The Mean of a Discrete Probability Distribution The mean of a probability distribution for a discrete random variable is where the sum is taken over all possible values of x.

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?

You win $200 if it comes up heads and lose $50 if it comes up tails. Find the expected outcome for this wager. $100 $25 $50 $75

Find the expected outcome for this wager. 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. $100 $125 $350 $275

How Can We Find Probabilities for Bell-Shaped Distributions? Section 6.2 How Can We Find Probabilities for Bell-Shaped Distributions?

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.

Normal Distribution

Z-Score Recall: The z-score for an observation is the number of standard deviations that it falls from the mean.

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-Score For z = 1: 68% of the area (probability) of a normal distribution falls between:

Z-Score For z = 2: 95% of the area (probability) of a normal distribution falls between:

Z-Score For z = 3: Nearly 100% of the area (probability) of a normal distribution falls between:

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.

Finding Normal Probabilities for Various Z-values Suppose we wish to find the probability within, say, 1.43 standard deviations of µ.

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.

Example: Find the probability within 1.43 standard deviations of µ

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

Example: Find the probability within 1.43 standard deviations of µ

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%

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.

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.

Example: Find the Value of z For a Cumulative Probability of 0.025

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.

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.

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

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:

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.

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?

Example: Finding Your Relative Standing on The SAT Find the z-score for x = 650.

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.

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.

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?

Example: What Proportion of Students Get A Grade of B? Calculate the z-score for 80 and for 90:

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.

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: Use a table of normal probabilities to get a cumulative probability. Convert it to the probability of interest.

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.

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?

What is the z-score for your SAT score of 650? For the SAT scores: µ = 500 and σ = 100. 2.15 1.50 -1.75 -1.25

What percentage of students scored higher than you? 10% 5% 2% 7%

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. 1.84 -1.56 1.91 -2.24

What percentage of students scored higher than your friend? 3% 6% 10% 1%

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.