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# Probability concepts and the normal distribution

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• 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.