1. STATISTICS “The Normal Probability Distribution” 11.0 The Normal Probability Distribution

2. Characteristics of a Normal Probability Distribution –A continuous random variables and a continuous probability distribution is known as the normal distribution –A Normal Process has the following characteristics: 1)The mean, median and mode are the same value 2)The distribution is bell shaped and symmetrical around the mean. 3)The total area under the curve is equal to 1 4)The left and right tails of the normal probability distribution extend indefinitely, never quite touching the horizontal axis

3. Characteristics of a Normal Probability Distribution –A smaller standard deviation results in a “skinner” curve that’s tighter and taller around the mean. –A larger standard deviation makes for a “fatter” curve that’s more spread out and not as tall. –The value of mean and standard deviation, completely describe the shape of the distribution. 11.0 The Normal Probability Distribution

4. The Normal Probability Distribution –To make Normal Probability Distribution, we need to define the standard normal distribution, which is a normal distribution with a μ=0 and σ=1. –This standard normal distribution is the basis for all normal probability calculation: z = x - μ σ z = the number of difference between x and μ, known as the standard z-score 11.0 The Normal Probability Distribution

5. The Normal Probability Distribution –Once obtain the z-score, use the Standard Normal Table to determine the probability. –In general, you can use the following two relationships for any value a when dealing with negative z-score: P[z >-a] = P[z ≤+a] P[z≤-a] = 1 – P[z ≤+a] 11.0 The Normal Probability Distribution

6. TRY THIS!!! 1) The amount of toxic spray use to kill Japanese beetle used each year follows a normal distribution with a mean of 60 liter and a standard deviation of 5 liter. What is the probability: a) Less than 64.3 liter (Answer: 0.8051) b) More than 62.5 liter (Answer: 0.3085) c) More than 54 liter (Answer: 0.8849) 2)The speed of cars passing through a checkpoint follows a normal distribution with μ = 62.6 m/h and σ = 3.7 m/h What is the probability of the next car passing will: a) Be exceeding 65.5 m/h (Answer: 0.2177) b) Be exceeding 58.1 m/h (Answer: 0.888) c) Be between 61 and 70 m/h (Answer: 0.6436) 11.0 The Normal Probability Distribution

7. Exercises The lengths of steel beams made by a particular steel mill is normally. Distributed with a mean of 10.35 metres and a standard deviation of 2.25 metres. a)Find the probability that the length of a steel beam will be over 10.86 metres. b)Find the probability that the length of a steel beam will be over 11.40 metres. c)Find the probability that the length of a steel beam will be over 9.41 metres. d)For a particular application, any beam less than 9.05 metres must be scrapped. What percentage of beams would expect to be scrapped? e)Find the probability that the length of a steel beam will less than 10.96 metres. f)Find the probability that the length of a steel beam will be between 10.1 and 11.10 metres. g)Find the probability that the length of a steel beam will be between 10.86 and 11.05 metres. h)Find the probability that the length of a steel beam will be between 9.01 and 10.08 metres.