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Probability Statistics

The Most Important Number. In the last section we looked at three related concepts: The mean of a sample, ?The mean of a population, ?The mean or expected value of a random variable, E(X)The mean is probably the single most important number that can be used to describe a sample, population, or p

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Probability Statistics

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    1. Probability & Statistics The Variance and Standard Deviation

    2. The Most Important Number In the last section we looked at three related concepts: The mean of a sample, Ż The mean of a population, µ The mean or expected value of a random variable, E(X) The mean is probably the single most important number that can be used to describe a sample, population, or probability distribution of a random variable.

    3. The Second Most Important Number In this section, we learn about the next most important number: the variance. Closely associated with the variance, and oftentimes used more, is the standard deviation.

    9. Example 1 Two golfers recorded their scores for 20 nine-hole rounds of golf. a.) Compute the sample mean and the variance of each golfer’s score. b.) Who is the better golfer? (Note: The lower the score, the better.) c.) Who is the more consistent golfer?

    12. Alternative Definitions Two alternative definitions for variance are and, for a binomial random variable with parameters n, p, and q,

    13. Example 2 Compute the variance and standard deviation of the probability distribution in the table at right.

    15. Example 3 Find the variance when a fair coin is tossed 5 times and X is the number of heads.

    16. More Formulas for Sample Variance and Standard Deviation (Use When Data Values Listed Individually)

    17. Example 4 The table at right gives the number of books (in millions) in the 10 largest public libraries in the United States. Determine the mean and the standard deviation for the number of books.

    19. Chebychev's Inequality Chebychev's Inequality Suppose that a probability distribution with numerical outcomes has expected value and standard deviation Then the probability that a randomly chosen outcome lies between - c and + c is at least

    20. Example 5 A drug company sells bottles containing 100 capsules of penicillin. Due to bottling procedure, not every bottle contains exactly 100 capsules. Assume that the average number of capsules in a bottle is 100 and the standard deviation is 2. If the company ships 5000 bottles, estimate the number of bottles having between 95 and 105 capsules, inclusive.

    21. Example 6 An electronics firm determines that the number of defective transistors in each batch averages 15 with standard deviation 10. Suppose that 100 batches are produced. Estimate the number of batches having between 0 and 30 defective transistors.

    22. Summary The variance of a random variable is the sum of the products of the square of each outcome's distance from the expected value and the outcome's probability. The variance of the random variable X can also be computed as E(X 2) - [E(X)] 2. A binomial random variable with parameters n and p has expected value np and variance np(1 - p).

    23. Summary The square root of the variance is called the standard deviation. Chebychev's Inequality states that the probability that an outcome of an experiment is within c units of the mean is at least , where is the standard deviation.

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