Chapter 3 Probability Distribution Normal Distribution

Numerous continuous variables have distribution closely resemble the normal distribution. The normal distribution can be used to approximate various discrete probability distribution.

CHARACTERISTICS OF NORMAL DISTRIBUTION ‘Bell Shaped’ Symmetrical Mean, Median and Mode are Equal Location is determined by the mean, μ Spread is determined by the standard deviation, σ The random variable has an infinite theoretical range: +  to   Mean = Median = Mode X f(X) μ σ

By varying the parameters μ and σ, we obtain different normal distributions Many Normal Distributions

The Standard Normal Distribution  Any normal distribution (with any mean and standard deviation combination) can be transformed into the standard normal distribution (Z)  Need to transform X units into Z units using  The standardized normal distribution (Z) has a mean of, and a standard deviation of 1,  Z is denoted by  Thus, its density function becomes

Calculating Probabilities for a General Normal Random Variable Mostly, the probabilities involved x, a normal random variable with mean, μ and standard deviation, σ Then, you have to standardized the interval of interest, writing it in terms of z, the standard normal random variable. Once this is done, the probability of interest is the area that you find using the standard normal probability distribution. Normal probability distribution, X~N ( μ, σ 2 ) Need to transform x to z using

Patterns for Finding Areas under the Standard Normal Curve

Example

a)Find the area under the standard normal curve of a)Find the area under the standard normal curve of

Exercise 3.1 Determine the probability or area for the portions of the Normal distribution described. Answer : a) , b) , c) , d) , e)

Example 3.2 Upper tail probabilities or areas of the distribution of Z have also been tabulated. An entry in the table specifies a value of Z α such that an area lies to its right. In other word, P(Z>Z α )

Exercise 3.2

Answers:

Suppose X is a normal distribution N(25,25). Find Example 3.3 Any normal distribution can be transformed into the standard normal distribution (Z)

Solution

Example

Exercise 3.3: 1.A normal random variable x has mean, 10, and standard deviation, 2. Find the probabilities below: (a) (b) (c) 2.Hupper Corporation produces many types of soft drinks including Orange Cola. It has been observed that the net amount of soda in such a can has a normal distribution with a mean of 12 ounces and a standard deviation of ounce. What Is the probability that a randomly selected can of Orange Cola contains between and ounces of soda? 3.The random variable X is normally distributed. Given μ= 54 and P ( X > 80 ) = Find the value of σ.

Normal Approximation of the Binomial Distribution  When the number of observations or trials n in a binomial experiment is relatively large, the normal probability distribution can be used to approximate binomial probabilities. A convenient rule is that such approximation is acceptable when

Continuous Correction Factor The continuous correction factor needs to be made when a continuous curve is being used to approximate discrete probability distributions. 0.5 is added or subtracted as a continuous correction factor according to the form of the probability statement as follows:

How do calculate Binomial Probabilities Using the Normal Approximation? Find the necessary values of n and p. Calculate μ = np and Write the probability you need in terms of X. Correct the value of x with appropriate continuous correction factor (ccf). Convert the necessary x-values to z-values using Use Standard Normal Table to calculate the approximate probability.

Example 3.5 In a certain country, 45% of registered voters are male. If 300 registered voters from that country are selected at random, find the probability that at least 155 are males. Solutions:

Exercise 3.5 Suppose that 5% of the population over 70 years old has disease A. Suppose a random sample of 9600 people over 70 is taken. What is the probability that less than 500 of them have disease A? Answer:

Normal Approximation of the Poisson Distribution  When the mean of a Poisson distribution is relatively large, the normal probability distribution can be used to approximate Poisson probabilities. A convenient rule is that such approximation is acceptable when

Example 3.6 A grocery store has an ATM machine inside. An average of 5 customers per hour comes to use the machine. What is the probability that more than 30 customers come to use the machine between 8.00 am and 5.00 pm? Solution:

Exercise 3.6 The average number of accidental drowning in United States per year is 3.0 per population. Find the probability that in a city of population there will be less than 10 accidental drowning per year. Answer :