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.
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
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 .
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.
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?
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.
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.