5 Random Variables Lesson 5 6 The Standard

5 Random Variables Lesson 5. 6 The Standard Normal Distribution Statistics and Probability with Applications, 3 rd Edition Starnes & Tabor Bedford Freeman Worth Publishers

The Standard Normal Distribution Learning Targets After this lesson, you should be able to: ü Use the 68– 95– 99. 7 rule to find approximate probabilities in a normal distribution. ü Use Table A to find a probability (area) from a z-score in the standard normal distribution. ü Use Table A to find a z-score from a probability (area) in the standard normal distribution. Statistics and Probability with Applications, 3 rd Edition 2

The Standard Normal Distribution Why are the Normal distributions important in statistics? • Normal distributions are good descriptions for some distributions of real data. • Normal distributions are good approximations of the results of many kinds of chance outcomes. • Many statistical inference procedures are based on Normal distributions. Statistics and Probability with Applications, 3 rd Edition 3

The Standard Normal Distribution Although there are many Normal curves, they all have properties in common. The 68 -95 -99. 7 Rule In the Normal distribution with mean µ and standard deviation σ: • Approximately 68% of the observations fall within σ of µ. • Approximately 95% of the observations fall within 2σ of µ. • Approximately 99. 7% of the observations fall within 3σ of µ. Statistics and Probability with Applications, 3 rd Edition 4

The Standard Normal Distribution All Normal distributions are the same if we measure in units of size σ from the mean µ as center. Standard Normal Distribution The standard Normal distribution is the Normal distribution with mean 0 and standard deviation 1. If a variable x has any Normal distribution with mean µ and standard deviation σ, then the standardized variable has the standard Normal distribution with µ = 0 and σ = 1. Statistics and Probability with Applications, 3 rd Edition 5

The Standard Normal Distribution The standard Normal Table (Table A) is a table of areas under the standard Normal curve. The table entry for each value z is the area under the curve to the left of z. Suppose we want to find the proportion of observations from the standard Normal distribution that are less than 0. 81. We can use Table A: Z . 00 . 01 . 02 0. 7 . 7580 . 7611 . 7642 0. 8 . 7881 . 7910 . 7939 0. 9 . 8159 . 8186 . 8212 Statistics and Probability with Applications, 3 rd Edition P(z < 0. 81) =. 7910 6

The Standard Normal Distribution Let’s find the 90 th percentile of the standard normal distribution. We’re looking for the z-score that has 90% of the area to its left Z . 07 . 08 . 09 1. 1 . 8790 . 8810 . 8830 1. 2 . 8980 . 8997 . 9015 1. 3 . 9147 . 9162 . 9177 Statistics and Probability with Applications, 3 rd Edition z = 1. 28 7

LESSON APP 5. 6 What’s a good batting average? 1. In baseball, a player’s batting average is the proportion of times that the player gets a hit out of his total number of times at bat. Suppose we select a Major League Baseball player at random. The random variable X = the player’s batting average can be modeled by a normal distribution with mean µ = 0. 261 and standard deviation σ = 0. 034. Use the 68– 95– 99. 7 rule to approximate: (a) The probability that a randomly selected player has a batting average greater than 0. 329 (b) P(0. 193 ≤ X ≤ 0. 295) 2. Suppose we convert the randomly selected player’s batting average to a z-score. Use Table A to find each of the following. Draw a standard normal distribution with the desired area shaded in each case. a) b) What’s the probability that the z-score is between 20. 58 and 1. 79? 45% of batting averages will have a z-score greater than what value? Statistics and Probability with Applications, 3 rd Edition 8

The Standard Normal Distribution Learning Targets After this lesson, you should be able to: ü Use the 68– 95– 99. 7 rule to find approximate probabilities in a normal distribution. ü Use Table A to find a probability (area) from a z-score in the standard normal distribution. ü Use Table A to find a z-score from a probability (area) in the standard normal distribution. Statistics and Probability with Applications, 3 rd Edition 9