139 You win $200 if it comes up heads and lose $50 if it comes up tails. Find the expected outcome for this wager. a. b. c. d. $100 $25 $50 $75 Agresti/Franklin Statistics, 1e, 3 of 139 You win $350 if it comes up head and lose your original $100 if it comes up tails. Find the expected outcome for this wager. a. b. c. d. $100 $125 $350 $275 Agresti/Franklin Statistics, 1e, 4 of 139 Section 6.2 How Can We Find Probabilities for Bell-Shaped Distributions? Agresti/Franklin Statistics, 1e, 5 of 139 Normal Distribution The normal distribution is symmetric, bell-shaped and characterized by its mean µ and standard deviation σ. The probability of falling within any particular number of standard deviations of µ is the same for all normal distributions. Agresti/Franklin Statistics, 1e, 6 of 139 Normal Distribution Agresti/Franklin Statistics, 1e, 7 of 139 Z-Score Recall: The z-score for an observation is the number of standard deviations that it falls from the mean. Agresti/Franklin Statistics, 1e, 8 of 139 Z-Score For each fixed number z, the probability within z standard deviations of the mean is the area under the normal curve between - z and z Agresti/Franklin Statistics, 1e, 9 of 139 Z-Score For z = 1: 68% of the area (probability) of a normal distribution falls between: - 1 and 1 Agresti/Franklin Statistics, 1e, 10 of 139 Z-Score For z = 2: 95% of the area (probability) of a normal distribution falls between: - 2 and 2 Agresti/Franklin Statistics, 1e, 11 of 139 Z-Score For z = 3: Nearly 100% of the area (probability) of a normal distribution falls between: - 3 and 3 Agresti/Franklin Statistics, 1e, 12 of 139 The Normal Distribution: The Most Important One in Statistics It’s important because… • Many variables have approximate normal • • distributions. It’s used to approximate many discrete distributions. Many statistical methods use the normal distribution even when the data are not bell-shaped. Agresti/Franklin Statistics, 1e, 13 of 139 Finding Normal Probabilities for Various Z-values Suppose we wish to find the probability within, say, 1.43 standard deviations of µ. Agresti/Franklin Statistics, 1e, 14 of 139 Z-Scores and the Standard Normal Distribution When a random variable has a normal distribution and its values are converted to z-scores by subtracting the mean and dividing by the standard deviation, the z-scores have the standard normal distribution. Agresti/Franklin Statistics, 1e, 15 of 139 Example: Find the probability within 1.43 standard deviations of µ Agresti/Franklin Statistics, 1e, 16 of 139 Example: Find the probability within 1.43 standard deviations of µ Probability below 1.43σ = .9236 Probability above 1.43σ = .0764 By symmetry, probability below -1.43σ = .0764 Total probability under the curve = 1 Agresti/Franklin Statistics, 1e, 17 of 139 Example: Find the probability within 1.43 standard deviations of µ Agresti/Franklin Statistics, 1e, 18 of 139 Example: Find the probability within 1.43 standard deviations of µ The probability falling within 1.43 standard deviations of the mean equals: 1 – 0.1528 = 0.8472, about 85% Agresti/Franklin Statistics, 1e, 19 of 139 How Can We Find the Value of z for a Certain Cumulative Probability? Example: Find the value of z for a cumulative probability of 0.025. Agresti/Franklin Statistics, 1e, 20 of 139 Example: Find the Value of z For a Cumulative Probability of 0.025 Look up the cumulative probability of 0.025 in the body of Table A. A cumulative probability of 0.025 corresponds to z = -1.96. So, a probability of 0.025 lies below µ - 1.96σ. Agresti/Franklin Statistics, 1e, 21 of 139 Example: Find the Value of z For a Cumulative Probability of 0.025 Agresti/Franklin Statistics, 1e, 22 of 139 Example: What IQ Do You Need to Get Into Mensa? Mensa is a society of high-IQ people whose members have a score on an IQ test at the 98th percentile or higher. Agresti/Franklin Statistics, 1e, 23 of 139 Example: What IQ Do You Need to Get Into Mensa? How many standard deviations above the mean is the 98th percentile? • The cumulative probability of 0.980 in the body of Table A corresponds to z = 2.05. • The 98th percentile is 2.05 standard deviations above µ. Agresti/Franklin Statistics, 1e, 24 of 139 Example: What IQ Do You Need to Get Into Mensa? What is the IQ for that percentile? • Since µ = 100 and σ 16, the 98th percentile of IQ equals: µ + 2.05σ = 100 + 2.05(16) = 133 Agresti/Franklin Statistics, 1e, 25 of 139 Z-Score for a Value of a Random Variable The z-score for a value of a random variable is the number of standard deviations that x falls from the mean µ. It is calculated as: z x- Agresti/Franklin Statistics, 1e, 26 of 139 Example: Finding Your Relative Standing on The SAT Scores on the verbal or math portion of the SAT are approximately normally distributed with mean µ = 500 and standard deviation σ = 100. The scores range from 200 to 800. Agresti/Franklin Statistics, 1e, 27 of 139 Example: Finding Your Relative Standing on The SAT If one of your SAT scores was x = 650, how many standard deviations from the mean was it? Agresti/Franklin Statistics, 1e, 28 of 139 Example: Finding Your Relative Standing on The SAT Find the z-score for x = 650. x- 650 - 500 z 1.50 100 Agresti/Franklin Statistics, 1e, 29 of 139 Example: Finding Your Relative Standing on The SAT What percentage of SAT scores was higher than yours? • Find the cumulative probability for the z• score of 1.50 from Table A. The cumulative probability is 0.9332. Agresti/Franklin Statistics, 1e, 30 of 139 Example: Finding Your Relative Standing on The SAT The cumulative probability below 650 is 0.9332. The probability above 650 is 1 – 0.9332 = 0.0668 About 6.7% of SAT scores are higher than yours. Agresti/Franklin Statistics, 1e, 31 of 139 Example: What Proportion of Students Get A Grade of B? On the midterm exam in introductory statistics, an instructor always give a grade of B to students who score between 80 and 90. One year, the scores on the exam have approximately a normal distribution with mean 83 and standard deviation 5. About what proportion of students get a B? Agresti/Franklin Statistics, 1e, 32 of 139 Example: What Proportion of Students Get A Grade of B? Calculate the z-score for 80 and for 90: z z x- x- 90 - 83 1.40 5 80 - 83 - 0.60 5 Agresti/Franklin Statistics, 1e, 33 of 139 Example: What Proportion of Students Get A Grade of B? Look up the cumulative probabilities in Table A. • For z = 1.40, cum. Prob. = 0.9192 • For z = -0.60, cum. Prob. = 0.2743 It follows that about 0.9192 – 0.2743 = 0.6449, or about 64% of the exam scores were in the ‘B’ range. Agresti/Franklin Statistics, 1e, 34 of 139 Using z-scores to Find Normal Probabilities If we’re given a value x and need to find a probability, convert x to a z-score using: z x- Use a table of normal probabilities to get a cumulative probability. Convert it to the probability of interest. Agresti/Franklin Statistics, 1e, 35 of 139 Using z-scores to Find Random Variable x Values If we’re given a probability and need to find the value of x, convert the probability to the related cumulative probability. Find the z-score using a normal table. Evaluate x = zσ + µ. Agresti/Franklin Statistics, 1e, 36 of 139 Example: How Can We Compare Test Scores That Use Different Scales? When you applied to college, you scored 650 on an SAT exam, which had mean µ = 500 and standard deviation σ = 100. Your friend took the comparable ACT in 2001, scoring 30. That year, the ACT had µ = 21.0 and σ = 4.7. How can we tell who did better? Agresti/Franklin Statistics, 1e, 37 of 139 What is the z-score for your SAT score of 650? For the SAT scores: µ = 500 and σ = 100. a. 2.15 b. 1.50 c. -1.75 d. -1.25 Agresti/Franklin Statistics, 1e, 38 of 139 What percentage of students scored higher than you? a. b. c. d. 10% 5% 2% 7% Agresti/Franklin Statistics, 1e, 39 of 139 What is the z-score for your friend’s ACT score of 30? The ACT scores had a mean of 21 and a standard deviation of 4.7. a. 1.84 b. -1.56 c. 1.91 d. -2.24 Agresti/Franklin Statistics, 1e, 40 of 139 What percentage of students scored higher than your friend? a. b. c. d. 3% 6% 10% 1% Agresti/Franklin Statistics, 1e, 41 of 139 Standard Normal Distribution The standard normal distribution is the normal distribution with mean µ = 0 and standard deviation σ = 1. It is the distribution of normal z-scores. Agresti/Franklin Statistics, 1e, 42 of 139