3.2 Measures of Central Tendency

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1 Math 166 Lecture Notes - S. Nite 9/22/2012 Page 1 of Measures of Central Tendency Mean The average, or mean, of the n numbers x = x 1 + x x n n x1,x2,...,xn is x (read x bar ), where Example 1: The following data give the number of cars observed waiting in line at the beginning of 2-minute intervals between 3 and 5 p.m. on a certain Friday at the drive-in teller of West wood Savings Bank and the corresponding frequency of occurrence. Find the average number of cars waiting in line at the bank s drive-in teller at the beginning of each 2-minute interval. Cars Frequency of Occurrence Expected Value Let X denote a random variable that assumes the values x1,x2,...,xn with associated probabilities p1,p2,...,pn respectively. Then the expected value of X, E(X), is given by E(X) x p + x p x n p = n. Note: The numbers x1,x2,...,xn may be positive, zero, or negative. For example, such a number will be positive if it represents a profit and negative if it represents a loss.

2 Math 166 Lecture Notes - S. Nite 9/22/2012 Page 2 of 5 Example 2: Re-solve Example 1 by using the probability distribution associated with the experiment, reproduced in the table below. x P(X = x) Example 3: Let X denote the random variable that gives the sum of the faces that fall uppermost when two fair dice are cast. Find the expected value, E(X), of X.

3 Math 166 Lecture Notes - S. Nite 9/22/2012 Page 3 of 5 Example 4: A group of private investors intends to purchase one of two motels currently being offered for sale in a certain city. The terms of sale of the two motels are similar, although the Regina Inn has 52 rooms and is in a slightly better location than the Merlin Motor Lodge, which has 60 rooms. Records obtained for each motel reveal that the occupancy rates, with corresponding probabilities, during the May-September tourist season are as shown in the following tables: Regina Inn Occupancy Rate Probability Merlin Motor Lodge Occupancy Rate Probability The average profit per day for each occupied room at the Regina Inn is $10, whereas the average profit per day for each occupied room at the Merlin Motor Lodge is $9. a. Find the average number of rooms occupied per day at each motel. b. If the investors objective is to purchase the motel that generates the higher daily profit, which motel should they purchase? (Compare the expected daily profit of the two motels.)

Math 166 Lecture Notes - S. Nite 9/22/2012 Page 4 of 5 Example 5: The Island Club is holding a fund-raising raffle. Ten thousand tickets have been sold for $2 each.

4 Math 166 Lecture Notes - S. Nite 9/22/2012 Page 4 of 5 Example 5: The Island Club is holding a fund-raising raffle. Ten thousand tickets have been sold for $2 each. There will be a first prize of $3000, 3 second prizes of $1000 each, 5 third prizes of $500 each, and 20 consolation prizes of $100 each. Let X denote the net winnings (that is, winnings less the cost of the ticket) associated with the tickets, find E(X). Interpret your results. Example 6: In the game of roulette as played in Las Vegas casinos, the wheel is divided into 38 compartments numbered 1 through 36, 0, and 00. One-half of the numbers 1 through 36 are red, the other half black, and 0 and 00 are green. Of the many types of bets that may be placed, one type involves betting on the outcome of the color of the winning number. For example, one may place a certain sum of money on red. If the winning number is red, one wins an amount equal to the bet placed and loses the best otherwise. Find the expected value of the winnings on a $1 bet placed on red. Example 7: Mike and Bill play a card game with a standard deck of 52 cards. Mike selects a card from a well-shuffled deck and received A dollars from Bill if the card selected is a diamond; otherwise, Mike pays Bill a dollar. Determine the value of A if the game is to be fair.

5 Math 166 Lecture Notes - S. Nite 9/22/2012 Page 5 of 5 Median and Mode The median of a set of numerical data is the middle number when the numbers are arranged in order of size. If there is an even number of entries, the median is the mean of the two middle numbers. The mode of a set of observations is the observation that occurs most frequently. If the frequency of two observations is the same and greater than the others, then the set is bimodal. If there is no single most frequent or two the same and greater than the others, there is no mode. Example 8: Find the median and mode of the following set of numbers. {3, 4, 5, 4, 7, 10, 13, 15, 4, 17, 3, 25, 4, 5} Example 9: Find the median and mode for the situation in Example 1.

4.3 Rules of Probability

4.3 Rules of Probability If a probability distribution is not uniform, to find the probability of a given event, add up the probabilities of all the individual outcomes that make up the event. Example: