1 7.5 CONTINUOUS RANDOM VARIABLES Continuous data occur when the variable of interest can take on anyone of an infinite number of values over some interval on the real number line. Much of the time, continuous data result from taking data on a measurement. Examples of this would be height of students or grade point averages (GPA).

2 Figure 7.5 looks at the binomial distribution for  = 0.50 and n= 10, 25, 50, and CONTINUOUS RANDOM VARIABLES

3 For continuous random variables, which can take on many more than 100 possible values, the probability distribution is called a probability density function, f(x), and is represented by a smooth curve such as the one in Figure CONTINUOUS RANDOM VARIABLES

4 A probability density function, f(x), is a smooth curve that represents probability distribution of a continuous random variable. We can talk about the probability that the random variable will take on anyone of the values over an interval of interest; that is, we can find P(x 1 <X<x 2 ). This probability is represented by the area under the probability density curve as shown in Figure CONTINUOUS RANDOM VARIABLES

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6 7.6 THE NORMAL DISTRIBUTION The Normal Curve The normal probability distribution is the symmetric, bell-shaped curve shown in Figure 7.8. It is usually referred to as the normal curve.

7 The actual formula for the probability density is Where e = and  = If you look at the equation you can see that the probability density for a value, x, relies on two parameters,  and . 7.6 THE NORMAL DISTRIBUTION

8 For a normal random variable, the parameter  is the mean of the normal random variable, X, and  is the standard deviation. When we refer to a normally distributed random variable we often use a special notation. 7.6 THE NORMAL DISTRIBUTION

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The Standard Normal Curve The standard normal random variable is known as Z and has a mean of 0 and a standard deviation of 1. We can use the shorthand notation to write that as 7.6 THE NORMAL DISTRIBUTION

11 Once we transform the problem we can use a single normal probability table, a standard normal table, to solve it. A Z random variable is normally distributed with a mean of 0 and a standard deviation of 1,. 7.6 THE NORMAL DISTRIBUTION

12 A standard normal table is a table of probabilities for a Z random variable. The transform from X to Z is then given by Where and. 7.6 THE NORMAL DISTRIBUTION

The Standard Normal Tables An interval probability gives the probability that a random variable will take a value between two given values P(xl  X  x2). There are several different types of standard normal tables that can be used. They each give probabilities for different types of intervals. 7.6 THE NORMAL DISTRIBUTION

14 The tables that we use in our textbook give P(Z<z); that is, the entries in the table are the probability that the random variable Z will take on a value that is less than some value z. 7.6 THE NORMAL DISTRIBUTION

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The Standard Normal Tables You now know how to use the table to find P(Z z) or P(Z1<Z<Z2)? The first problem is not difficult to figure out. Since the tables give P(Z z) by using the definition of the complement of an event. That is, P(Z>z)= 1 - P(Z<z) 7.6 THE NORMAL DISTRIBUTION

17 Since the normal probability distribution is symmetric about the mean, it must be true that the area to the right of a Z value must be equal to the area to the left of the negative of that Z value, or P(Z>z) = P(Z<-z) 7.6 THE NORMAL DISTRIBUTION

18 Using the tables to find P(z1<Z<z2) is not difficult either. Figure 7.13 illustrates the process. P(z1<Z<z2)=P(Z<Z2)-P(Z<z1) 7.6 THE NORMAL DISTRIBUTION

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