Continuous Random Variables and the Normal Distribution Last time: We looked at discrete random variables. We looked at what a discrete probability distribution looks like. (a table or a bar graph) We looked at how to find probabilities for a discrete random variable. Today we will look at: What does a probability distribution for a continuous random variable X look like? How do you find the probabilities of particular values of a continuous random variable X? Since continuous random variable X can assume INFINITELY many values, the probability that X is equal to any ONE particular value, a, will ALWAYS equal to 0. P(X = a) = (1/infinity) = 0 Example: X = height of a random person The probability that a randomly selected person will be exactly 5.00 feet tall is 0%. P(X=5ft) = 0 For continuous random variables X, we are only interested in whether X is Greater than a particular value a, P (X > a) Less than a particular value a, P (X < a) Is in between two values a and b, P ( a < X < b) Ex 11.1: Let us say we are interested in the weights of high risk newborn babies. Experiment: Choose a random high-risk newborn baby X = the weight of the randomly chosen high-risk newborn baby Continuous Probability Distribution 1/8
A Continuous Probability Distribution (definition) - is a graph that indicates on the horizontal axis the range of values that the continuous random variable X can take, and above which is drawn a curve, called the density curve. Probabilities for Continuous Distributions are represented by the geometric area under the curve above the interval of interest. Example: The probability that a randomly selected infant weights more than three pounds is given by: P(X>3) = area under the curve to the right of X=3 PROBABILITY = AREA For any Continuous Probability Distribution: The total area under the density curve will always be equal to 1.00 Ex 11.2: Suppose that the students at a certain university will be named to the Dean s List if the maintain a grade point average of at least 3.0. Using the probability distribution of student GPAs, shade in the area that represents the probability that X is between 3.0 and 4.0 2/8
Normal Probability Distribution a continuous probability distribution with a symmetric bellshaped curve. The curve is concentrated in the center and decreases on either side. Normal distributions can be used to model many sets of measurements in nature, industry and business. It is therefore considered the most important probability distribution in the world. Examples: Systolic blood pressures of humans Lifetimes of products Amount of rainfall Standardized test scores The mean µ and the standard deviation σ constitute the parameters of the Normal Distribution. Mean µ Can be any number, positive, negative or zero. Determines the center of the distribution on the number line Standard Deviation σ Standard deviation can never be negative Determined the spread or the shape of the distribution curve 3/8
Normal Curve Properties of the Normal Curve 1. It is symmetric about the mean with the highest point occurring at the X = µ 2. The total area under the curve is 1.00. 3. The area under the curve to the left of µ and the area under the curve to the right of µ are both equal to 0.50. 4. The normal distribution is defined for values of X extending indefinitely in both the positive and negative directions. 11.3: a) The two normal distributions below have the same standard deviations of 5 but different means. Which normal distribution has mean 10 and which has mean 15. b) The two normal distributions in the figure below have the same mean of 100 but different standard deviations. Which normal distribution has standard deviation 3 and which has standard deviation 6. 4/8
c) Use the below graph of a normal distribution to determine the mean and the standard deviation. Ex 11.4: The National Center for Education Statistics reports that the scores on a particular standardized mathematics test for eighth-graders are normally distributed with the mean 273 and standard deviation of 7. a) Sketch the graph of the normal curve b) Find the probability that a randomly chosen student receives a score above 273. Empirical Rule: About 68% of the area under the curve lies within one standard deviation of the mean. About 95% of the area under the curve lies within two standard deviations of the mean. About 99.7% of the area under the curve lies within three standard deviations of the mean. Ex 11.5: GPAs at a particular high school are normally distributed with mean µ = 2.6 and standard deviation σ = 0.46. a) Sketch the graph of the distribution of GPAs. 5/8
b) What is the probability that a randomly chosen GPA at the high school will be between 3.06 and 3.52? c) What is the probability that a randomly chosen GPA will be greater than 3.52? Ex 6: A study of Pennsylvania hospitals has found that the mean patient length of stay is 4.87 days with the standard deviation of 0.97 days. Assume that the distribution of patient length of stays is normal. Find the probability that a randomly selected patient stays for more than 3.9 days. The Standard Normal (Z) Distribution (def) is a special case of normal distribution with mean µ = 0 and standard deviation σ = 1 The standard normal random variable is always denoted by a Z. 6/8
The standard normal random variable is always denoted by a Z. Example 11.7 Z is a standard normal variable. Calculate the following probabilities. a) P(Z>0) b) P(Z<0) c) P( -1<Z<1) 7/8