Slide 12 of 18 THE NORMAL DISTRIBUTION Interpreting any partial area under the curve: o Any score above 75 has an area of 0.1587 which is interpreted as 15.87% probability. a If these are student test scores, a student with a score of 75 or above is part of the 15.87% of the class with high 75 scores. THE NORMAL DISTRIBUTION Interpreting any partial area under the curve: o So, if 15.87% are students with high scores, how many among the 40 students in the class got high scores? a Clue: What is 15.87% of 402 o Answer: (0.1587)(40) = 6.34 75 Or simply, about 6 students. THE NORMAL DISTRIBUTION Interpreting any partial area under the curve: o Consequently, a score below 65 also has an area of 0.1587. o This means there are also about 6 students (among the 40) with a score of 65 or below. 15THE NORMAL DISTRIBUTION Example: # = 70 (the mean) 0 =5 (the standard deviation) The mean value is located at the middle of the curve. The standard deviation value is added or subtracted as you move away from the mean. un The whole area under the curve is always equal to 1. Any partial greg represents the probability on which a score range occurs. (This is explained in detail on the next side.) 60 65 70 75 80 THE NORMAL DISTRIBUTION Interpreting any partial area under the curve: o The whole area is equal to 1 which is Interpreted as 100% probability. 75 B 12 THE NORMAL DISTRIBUTION Interpreting any partial area under the curve: o Any score above 70 covers one-half the area which is interpreted as 50% probability. In other words, there is a 50% chance that a score will be 70 or higher.MODULE 4 THE NORMAL DISTRIBUTION Interpreting any partial area under the curve: o When computed, the area between scores of 68 to 72 is 0.3108. o This means about 31% of the class got scores between 68 to 72. In a class of 40, how many students belong to this 55 60 65 70 75 85 group? o Answer: (0.3108)(40) = 12.43 Or 12 students. MODULE 4 17 THE NORMAL DISTRIBUTION Computing for the area under the normal curve: https://onlinestatbook.com/2/calculatorsormal dist.html o Using this online calculator will be discussed in detail during our synchronous meeting of the week. o In the meantime, access and explore the site of the link. D7. How many ways can you form a 6- member superhero team from a pool of 13 superhero characters? Show the Excel formula you used as well as the nal answer. [Delete this then place your answer inside this box] 8. How many ways can you create a number- code for a 3-digit lock with repetitions allowed? How many with no repetitions allowed? For each question, show the Excel formula you used as well as the nal answers. [Delete this then place your answer inside this box] Slide 6 of 18 COUNTING TECHNIQUES PERMUTATION RULE is the ordered arrangement of a objects taken r at a time. Excel formulas are: o PERMUT(n.r) with no repetitions o PERMUTATIONA(n.r) with repetitions Examples #1: How many 4-digit PINs are there when you know that no digit repeats? n = 10 r=4 PERMUT (10,4) = 5040 #2: How many 4-digit PINs are there when repetition is allowed? n = 10 r =4 PERMUTATIONA(10,4) = 10000 Moout 4 COUNTING TECHNIQUES COMBINATION RULE is the selection of r objects taken from , without regard to order. Excel formulas are: o COMBIN(n,r) with no repetitions . COMBINA(n, r) with repetitions Examples #1: How many possibilities are there for the winning ticket of the 6/58 lotto game? n = 58 r =6 COMBIN(58,6) = 40,475,358 #2: How many ways can you have a two-letter combination with repetition? n = 26 T =2 COMBINA(26,2) = 351 Mobat A THE NORMAL DISTRIBUTION o The single most important probability distribution in statistics. o The graph is bell-shaped, and the total area is equal to 1. o The formula is dependent on the mean and the standard deviation. Non- Performers Average performers Top Performers Example: Student performanceDEFINITIONS PROBABILITY is the chance that a particular event will occur. o Classical definition: number of ways X can occur P(X) = total number of equally likely possible outcomes o Empirical definition: P(X) = number of times X occurred number of times experiment was repeated EXAMPLES o Tossing a coin: P(getting tail) = o Tracking lifespan: P(cancer patients reaching 60 years) = number of patients still alive number of patients considered COUNTING TECHNIQUES MULTIPLICATION PRINCIPLE o The number of "combo meals" you can create from five choices of burgers and four choices of drinks. (5)(4) = 20 o The number of attires you can have with 14 shirts, 8 pairs of pants, and 5 pairs of shoes. (14)(8) (5) = 560