Unit 1B-Modelling with Statistics. By: Niha, Julia, Jankhna, and Prerana

Transcription

1 Unit 1B-Modelling with Statistics By: Niha, Julia, Jankhna, and Prerana

2 [ Definitions ] A population is any large collection of objects or individuals, such as Americans, students, or trees about which information is desired. A population parameter is any summary number, like an average or percentage, that describes the entire population. A sample statistic is a representative group drawn from the population. A statistic is any summary number, like an average or percentage, that describes the sample. Example: In a USA Today Internet poll, readers responded voluntarily to the question How many caffeinated beverages do you consume everyday? Population: The readers Population Parameter: Average number of daily consumed beverages Sample: Those who answered the poll Sample Statistic: Average number of daily consumed beverages of those who answered the poll

3 [ Parameter vs statistic ] The parameter represents the entire population. A statistic is a sample of that population. Identify if the underlined are statistics or parameters. a.of all US kindergarten teachers,32% say that knowing the alphabet is an essential skill. b.of the 800 US kindergarten teachers polled, 34% say that knowing the alphabet is an essential skill.

4 [ Parameter vs statistic ] Determine whether the numerical value is a parameter or a statistic: a) A survey of 1103 students was taken from the university with 19,500 students. b) The 2006 team payroll of the New York Mets was $101,084,963. c) In a recent study of physics majors at the university, 15 students were double majoring in math.

5 [ Parameter vs statistic ] Determine whether the numerical value is a parameter or a statistic: d) A recent survey by the alumni of a major university indicated that the average salary of 10,000 of its 300,000 graduates was $125,000. e) The average salary of all assembly-line employees at a certain car manufacturer is $33,000. f) The average late fee for 360 credit card holders was found to be $56.75

6 [ Margin of error ] Suppose that 900 American teens were surveyed about their favorite ski category of the 2002 Winter Olympics in Park City, Utah. Ski jumping was the favorite for 20% of those surveyed. This result can be used to predict how many of all 31 million American teens favor ski jumping. How? To determine how accurately the results of surveying 900 American teens truly reflect the results of surveying all 31 million American teens, a margin of error should be given. When pollsters report the margin or error for their surveys, they are stating their confidence mathematically in the data they have collected. The margin of error can be calculated by using the formula 1/ n, where n is the number in a sample size. For the above sample, the margin of error can be expressed as ± 3%.

7 [ Designing simulations ] 1.Describe the possible outcomes for the problem you are stimulating 2.Choose a random device to source your numbers 3.Choose a number of trials and conduct them 4.Record and analyze the results 5.Record conclusions and create a prediction Random devices include: Coins, random number table, cards, spinners, dice, random

8 [ Expected Value ] Expected Value: a predicted value of a variable, calculated as the sum of all possible values each multiplied by the probability of its occurrence. (mean amount you should recieve) E(x) = P 1 (n 1 ) + P 2 (n 2 ) + P 3 (n 3 ) + + P k (n k ) P = probability of each event occurring n = amount earned k = number of trials The expected value is a calculation that serves as the best prediction of a value. It is the probability-weighted average of all possible outcomes.

9 [ EXAMPLE ] A player pays $3 to play the following game: Win $7 by rolling a 6 Win $1 by rolling any other number $ earned Expected value, E(x) = $10/6 + $4/6 Expected value, E(x) = $1

10 [ Real world connection to expected value ] Expected value can be used in: making investments determining a price for numerous services prioritizing events calculating Return on Investment (gain/loss of an investment) Insurance companies casinos

11 Fair Games Fair game is a game in which the expected value is 0, this means the game doesn t favor the player or the house. If the winnings are negative, it favors the house (against the player) If the winnings are positive, it favors the player

12 Fair games example Using the previous example: A player pays $3 to play the following game: Win $7 by rolling a 6 Win $1 by rolling any other number Expected value, E(x) = $1 This is not a fair game! It is unfair since the expected value is negative, it favors the house.

13 Both of us have a ½ chance of winning or losing, so this game is fair. [ Fair Games ] Remember! Fair game- a game which is not biased towards any player Ex. Tim and Lisa roll a number cube. If a 3 or a 4 is rolled, Tim wins. If a 5 or a 6 is rolled, Lisa wins. Tim and Lisa both have a ⅓ chance of winning and ⅔ chance of losing, so this game is fair. Ex. Toss a coin. If it lands on heads, I win. If it lands on tails, you win.

14 [ Fair Games ] *Fair games have expected values of zero!!* Ex. Pablo was playing a card game, where he earns $3 if he picks a card that s suit is a diamond. He earns $2 if the number is 7. He earns $5 if he picks a 7 of diamonds. He loses $1 for every other card he picks. Is this game fair? 3(13/52)+ 2(4/52)+ 5(1/52)- 1(34/52)= 39/52 + 8/52 + 5/52-34/52= 18/52= 9/26 This game is not fair.