STATISTICS-I

COURSE INSTRUCTOR: TEHSEEN IMRAAN

CHAPTER 7

CONTINUOUS PROBABILITY DISTRIBUTION

INTRODUCTION

• This chapter continues our study of probability distributions by examining the continuous probability distribution. Recall that a continuous probability distribution can assume an infinite number of values within a given range. As an example, the weights for a sample of small engine blocks are: 54.3, 52.7, 53.1 and 53.9 pounds.

INTRODUCTION

• We consider two families of continuous probability distributions, the uniform probability distribution and the normal probability distribution. Both of these distributions describe the likelihood of a continuous random variable that has an infinite number of possible values within a specified range.

INTRODUCTION

• An example of a uniform probability distribution is the flight time between NY and Chicago. Suppose the time to fly from NY to Chicago is uniformly distributed within a range of 55 minutes to 75 minutes. We can determine the probability that we can fly from NY to Chicago in less than 60 minutes. Flight time is measured on a continuous scale.

INTRODUCTION

• The normal probability distribution is described by its mean and standard deviation. Suppose the life of an automobile battery follows the normal distribution with a mean of 36 months and a standard deviation of 3 months. We can determine the probability that a battery will last between 36 and forty months. Life of a battery is measured on a continuous scale.

Characteristics of the Uniform Probability Distribution

• A continuous probability distribution with its values spread evenly over a range of values that are rectangular in shape and are defined by minimum and maximum values.

• A uniform distribution is shown in Chart 7-1. The distribution’s shape is rectangular and has a minimum value of "a" and a maximum value of "b". The height of the distribution is uniform for all values between "a " and "b". This implies that all the values in the range are equally likely.

CHART 7-1

Mean of a Uniform Distribution

Mean of a Uniform Distribution

• The mean of a uniform distribution is located in the middle of the interval between the minimum value of "a " and a maximum value of "b".

• For example: Suppose that the time to fly from NY to Chicago is uniformly distributed within a range of 55 minutes minimum to 75 minutes maximum. The mean is found by using Formula [7–1]:

Mean of a Uniform Distribution

Standard Deviation of a Uniform Distribution

For the flight time from NY to Chicago example the standard deviation is calculated using Formula [7–2]

Another key element of the uniform distribution is the height, P(x). The height is the same for all values of the random variable "x ". It is calculated using Formula[7–3]:

• In Chapter 6, we discussed the fact that probability distributions are useful when making probability statements concerning the values of a random variable. Also for continuous random variables, areas within the distribution represent probabilities. Recall that:

FOR ALL VALUES OF x

The relationship between area and probabilities is applied to the uniform distribution and its rectangular shape using the area of a rectangle formula. Recall that:

Thus for any uniform distribution, the area under the curve is always 1.For the flight time from NY to Chicago example the area is:

Normal probability distribution

• A continuous probability distribution uniquely determined by m and s.

• The major characteristics of the normal distribution are:– The normal distribution is "bell-shaped" and the

mean, median, and mode are all equal and are located in the center of the distribution. Exactly one-half of the area under the normal curve is above the center and one-half of the area is below the center.

Normal probability distribution– The distribution is symmetrical about the mean. A

vertical line drawn at the mean divides the distribution into two equal halves and these halves possess exactly the same shape.

– It is asymptotic. That is, the tails of the curve approach the X-axis but never actually touch it.

– A normal distribution is completely described by its mean and standard deviation. This indicates that if the mean and standard deviation are known, a normal distribution can be constructed and its curve drawn.

Normal probability distribution

– There is a "family" of normal probability distributions. This means there is a different normal distribution for each combination of m and s.

These characteristics are summarized in the graph.

The Standard Normal Probability Distribution

• Standard normal distribution: A normal distribution with a mean of 0 and a standard deviation of 1.

• z value: The signed distance between a selected value designated X, and the population mean, m , divided by the population standard deviation, s .

The formula for a specific standardized z value is text formula [7–5]:

Where:

X is the value of any particular observation or measurement.m is the mean of the distribution.s is the standard deviation of the distribution.z is the standardized normal value, usually called the z value.

Applications of the Standard Normal Distribution

• To obtain the probability of a value falling in the interval between the variable of interest (X) and the mean (m ), we first compute the distance between the value (X) and the mean (m ). Then we express that difference in units of the standard deviation by dividing (X - m ) by the standard deviation. This process is called standardizing.

• To illustrate the probability of a value being between a selected X value and the mean m , suppose the mean useful life of a car battery is 36 months, with a standard deviation of 3 months. What is the probability that such a battery will last between 36 and 40 months?

The first step is to convert the 40 months to an equivalent standard normal value, using formula [7–5]. The computation is:

a table for the areas under the normal curve

z 0.00 0.01 0.02 0.03 0.04 0.05    ! ! ! !      ! ! ! !  

    ! ! ! !  1.0            

1.1   0.3665 0.3686 0.3708 0.3729  

1.2   0.3869 0.3888 0.3907 0.3925  1.3   0.4049 0.4066 0.4082 0.4099  1.4   0.4207 0.4222 0.4236 0.4251  

• To use the table, the z value of 1.33 is split into two parts, 1.3 and 0.03. To obtain the probability go down the left-hand column to 1.3, then move over to the column headed 0.03 and read the probability. It is 0.4082.

• The probability that a battery will last between 36 and 40 months is 0.4082. Other probabilities may be calculated, such as more than 46 months, and less than 33 months.

Empirical Rule

• About 68 percent of the area under the normal curve is within plus one and minus one standard deviation of the mean. This can be written as m ± 1s.

• About 95 percent of the area under the normal curve is within plus and minus two standard deviations of the mean, written m ± 2s.

• Practically all of the area under the normal curve is within three standard deviations of the mean, written m ± 3s.

Empirical Rule

The Normal Approximation to the Binomial

• A binomial probability can be estimated using the normal distribution.

• To apply the normal approximation to the binomial, both np and nq must be greater than 5. The sample size, or the number of trials, is designated by n, and p is the probability of a success. The mean and the standard deviation of the binomial are computed by:

Example

• To illustrate, suppose 60 percent of the applications for an exclusive credit card are approved. In a sample of 200 applications, what is the probability that 130 or more applications are approved?

• First verify that both n p and nq exceed 5. For n = 200 and p = 0.6

Thus the normal approximation to the binomial may be used.

The mean and standard deviation are computed as follows:

This distribution is standardized by formula [7–5] and m = 120, s = 6.93 and letting X

=129.5 (Not 130)

• Why is 129.5 used instead of 130? It is used to "correct" for the fact that a continuous distribution (the normal) is used to approximate a discrete distribution (the binomial). On a continuous scale the value 130 would range from 129.5 to 130.5. On a discrete scale there would be a "gap" between 129 and 130 where there would not be any probability. The 0.50 is called the correction for continuity.

Continuity correction factor

• The value 0.5 subtracted or added, depending on the question, to a selected value when a discrete probability distribution is approximated by a continuous probability distribution

• The probability of a z value between 0 and 1.37 is 0.4147 (See Appendix D). Therefore the probability of a z value greater than 1.37 is 0.0853, found by 0.5000 - 0.4147. So, the probability that 130 or more applications will be approved is 0.0853.