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# Probability concepts and the normal distribution

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### Probability concepts and the normal distribution

1. 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. 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. 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. 4. <ul><li>Probabilities can be approached in the following ways: </li></ul><ul><li>Classical Probability </li></ul><ul><li>Empirical Probability </li></ul><ul><li>c) Subjective Probability – uses a probability value based on an educated guess or estimate employing opinions and in exact information. </li></ul>
5. 5. <ul><li>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. </li></ul><ul><li>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. </li></ul><ul><li>Note: </li></ul>
6. 6. The Normal Distribution <ul><li>- the central idea in the study of parametric statistical inference. </li></ul><ul><li>- the basis and assumption of common parametric tests; </li></ul><ul><li>namely, the t – test or z – test. </li></ul><ul><li>Definition: The normal distribution is defined by the density function: </li></ul><ul><li>where and are respectively the population mean and the population standard deviation of the distribution, and . </li></ul>
7. 7. <ul><li>Remark: For a given value of and , the density function generates a graph, called the normal curve. </li></ul><ul><li>Illustration: </li></ul>
8. 8. <ul><li>Properties of the Normal Curve </li></ul><ul><li>1. The curve is bell – shaped and symmetric about a vertical axis through . </li></ul><ul><li>2. The total area under the curve is 1. </li></ul><ul><li>3. </li></ul><ul><li>4. The curve extends from the mean in both directions towards </li></ul><ul><li>infinity. </li></ul><ul><li>5. The curve has a single peak. </li></ul><ul><li>6. The mean lies at the center of the distribution. </li></ul>
9. 9. <ul><li>Various Normal Curves for Different Mean and Standard Deviations </li></ul><ul><li>I. </li></ul><ul><li>II. but </li></ul><ul><li>III. </li></ul>
10. 10. <ul><li>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. </li></ul><ul><li>The following transformation transformed the </li></ul><ul><li>X – score into a Z – score. Z is called the standard normal random variable with mean and . </li></ul><ul><li>Remark: A Z – score tells us the number of standard deviations a score lies above or below its mean. </li></ul>
11. 11. Applications <ul><li>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? </li></ul>
12. 12. <ul><li>Definition : If then the random variable Z will fall between the corresponding values </li></ul><ul><li>and . Thus, </li></ul><ul><li>. </li></ul><ul><li>Example : Using Excel, find the area under the standard normal curve corresponding to . </li></ul><ul><li>1. P (Z < 1.56) 3 . P ( Z > 2.46) </li></ul><ul><li>2. P (-1.32 < Z < 1.54) </li></ul><ul><li> </li></ul>
13. 13. Applications <ul><li>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. </li></ul><ul><li>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. </li></ul>
14. 14. <ul><li>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. </li></ul><ul><li>a. Less than 6 minutes or more than 9 minutes </li></ul><ul><li>b. Between 5 and 10 minutes. </li></ul>