Applied Business Statistics Week 5 Uniform Normal Distributions Next Week Monday 2 13 Discuss Midterm of questions on exam Exam format How to prepare Review session information Today s Agenda Today Uniform Distribution Normal Distribution Empirical Rule Use Excel to introduce NORM DIST NORM INV NORM S DIST Wednesday Sampling Distribution Chebyshev s Theorem Central Limit Theorem Z scores Get more practice in Excel Previous Topics of Emphasis This Chapter Week Mean Standard deviation Probability Discrete Probability 0 3 0 25 0 2 0 15 0 1 0 05 0 Probability Distribution Function PMF 1 6 0 167 1 2 3 4 5 6 Outcome PMF f x P X x means probability our random variable is equal to a value x Therefore P X 2 1 6 Uniform Distribution p x probability density function area under the region 1 f x 1 b a The pdf for values uniformly distributed across a b is given by Calculates height not probability rsample Normal Distribution Note constants cid 0 3 14159 e 2 71828 The Normal Distribution The larger the standard deviation the greater spread within a given distribution f X Changing shifts the distribution left or right o Changing increases or decreases the spread X The Empirical Rule The Empirical Rule Standard Normal Distribution 0 1 99 7 of data are within 3 standard deviations of the mean 95 within 2 standard deviations 68 within 1 standard deviation S 0 1 2 4 34 34 2 4 13 5 13 5 3s 2s 1s 1s 2s 3s 0 1 Shifting the Mean Shifting the Standard Deviation Normal Distribution S D If the data are normally distributed then the interval contains about 68 of the values in the population or the sample 68 Normal Distribution S D contains about 95 of the values in the population or the sample contains about 99 7 of the values in the population or the sample 95 99 7 Example If thermometers have an average mean 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 freezing water between 0 degrees and 1 58 degrees Area 0 4429 p 0 X 1 58 0 4429 0 1 58 The probability that the chosen thermometer will measure freezing water between 0 and 1 58 degrees is 0 4429 Finding the Area to the Right of z 1 27 Area from mean to z 0 3980 This area is 0 5 0 3980 0 1020 z 1 27 0 15 Finding the Area Between z 1 20 and z 2 30 0 4893 Area from mean to z 2 30 Area A is 0 4893 0 3849 0 1044 0 3849 A z 1 20 z 2 30 0 Z score Transformation The distribution of z scores is called a standardized distribution When you standardize a distribution it is much easier to compare scores across distributions The standard normal distribution has a mean of 0 and a standard deviation of 1 Standard Normal Distribution Z score Transformation A z score states the number of standard deviations by which the original score lies above or below the mean When you standardize a distribution it is much easier to compare scores across distributions Any score can be converted to a z score as follows Z score Transformation Example Question