Chapter 6: Normal Probability Distributions PDF

The Standard Normal Distribution Applications of Normal Distributions Chapter 6: Normal Probability Distributions International Business School BEIJING FOREIGN STUDIES UN...

The Standard Normal Distribution Applications of Normal Distributions Chapter 6: Normal Probability Distributions International Business School BEIJING FOREIGN STUDIES UNIVERSITY 1 / 41 The Standard Normal Distribution Applications of Normal Distributions Continuous Random Variable Suppose 5,000 female students are enrolled at a university, and x is the continuous random variable that represents the heights of these female students. Table 1 lists the frequency and relative frequency distributions of x. Table 1: Table of Frequency and Relative Frequency 2 / 41 The Standard Normal Distribution Applications of Normal Distributions Continuous Random Variable The relative frequencies given in the Table 1 can be used as the probabilities of the respective classes. Figure 1: Histogram and Polygon for Table 1 Figure 1 displays the histogram and polygon for the relative frequency distribution of the Table 1. 3 / 41 The Standard Normal Distribution Applications of Normal Distributions Continuous Random Variable Density Curve A density curve is the graph of a continuous probability distribution. It must satisfy the following properties: 1 The total area under the curve must equal to 1. 2 Every point on the curve must have a vertical height that is 0 or greater. (That is, the curve cannot fall below the x-axis.) The smoothed polygon (refer to Figure 1) is an approximation of the probability distribution curve of the continuous random variable x, which is also called as probability density function. 4 / 41 The Standard Normal Distribution Applications of Normal Distributions The Standard Normal Distribution Key Concept This section presents the standard normal distribution which has three properties: It’s graph is bell-shaped. It’s mean is equal to 0 (µ = 0). It’s standard deviation is equal to 1 (σ = 1). 5 / 41 The Standard Normal Distribution Applications of Normal Distributions Normal Distribution Definition If a continuous random variable has a distribution with a graph that is symmetric and bell-shaped, and it can be described by the equation below, we say it has a normal distribution. 1 x−µ 2 e− 2 ( σ ) f (x) = √ σ 2π 6 / 41 The Standard Normal Distribution Applications of Normal Distributions Normal Distribution Uniform Distribution A continuous random variable has a uniform distribution if its values are spread evenly over the range of probabilities. The graph of a uniform distribution results in a rectangular shape. Area and Probability Because the total area under the density curve is equal to 1, there is a correspondence between area and probability 7 / 41 The Standard Normal Distribution Applications of Normal Distributions Normal Distribution Example Using Area to Find Probability: Given the uniform distribution illustrated, find the probability that a randomly selected voltage level is greater than 124.5 volts. Shaded area represents voltage levels greater than 124.5 volts. Correspondence between area and probability: 0.25. 8 / 41 The Standard Normal Distribution Applications of Normal Distributions Standard Normal Distribution Definition The standard normal distribution is a normal probability distribution with µ = 0 and σ = 1. The total area under its density curve is equal to 1, i.e. Z ∼ N(0, 1) 9 / 41 The Standard Normal Distribution Applications of Normal Distributions Finding Probabilities when given z Scores We can find areas (or probabilities) for many different regions using : Table for Normal (Z ) Distribution1 Formulas and Tables insert card Other methods: STATDISK Minitab Excel TI/83/84 Plus 1 page 48-49 of Mathematical Formulae book 10 / 41 The Standard Normal Distribution Applications of Normal Distributions Methods for Finding Normal Distribution Areas Z -Table, STATDISK, Minitab, Excel Method Description Gives the cumulative area Z -Table The procedure for using Z - Table is described in the next from the left up to a vertical slide. line above a specific value of Minitab Select Calc, Probability Dis- z. tributions, Normal. In the dialog box, select Cumulative Probability, Input Constant. Excel Select fx, Statistical, NOR- MDIST. In the dialog box, en- ter the value and mean, the standard deviation, and “true”. 11 / 41 The Standard Normal Distribution Applications of Normal Distributions Using Table for Normal (Z ) Distribution (Z -Table) 12 / 41 The Standard Normal Distribution Applications of Normal Distributions Using Z -Table Z -Table 1 It is designed only for the standard normal distribution, which has a mean of 0 and a standard deviation of 1. 2 It is on two pages, with one page for negative z scores and the other page for positive z scores. 3 Each value in the body of the table is a cumulative area from the left up to a vertical boundary above a specific z score. 13 / 41 The Standard Normal Distribution Applications of Normal Distributions Using Z -Table Z -Table 4 When working with a graph, avoid confusion between z scores and areas. z Score Distance along horizontal scale of the standard normal distribution; refer to the leftmost column and top row of Z -Table. Area Region under the curve; refer to the values in the body of Z -Table. 5 The part of the z score denoting hundredths is found across the top row. 14 / 41  The Standard Normal Distribution Applications of Normal Distributions Using Z -Table Example A bone mineral density test can be helpful in identifying the presence or likelihood of osteoporosis, a disease causing bones to become more fragile and more likely to break. The result of a bone density test is commonly measured as a z score. The population of z scores is normally distributed with a mean of 0 and a standard deviation of 1. A randomly selected adults undergoes a bone density test. Find the probability that the result is a reading less than 1.27. P(z < 1.27) = Excel formula: =NORM.S.DIST(1.27,TRUE) 15 / 41 The Standard Normal Distribution Applications of Normal Distributions Using Z -Table From the Z -Table (continue from the previous example), 16 / 41 The Standard Normal Distribution Applications of Normal Distributions Using Z -Table (continue from the previous example) Interpretation 1: The probability that a randomly selecting person has a bone density test result below 1.27 is 0.8980. Interpretation 2: 89.80% of people have bone density levels below 1.27. P(z < 1.27) = 0.8980 17 / 41 The Standard Normal Distribution Applications of Normal Distributions Using Z -Table Example If thermometers have an average reading of 0 degrees and a standard deviation of 1 degree for freezing water, and if one thermometer is randomly selected, find the probability that it reads (at the freezing point of water) above -1.23 degrees. P(z > −1.23) =? 18 / 41 The Standard Normal Distribution Applications of Normal Distributions Using Z -Table Solution P(z > −1.23) = 0.8907 Probability of randomly selecting a thermometer with a reading above -1.23° is 0.8907. 89.07% of the thermometers have readings above -1.23 degrees. Excel formula: =1- NORM.S.DIST(-1.23,TRUE) 19 / 41 The Standard Normal Distribution Applications of Normal Distributions Using Z -Table Example A thermometer is randomly selected. Find the probability that it reads (at the freezing point of water) between -2.00 and 1.50 degrees. P(−2.00 < z < 1.50) =? OR P(z < 1.50) − P(z < −2.00) =? 20 / 41 The Standard Normal Distribution Applications of Normal Distributions Using Z -Table Solution P(z < −2.00) = 0.0228 P(z < 1.50) = 0.9332 P(−2.00 < z < 1.50) = 0.9332 − 0.0228 = 0.9104 21 / 41 The Standard Normal Distribution Applications of Normal Distributions Using Z -Table Notation P(a < z < b) ⇒ denotes the probability that the z score is between a and b. P(z > a) ⇒ denotes the probability that the z score is greater than a. P(z < a) ⇒ denotes the probability that the z score is less than a. 22 / 41 The Standard Normal Distribution Applications of Normal Distributions Using Z -Table Example Find the following probabilities for the standard normal curve: 1. P(z < 1.56) 2. P(z > 1.56) 3. P(z < −1.56) 4. P(z > −1.56) 5. P(z > −0.85) 6. P(−0.85 < z < 1.56) 7. P(−1.85 < z < 0.85) 8. P(1.85 < z < 2.85) 9. P(z < −1.65 or z > 1.65) Excel formula: =NORM.S.DIST(1.56,TRUE) = 0.9406 23 / 41 The Standard Normal Distribution Applications of Normal Distributions Using Z -Table Solution 1. P(z < 1.56) = 0.9406 2. P(z > 1.56) = 0.0594 3. P(z −1.56) = 0.9406 5. P(z >−0.85) = 0.8023 6. P(−0.85