Section 6-2 I. Continuous Probability Distributions A continuous random variable is one that can theoretically take on any value on some line interval. We use f ( x) to represent a probability density function. Unfortunately, f ( x ) does not give us the probability that the value x will be observed. To understand how a probability density function for a continuous random variable enables us to find probabilities, it is important to understand the relationship between probability and area. For the following given histogram, what is the probability that x is in between 2.5 to 5.5? Frequency Histogram Relative Frequency Histogram 5 20 4 Frequency 3 2 Percent 10 1 0 0 1 2 3 4 5 6 7 8 C1 0 0 1 2 3 4 5 6 7 8 C1 Use the given frequency histogram to calculate P(2.5 < x < 5.5): Use the corresponding relative frequency histogram to calculate P(2.5 < x < 5.5): For a continuous probability distribution, 1) 2) 3) Note: P(x = a) = 0 for continuous random variables. This implies P(a x b) = P(a < x < b); P(x a) = P(x > a); and P(x a) = P(x < a).
Section 6-3 I. The Normal Distribution Continuous probability distributions can assume a variety of shapes. However, the most important distribution of continuous random variables in statistics is the normal distribution that is approximately mound-shaped. Many naturally occurring random variables such as IQs, height of humans, weights, times, etc. have nearly normal distributions. The formula that generates this distribution is shown below. 1 2 2 ( x μ) /(2 σ ) f( x) = e x σ 2π μ 1 The mean μ is located at the of the distribution. The distribution is about its mean μ. There is a correspondence between and. Normal probabilities can be obtained by the use of a. Since the total under the normal probability distribution is equal to, the symmetry implies that the area to the right of μ is 0.5 and the area to the left of μ is also 0.5. values of σ the height of the curve and the spread. values of σ the height of the curve and the spread. values of a normal random variable lie in the interval μ ± 3σ. II. The Standard Normal Distribution The standard normal distribution is a normal probability distribution that has a mean of and a standard deviation of. The symbol is used to represent a standard normal random variable.
Probabilities associated with the standard normal variable can be found by the use of Appendix Table E. Each value in the body of the table is a cumulative from the center to a specific positive value of z. Example 1: You must include a sketch in your solution. (a) Find P(0 < z < 1.63) (b) Find P(-2.48 < z < 0) (c) Find P(-2.02 < z < 1.74) (d) Find P(1.02 < z < 1.84) (e) Find the probability that z is between -.58 and -.10. (f) Find the probability that z is larger than 1.76. (g) Find the probability that z is less than 2.04. (h) Find the probability that z is within two standard deviations of the mean.
Example 2: Assume the standard normal distribution. Fill in the blanks. (a) P( 0 < z < ) =.4279 (b) P( 0 < z < ) =.4997 (c) P( < z < 0 ) =.4370 (d) P( z < ) =.9846 (e) P( z < ) =.1190 (f) Find the z value to the right of the mean so that 71.90% of the area under the distribution curve lies to the left of it. (g) Find two z values, one positive and one negative, so that the areas in the two tails total to 12%.
Section 6 4 I. Calculating Probabilities for a Non-standard Normal Distribution Consider a normal variable x with mean μ and standard deviation σ. 1. Standardize from x to z x μ z = σ 2. Use Table E to find the central area corresponding to z 3. Adjust the area to answer the question Example 1: Let x be a normal random variable with mean 80 and standard deviation 12. What percentage of values are (a) larger than 56? (b) less than 62? (c) between 85 and 98 (d) outside of 1.5 standard deviations of the mean?
Example 2: (Ref: General Statistics by Chase/Bown, 4 th Ed.) The length of times it takes for a ferry to reach a summer resort from the mainland is approximately normally distributed with mean 2 hours and standard deviation of 12 minutes. Over many past trips, what proportion of times has the ferry reached the island in (a) less than 1 hour, 45 minutes? (b) more than 2 hours, 5 minutes? (c) between 1 hour, 50 minutes and 2 hours, 20 minutes? II. Calculating a Cutoff Value Backward steps for calculating probabilities of a non-standard normal distribution. 1. Adjust to the corresponding central area. 2. Use Table E to find the corresponding z cutoff value. 3. Non-standardize from z to x x μ z = σ
Example 1: Employees of a company are given a test that is distributed normally with mean 100 and variance 25. The top 5% will be awarded top positions with the company. What score is necessary to get one of the top positions? Example 2: Quiz scores were normally distributed with μ = 14 and σ = 2.8, the lower 20% should receive tutorial service. Find the cutoff score.