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:

E1 = event of getting a prime number

E2 = event of getting an odd number

E3 = event of getting an even number

6 ,5 ,4 ,3 ,2 ,1S

5 ,3 ,21 E

5 ,3 ,12 E

6 ,4 ,23 E

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 .

10 AP

BPAPBAP

0PcA APAP c 1

BAPBPAPBAP

Probabilities can be approached in the following ways:

a)Classical Probability

b)Empirical Probability

c) Subjective Probability – uses a probability value based on an educated guess or estimate employing opinions and in exact information.

Sin points sample ofnumber

Ain points sample ofnumber AP

conducted wasexperiment the timesof no.

occuredA event timesof no.AP

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: )!(!

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rnr

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rn

nn

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 .

2

2

1

2

1

X

eXf

14159.3

718.2e

Remark: For a given value of and , the density function generates a graph, called the normal curve.

Illustration:

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.

MoMdX

Various Normal Curves for Different Mean and Standard Deviations

I.

II. but

III.

2121 but

21 21

2121 and

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.

X

Z

0 1

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?

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)

21 XXX

11

XZ

2

2

XZ

2121 ZZZPXXXP

zZP

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.

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.