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Data & Probability

Data & Probability. Data/Statistics. Terms. Mean: Average Median: Middle of an ordered list Exact middle for an odd # of items Average of the middle two for an even # of items Mode: Most frequent Range: Highest - Lowest. Box & Whisker.

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Data & Probability

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  1. Data & Probability

  2. Data/Statistics

  3. Terms • Mean: Average • Median: Middle of an ordered list • Exact middle for an odd # of items • Average of the middle two for an even # of items • Mode: Most frequent • Range: Highest - Lowest

  4. Box & Whisker • Helps you to see where the majority of the data lies, as each part is 25% of the data • Lowest and highest values = endpoints • Median of the data = center of the box • Median of the lower part and upper part = edges of the box

  5. Box & Whisker Plot low Q1 median Q3 high lowest 25% 2nd 25% 3rd 25% highest 25% the box contains 50% of the data Outliers are 1.5 . IQR from the ends of the box • IQR = Q3 – Q2 Extreme Outliers are 3∙IQR from the ends of the box The high and the low are not always Outliers, not all data sets contain outliers.

  6. Box & Whisker • Relatively evenly distributed (normal) data • Skewed left (longer left tail) • Skewed right (longer right tail) • Skew is determined by the tail

  7. Boxplot Draw boxplot for the following test scores: 98, 75, 80, 74, 92, 88, 83, 60, 72, 99 Ordered list: 60, 72, 74, 75, 80, 83, 88,92, 98, 99 Draw a number line Plot the end points Find the median Find the median of the first half Find the median of the second half Draw the box around the “three” medians Connect the box with “whiskers” to the endpoints 60 70 80 90 100

  8. Stem & Leaf • Displays all data • Stem Leaf 1st #(s) Last #

  9. Dot Plot • Similar to a stem and leaf plot but does not necessarily retain the precise values of the data • Given: 10, 18, 21, 26, 30, 31, 38, 40 • Stem and Leaf Dot Plot 1 0, 8 2 1, 6 3 0, 1, 8 1 2 3 4 4 0

  10. the median • the middle of the 17 values or 309 • the first quartile • the middle of the first half or (201+206)/2=203.5 • the third quartile • the middle of the second half or (407+408)/2=407.5 • the inter-quartile range • the difference of the quarter points 407.5-203.5=204 • the mode • the most frequent 309 • the percentile for 305 • 305 if the 5th item, 5/17=.294 * 100= 29.4 • or the 29th percentile • the value closest to the 60thpercentile • 60/100=x/17 • .6 = x/17 • .6*17 = x • 10.2 = x the 10th item (402) is closest to the 60th Percentile • Find the standard deviation • enter all the data in L1 • press STAT  calc, choose one-var stat St. dev. =Ϭx EXAMPLE: Given a stem and leaf plot FIND: • 2 5 7 • 1 6 • 5 8 9 9 • 2 3 5 7 8 • 2 • 60 3 6

  11. Scatter Plot • Shows how many and approximate values of the data • If the points follow a pattern, you can find the regression line

  12. To enter and plot points: • Press 2nd + 7 1 2 (clears everything) • Press 2nd 0 x-1 find diagnostics on press enter • Press Stat enter • X’s go in L1 • Y’s go in L2 • Press Y=, arrow up press enter, zoom 9

  13. To find the regression line: • Decide what pattern the point appear to be following • Press STAT arrow over to calc • Choose the correct pattern • 4 for linear • 5 for quadratic • 0 for exponential • Press variable, arrow to y-vars, press 1, press 1, enter • Write down the value of r • Press Y= write down the equation • Press graph to see the fit

  14. Predicting knowing x • Set the window to be large enough for the given value • Graph • Press 2nd trace (calc) • Choose 1 (value) • Enter the value and press enter • Estimating knowing y • Set the window to be large enough for the given value • Enter the value in Y2= • Press 2nd trace (calc) • Choose 5 (intersect) • Press enter three times • You may also substitute values into the equation

  15. Regression Equations • Find the equation for the following data and determine the value when x = 2 and when x = 7 Now try it for your self, checking along the way to see if you have the same values/screen shots as below—click each time you are ready to check your calculations. Scatterplot—enter data in stat edit Linear regression values Graph to make sure the line fits the pattern Use the calculations and enter a value of 2 Click on the calculator to see how to find a regression line Use the calculations and enter a value of 7

  16. Probability

  17. How can we determine all the possible outcomes of a given situation? TREE DIAGRAM—an illustrative method of counting all possible outcomes. List all the choices for the 1st event Then branch off and list all the choices for the second event for each 1st event, etc.

  18. A restaurant offers a salad for $3.75. You have a choice of lettuce or spinach. You may choose one topping, mushrooms, beans or cheese. You may select either ranch or Italian dressing. How many days could you eat at the restaurant before you repeat the salad? ranch mushrooms Italian ranch beans Italian Lettuce ranch cheese Italian ranch mushrooms Italian spinach ranch beans Italian ranch cheese Italian

  19. While the tree diagram is beneficial in that it lists every possible outcome, the more options you have the more difficult it is to draw the diagram. Fundamental counting Principle—is a mathematical version of the tree diagram, it gives the # of possible ways something can be accomplished but not a list of each way.

  20. Example: Jani can choose from gray or blue jeans, a navy, white, green or stripped shirt and running shoes, boots or loafers? How many outfits can she wear? _______ _______ _______ pants shirts shoes 2 3 3 =18

  21. Permutations—all the possible ways a group of objects can be arranged or ordered Example: There are four different books to be placed in order on a shelf. A history book (H), a math book (M), a science book (S), and an English book (E). How many ways can they be arranged? 24 WAYS 4 • 3 • 2 • 1 = 24 H, M, S, E M, E, S, H S, M, E, H E, M, S, H H, M, E, S M, E, H, S S, M, H, E E, M, H, S H, S, E, M M, S, H, E S, H, M, E E, H, M, S H, S, M, E M, S, E, H S, H, E, M E, H, S, M H, E, M, S M, H, E, S S, E, M, H E, S, M, H H, E, S, M M, H, S, E S, E, H, M E, S, H, M

  22. A permutation of n objects r at a time follows the formula Example: • This can be done on your calculator with the following keystrokes: • Type the number before the P • Press math • Over to prb • Choose number 2 nPr • Enter the number after the P • Press enter.

  23. Combinations P-3 How can you determine the difference between a permutation and a combination?

  24. Combinations—the number of groups that can be selected from a set of objects --the order in which the items in the group are selected does not matter

  25. Example: How many three person committees can be formed from a group of 4 people—Joe, Jim, Jane, and Jill Formula: Joe, Jim , Jill Joe, Jill, Jane Joe, Jim Jane Is Joe, Jane, Jim A different committee Jim, Jane, Jill

  26. Repetitions and Circular Permutations P-2 • What is the difference between replacement and repetition?

  27. Combinations • This can be done on your calculator with the following keystrokes: • Type the number before the C • Press math • Over to prb • Choose number 3 nCr • Enter the number after the C • Press enter.

  28. Replacement—using the same object again (nr) Example: The keypad on a safe has the digits 1- 6 on it how many: a) four digit codes can be formed _____ _____ _____ _____ b) four digit codes can be formed if no 2 digits can be the same _____ _____ _____ _____ 6 6 6 6 6 5 4 3

  29. Repetition—occurs when you have identical items in a group Example: Find all arrangements for the letters in the word TOOL ____ ____ ____ ____ TOOL OLOT LOTO TOLO OLTO LOOT TLOO OTOL LTOO OTLO OOTL OOLT We would expect 24 but since you can’t distinguish between the two O’s all possibilities with the O’s switched are removed 4 3 2 1

  30. Formula for repetitions: where s and t represent the number of times an item is repeated EXAMPLE: How many ways can you arrange the letters in BANANAS A N The factorial key is also found my pressing math and arrowing over to PRB

  31. Circular Permutation—arranging items in a circle when no reference is made to a fixed point Example: How many ways can you arrange the numbers 1-4 on a spinner? We would expect 4! Or 24 ways but we only have 6 Circular permutations are always (n-1)! 4 2 3 1 3 2 4 1 1 1 1 2 2 1 1 1 A ? ? B C D B E 3 2 3 3 4 4 2 3 3 2 4 2 4 3 4 4 D

  32. If all outcomes are successful, the probability will be 1 If no outcomes are successful, the probability will be 0 So Probability is 0 ≤ P ≤ 1

  33. Examples: What is the probability of getting an ace from a deck of 52 cards? 4 aces so What is the probability of rolling a 3 on a 6 sided die? there is 1 3 on 6 sides so

  34. What is the probability of rolling an even number? 2,4, 6 are even so What is the probability of getting 2 spades when 2 cards are dealt at the same time? at the same time indicates use of a combination —hint there are 13 spades

  35. What is the probability of getting a total of 5 when a pair of dice is rolled? Draw the following chart for the sum of all rolls and count how many have a sum of 5

  36. Compound Probability P-5 • What is meant by compound probability?

  37. OR: P(A or B) = P(A) + P(B) – P(A and B) Example: What is the probability of getting a 2 or a 5 on the roll of a die? Exclusive Events: events that do not have bearing on each other

  38. What is the probability of drawing an ace or a heart? ace + heart – ace of hearts + - = Events are inclusive if they have overlap!

  39. AND: indicates multiplication Examples: What is the probability of tossing a three of the roll of a die and getting a head when you toss a coin? three and a head * = These events are independent—have no effect on the outcome of the other

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