Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 1 of 32 Chapter 7 Section 2 The Standard Normal Distribution

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 2 of 32 Chapter 7 – Section 2 ●Learning objectives  Find the area under the standard normal curve  Find Z-scores for a given area  Interpret the area under the standard normal curve as a probability 1 2 3

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 3 of 32 Chapter 7 – Section 2 ●Learning objectives  Find the area under the standard normal curve  Find Z-scores for a given area  Interpret the area under the standard normal curve as a probability 1 2 3

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 4 of 32 Chapter 7 – Section 2 ●The standard normal curve is the one with mean μ = 0 and standard deviation σ = 1 ●We have related the general normal random variable to the standard normal random variable through the Z-score ●In this section, we discuss how to compute with the standard normal random variable

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 5 of 32 Chapter 7 – Section 2 ●There are several ways to calculate the area under the standard normal curve  What does not work – some kind of a simple formula  We can use a table (such as Table IV on the inside back cover)  We can use technology (a calculator or software) ●Using technology is preferred

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 6 of 32 Chapter 7 – Section 2 ●Three different area calculations  Find the area to the left of  Find the area to the right of  Find the area between ●Three different area calculations  Find the area to the left of  Find the area to the right of  Find the area between ●Three different methods shown here  From a table  Using Excel  Using StatCrunch

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 7 of 32 Chapter 7 – Section 2 ●"To the left of" – using a table ●Calculate the area to the left of Z = 1.68 ●"To the left of" – using a table ●Calculate the area to the left of Z = 1.68  Break up 1.68 as ●"To the left of" – using a table ●Calculate the area to the left of Z = 1.68  Break up 1.68 as  Find the row 1.6 Enter ●"To the left of" – using a table ●Calculate the area to the left of Z = 1.68  Break up 1.68 as  Find the row 1.6  Find the column.08 Enter ●"To the left of" – using a table ●Calculate the area to the left of Z = 1.68  Break up 1.68 as  Find the row 1.6  Find the column.08 ●The probability is Read Enter

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 8 of 32 Chapter 7 – Section 2 ●"To the left of" – using Excel ●The function in Excel for the standard normal is =NORMSDIST(Z-score)  NORM (normal) S (standard) DIST (distribution) Enter Read Enter

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 9 of 32 Chapter 7 – Section 2 ●To the left of 1.68 – using StatCrunch ●The function is Stat – Calculators – Normal Enter Read Enter

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 10 of 32 Chapter 7 – Section 2 ●"To the right of" – using a table ●The area to the left of Z = 1.68 is Read Enter ●"To the right of" – using a table ●The area to the left of Z = 1.68 is ●The right of … that’s the remaining amount ●The two add up to 1, so the right of is 1 – =

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 11 of 32 Chapter 7 – Section 2 ●"To the right of" – using Excel Read Enter ●"To the right of" – using Excel ●The right of … that’s the remaining amount of to the left of, subtract from 1 1 – =

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 12 of 32 Chapter 7 – Section 2 ●"To the right of" – using StatCrunch Read Enter ●"To the right of" – using StatCrunch ●Change the = ●"To the right of" – using StatCrunch ●Change the = (the picture and number change) Read Enter

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 13 of 32 Chapter 7 – Section 2 ●“Between” ●Between Z = – 0.51 and Z = 1.87 ●This is not a one step calculation

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 14 of 32 Chapter 7 – Section 2 ●The left hand picture … to the left of 1.87 … includes too much ●It is too much by the right hand picture … to the left of Included too much

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 15 of 32 Chapter 7 – Section 2 ●Between Z = – 0.51 and Z = 1.87 We want We start out with, but it’s too much We correct by

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 16 of 32 Chapter 7 – Section 2 ●We can use any of the three methods to compute the normal probabilities to get ●The area between and 1.87 ●We can use any of the three methods to compute the normal probabilities to get ●The area between and 1.87  The area to the left of 1.87, or … minus ●We can use any of the three methods to compute the normal probabilities to get ●The area between and 1.87  The area to the left of 1.87, or … minus  The area to the left of -0.51, or … which equals ●We can use any of the three methods to compute the normal probabilities to get ●The area between and 1.87  The area to the left of 1.87, or … minus  The area to the left of -0.51, or … which equals  The difference of ●We can use any of the three methods to compute the normal probabilities to get ●The area between and 1.87  The area to the left of 1.87, or … minus  The area to the left of -0.51, or … which equals  The difference of ●Thus the area under the standard normal curve between and 1.87 is

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 17 of 32 Chapter 7 – Section 2 ●A different way for “between” We want We delete the extra on the left We delete the extra on the right

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 18 of 32 Chapter 7 – Section 2 ●Again, we can use any of the three methods to compute the normal probabilities to get ●The area between and 1.87  The area to the left of -0.51, or … plus  The area to the right of 1.87, or.0307 … which equals  The total area to get rid of which equals ●Again, we can use any of the three methods to compute the normal probabilities to get ●The area between and 1.87  The area to the left of -0.51, or … plus  The area to the right of 1.87, or.0307 … which equals  The total area to get rid of which equals ●Thus the area under the standard normal curve between and 1.87 is 1 – =

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 19 of 32 Chapter 7 – Section 2 ●Learning objectives  Find the area under the standard normal curve  Find Z-scores for a given area  Interpret the area under the standard normal curve as a probability 1 2 3

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 20 of 32 Chapter 7 – Section 2 ●We did the problem: Z-Score  Area ●Now we will do the reverse of that Area  Z-Score ●We did the problem: Z-Score  Area ●Now we will do the reverse of that Area  Z-Score ●This is finding the Z-score (value) that corresponds to a specified area (percentile) ●And … no surprise … we can do this with a table, with Excel, with StatCrunch, with …

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 21 of 32 Chapter 7 – Section 2 ●“To the left of” – using a table ●Find the Z-score for which the area to the left of it is 0.32 ●“To the left of” – using a table ●Find the Z-score for which the area to the left of it is 0.32  Look in the middle of the table … find 0.32 Find Read ●“To the left of” – using a table ●Find the Z-score for which the area to the left of it is 0.32  Look in the middle of the table … find 0.32  The nearest to 0.32 is … a Z-Score of -.47

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 22 of 32 Chapter 7 – Section 2 ●"To the left of" – using Excel ●The function in Excel for the standard normal is =NORMSINV(probability)  NORM (normal) S (standard) INV (inverse) Read Enter

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 23 of 32 Chapter 7 – Section 2 ●"To the left of" – using StatCrunch ●The function is Stat – Calculators – Normal Read Enter

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 24 of 32 Chapter 7 – Section 2 ●"To the right of" – using a table ●Find the Z-score for which the area to the right of it is ●Right of it is.4332 … left of it would be.5668 ●A value of.17 Enter Read

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 25 of 32 Chapter 7 – Section 2 ●"To the right of" – using Excel ●To the right is.4332 … to the left would be.5668 (the same as for the table) Read Enter

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 26 of 32 Chapter 7 – Section 2 ●"To the right of" – using StatCrunch ●Change the = (watch the picture change) Read Enter

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 27 of 32 Chapter 7 – Section 2 ●We will often want to find a middle range, to find the middle 90% or the middle 95% or the middle 99%, of the standard normal ●The middle 90% would be

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 28 of 32 Chapter 7 – Section 2 ●90% in the middle is 10% outside the middle, i.e. 5% off each end ●These problems can be solved in either of two equivalent ways ●We could find  The number for which 5% is to the left, or  The number for which 5% is to the right

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 29 of 32 Chapter 7 – Section 2 ●The two possible ways  The number for which 5% is to the left, or  The number for which 5% is to the right 5% is to the left 5% is to the right

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 30 of 32 Chapter 7 – Section 2 ●The number z α is the Z-score such that the area to the right of z α is α ●Some useful values are  z.10 = 1.28, the area between and 1.28 is 0.80  z.05 = 1.64, the area between and 1.64 is 0.90  z.025 = 1.96, the area between and 1.96 is 0.95  z.01 = 2.33, the area between and 2.33 is 0.98  z.005 = 2.58, the area between and 2.58 is 0.99

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 31 of 32 Chapter 7 – Section 2 ●Learning objectives  Find the area under the standard normal curve  Find Z-scores for a given area  Interpret the area under the standard normal curve as a probability 1 2 3

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 32 of 32 Chapter 7 – Section 2 ●The area under a normal curve can be interpreted as a probability ●The standard normal curve can be interpreted as a probability density function ●The area under a normal curve can be interpreted as a probability ●The standard normal curve can be interpreted as a probability density function ●We will use Z to represent a standard normal random variable, so it has probabilities such as  P(a < Z < b)  P(Z < a)  P(Z > a)

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 33 of 32 Summary: Chapter 7 – Section 2 ●Calculations for the standard normal curve can be done using tables or using technology ●One can calculate the area under the standard normal curve, to the left of or to the right of each Z-score ●One can calculate the Z-score so that the area to the left of it or to the right of it is a certain value ●Areas and probabilities are two different representations of the same concept

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 34 of 32 Example: Chapter 7 – Section 2 ●Determine the area under the standard normal curve that lies  a. to the left of Z = –2.31. (0.0104)  b. to the right of Z = –1.47. (0.9292)  c. between Z = –2.31 and Z = 0. (0.4896)  d. between Z = –2.31 and Z = –1.47. (0.0603)  e. between Z = 1.47 and Z = (0.0603)  f. between Z = –2.31 and Z = (0.9188)  g. to the left of Z = –2.31 or to the right of Z = (0.0812)

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 2 – Slide 35 of 32 Example: Chapter 7 – Section 2 ●The Graduate Record Examination (GRE) is a test required for admission to many U.S. graduate schools. The Department of Molecular Genetics at Ohio State University requires a GRE score no less than the 60th percentile. (Source: state.edu/~molgen/html/admission_criteria.html.)  a. Find the Z-score corresponding to the 60th percentile. In other words, find the Z-score such that the area under the standard normal curve to the left is (0.25)  b. How many standard deviations above the mean is the 60th percentile? (0.25)