Continuous Probability Distributions Chapter 6 PDF

Chapter 6 Introduction to Continuous Probability Distributions ALWAYS LEARNING Copyright © 2018 Pearson Education, Ltd. Slide - 1 Learning Outcomes Outcome 1. Convert a normal distribution to a standard normal distribution. Outcome 2. Determine probabilities using the standard norma...

Chapter 6 Introduction to Continuous Probability Distributions ALWAYS LEARNING Copyright © 2018 Pearson Education, Ltd. Slide - 1 Learning Outcomes Outcome 1. Convert a normal distribution to a standard normal distribution. Outcome 2. Determine probabilities using the standard normal distribution. Outcome 3. Calculate values of the random variable associated with specified probabilities from a normal distribution.. ALWAYS LEARNING Copyright © 2018 Pearson Education, Ltd. Slide - 2 6.1 The Normal Probability Distribution Is a bell-shaped distribution with the following properties: 1. It is unimodal; that is, the normal distribution peaks at a single value. 2. It is symmetrical; this means that the two areas under the curve between the mean and any two points equidistant on either side of the mean are identical. 3. The mean, median, and mode are equal. ALWAYS LEARNING Copyright © 2018 Pearson Education, Ltd. Slide - 3 The Normal Distribution Shape ALWAYS LEARNING Copyright © 2018 Pearson Education, Ltd. Slide - 4 Normal Distribution Variations Different normal distributions can be obtained changing μ and σ. Changing μ shifts the distribution left or right. Changing σ increases or decreases the spread. ALWAYS LEARNING Copyright © 2018 Pearson Education, Ltd. Slide - 5 Approximate Areas under the Normal Distribution ALWAYS LEARNING Copyright © 2018 Pearson Education, Ltd. Slide - 6 Finding Normal Probabilities Because x is a continuous random variable, the probability, P(x)≈0 for any particular x The probability for a range of values between x1 and x2 is defined by the area under the curve between these two values. P ( x1  x  x 2 ) f(x) x1 x2 x ALWAYS LEARNING Copyright © 2018 Pearson Education, Ltd. Slide - 7 The Standard Normal Distribution A normal distribution that has a mean = 0.0 and a standard deviation = 1.0 The horizontal axis is scaled in z-values that measure the number of standard deviations a point is from the mean Values above the mean have positive z-values Values below the mean have negative z-values ALWAYS LEARNING Copyright © 2018 Pearson Education, Ltd. Slide - 8 The Standard Normal Distribution Standardized Normal z-Value ALWAYS LEARNING Copyright © 2018 Pearson Education, Ltd. Slide - 9 The Standard Normal Distribution Any normal distribution (with any mean and standard deviation combination) can be scaled into the standard normal distribution (z) Any specified value, x, from the population distribution can be converted into a corresponding z-value ALWAYS LEARNING Copyright © 2018 Pearson Education, Ltd. Slide - 10 Standard Normal Distribution - Example The gas mileage for a particular new SUV automobile is normally distributed with µ = 22 mpg and σ = 4. Find the probability of one SUV getting between 23 and 27 mpg?   22  4 Probability of Interest P(23  x  27)  ? 23 27 Need to convert to the standard normal distribution ALWAYS LEARNING Copyright © 2018 Pearson Education, Ltd. Slide - 11 Standard Normal Distribution – Gas Mileage Example   22 x 23  22 z   0.25  4  4 x   27  22 P(23  x  27)  ? z   1.25  4 Probability of Interest z 0.25 1.25 ALWAYS LEARNING Copyright © 2018 Pearson Education, Ltd. Slide - 12 Using The Standard Normal Distribution Table The column gives the value of z to the second decimal point z 0.00 0.01 0.02 … 0.1 The row shows the value of z to the first 0.2 decimal point The value in the 2.0.4772 intersection gives the probability from z = 0 up to the desired z value P(0 < z < 2.00)2.0 = 0.4772 ALWAYS LEARNING Copyright © 2018 Pearson Education, Ltd. Slide - 13 Standard Normal Distribution – Gas Mileage Example P (0 ≤ Z ≤ 0.25) = 0.0987 P (0 ≤ Z ≤ 1.25) = 0.3944 P (0.25 ≤ Z ≤ 1.25) = 0.3944-0.0987 = 0.2957 ALWAYS LEARNING Copyright © 2018 Pearson Education, Ltd. Slide - 14 Standard Normal Distribution – Gas Mileage Example 0.2957 z 0.25 1.25 0.2957   22  4 P (23  x  27)  0.2977 0.2957 23 27 = 0.2957 ALWAYS LEARNING Copyright © 2018 Pearson Education, Ltd. Slide - 15 Standard Normal Distribution - Restaurant Tip Example The tips left by customers at a local restaurant are normally distributed with a mean equal to $12.00 and a standard deviation equal to $3.00. What is the probability that a customer will leave a tip of less than $8.00?   12  3 Probability P ( x  8)  ? of Interest 8 Need to convert to the standard normal distribution ALWAYS LEARNING Copyright © 2018 Pearson Education, Ltd. Slide - 16 Standard Normal Distribution - Restaurant Tip Example   12 x 8  12  3 z   1.33  3 P( x  8)  ? Probability of Interest z -1.33 ALWAYS LEARNING Copyright © 2018 Pearson Education, Ltd. Slide - 17  Probabilities associated with negative z values equal the probabilities for the corresponding positive z values. P( z  1.33)  0.5000  0.4082 P( z  1.33)  0.0918 P( x  $8)  0.0918 0.4082 0.0918 z -1.33 ALWAYS LEARNING Copyright © 2018 Pearson Education, Ltd. Slide - 18