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Probability concepts and the normal distribution
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Probability concepts and the normal distribution



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  • N


  • 1. Definition: Mathematical or Random Experiments – any procedure or process of obtaining a set of observations which may be repeated under basically the same conditions which lead to well – defined outcomes. Examples: Tossing of a Coin Rolling a die Definition: Sample Space – the set of all possible outcomes in a mathematical or random experiment. Definition: Event – any subset of the sample space.
  • 2. Example: Consider the experiment of rolling a die. Then, the sample space S is the set . Now, define the following events as follows: E 1 = event of getting a prime number E 2 = event of getting an odd number E 3 = event of getting an even number
  • 3. Definition: Let S be a sample space and let A be any event of S. The probability of A denoted by P(A) is a real number satisfying the following axioms: i) ii) P(S) = 1 iii) If A and B are mutually exclusive, then . From the preceding axiom, the following results are immediate: i) If is a null event, then . ii) If is the complement of an event A , then . iii) If A and B are any events, then .
  • 4.
    • Probabilities can be approached in the following ways:
    • Classical Probability
    • Empirical Probability
    • c) Subjective Probability – uses a probability value based on an educated guess or estimate employing opinions and in exact information.
  • 5.
    • Example: If there are 50 tickets sold for a raffle and one person buys 7 tickets, what is the probability of that person winning the price.
    • Example: In an office there are seven women and nine men. If one person is promoted, find the probability that the person is a man.
    • Note:
  • 6. The Normal Distribution
    • - the central idea in the study of parametric statistical inference.
    • - the basis and assumption of common parametric tests;
    • namely, the t – test or z – test.
    • Definition: The normal distribution is defined by the density function:
    • where and are respectively the population mean and the population standard deviation of the distribution, and .
  • 7.
    • Remark: For a given value of and , the density function generates a graph, called the normal curve.
    • Illustration:
  • 8.
    • Properties of the Normal Curve
    • 1. The curve is bell – shaped and symmetric about a vertical axis through .
    • 2. The total area under the curve is 1.
    • 3.
    • 4. The curve extends from the mean in both directions towards
    • infinity.
    • 5. The curve has a single peak.
    • 6. The mean lies at the center of the distribution.
  • 9.
    • Various Normal Curves for Different Mean and Standard Deviations
    • I.
    • II. but
    • III.
  • 10.
    • Remark: The mean and the standard deviation vary in values and hence several normal curves can be possibly drawn. Thus, a standard normal distribution with a fixed mean and a fixed standard deviation must be established.
    • The following transformation transformed the
    • X – score into a Z – score. Z is called the standard normal random variable with mean and .
    • Remark: A Z – score tells us the number of standard deviations a score lies above or below its mean.
  • 11. Applications
    • Example: A graduate student got a score of 58 in professional course and 49 research. In his major course, the mean score was 55 with a standard deviation of 6. On the other hand, the mean score of his research course was 45 with a standard deviation of 4. In which of the two courses did he perform better?
  • 12.
    • Definition : If then the random variable Z will fall between the corresponding values
    • and . Thus,
    • .
    • Example : Using Excel, find the area under the standard normal curve corresponding to .
    • 1. P (Z < 1.56) 3 . P ( Z > 2.46)
    • 2. P (-1.32 < Z < 1.54)
  • 13. Applications
    • 1. If the scores for the test have a mean of 100 and a standard deviation of 15, find the percentage of scores that will fall below 112.
    • 2. An advertising company plans to market a product to low – income families. A study states that for a particular area, the average income per family is $24,596 and the standard deviation is $6,256. If the company plans to target the bottom 18% of the families based on income, find the cutoff income. Assume the variable is normally distributed.
  • 14.
    • 3. The average waiting time for a drive – in window at a local bank is 9.2 minutes, with a standard deviation of 2.6 minutes. When a customer arrives at the bank, find the probability that the customer will have the following time. Assume that the variable is normally distributed.
    • a. Less than 6 minutes or more than 9 minutes
    • b. Between 5 and 10 minutes.