Math 166: Topics in Contemporary Mathematics II

Transcription

1 Math 166: Topics in Contemporary Mathematics II Ruomeng Lan Texas A&M University October 15, 2014 Ruomeng Lan (TAMU) Math 166 October 15, / 12

2 Mean, Median and Mode Definition: 1. The average or mean of n numbers x 1, x 2,..., x n, denoted by µ, is given by µ = x 1 + x x n n 2. The median of n numbers x 1, x 2,..., x n is the middle number when the numbers are arranged in increasing (or decreasing) if n is odd. If n is the even, it is the average of the middle two numbers. 3. The mode of n numbers x 1, x 2,..., x n is the number that occurs the most frequently. If two numbers occur the same number of times and more frequent than all other numbers, we say the set is bimodal and has two modes. If no one or two numbers occur more frequently than the rest, we say there is no mode. Ruomeng Lan (TAMU) Math 166 October 15, / 12

3 Example: Find the mean, median and mode of the following sets of numbers a. {1, 4, 5, 4, 2, 3, 1, 4} b. {2, 3, 3, 5, 5, 6, 676} c. {x x is the one of first 6 positive odd number} Ruomeng Lan (TAMU) Math 166 October 15, / 12

4 Using Calculator Calculate the mean and median (CANNOT FIND MODE) 1. Press STAT, and then Enter to select Edit 2. Input the values in the list L1. 3. Press STAT again. Move the cursor to the right to the CALC menu and then select 1 - Var Stats. 4. Press 2ND and 1 to select list L1. The home screen should shows 1 - Var Stats L1. 5. Press Enter. x is the mean and scroll down to see the median (Med). To clear a list: Scroll up to the list name, press CLEAR, and then press ENTER. Make sure you do this every time before you start a new problem. Ruomeng Lan (TAMU) Math 166 October 15, / 12

5 Example with Frequency Distribution The grade distribution in a certain math class is given in the following table. Grade in Letter A B C D F Grade in Number Number of Students Find the mean, median and mode. Ruomeng Lan (TAMU) Math 166 October 15, / 12

6 Using Calculator Calculate the mean and median for the data with frequencies. 1. Press STAT, and then Enter to select Edit 2. Input the values in the list L1. Press the right arrow to input the frequencies in the list L2. 3. Press STAT again. Move the cursor to the right to the CALC menu and then select 1 - Var Stats. 4. Press 2ND and 1 to select list L1. Press comma,. And then press 2ND and 2 to selet list L2. This time, the home screen should shows 1 - Var Stats L1, L2. 5. Press Enter. x is the mean and scroll down to see the median (Med). Remind: Do not forget to clear the previous lists before you start a new problem. Ruomeng Lan (TAMU) Math 166 October 15, / 12

7 Expected Value Give the following empirical probability distribution table Find the mean. Grade in Letter A B C D F Grade in Number Probability Ruomeng Lan (TAMU) Math 166 October 15, / 12

8 Expected Value Definition: Let X denote the random variable that has values x 1, x 2,..., x n, and let the associated (empirical) probabilities be p 1, p 2,..., p n. Then expected value or mean of the random variable X, denoted by E(X ), is E(X ) = x 1 p 1 + x 2 p x n p n Note: 1. E(X ) is what we expect over the long term, but E(X ) need not be an actual outcome (x 1, x 2,..., x n ). 2. The expected value and mean are actually the same number; we just use the different words in different contexts. Ruomeng Lan (TAMU) Math 166 October 15, / 12

9 Examples A hurricane and flood insurance policy costs 100 dollars a year. The company will pay 4000 dollars if there is hurricane and flood damage to the house. If the chance that a hurricane or a flood will damage a house is 2% in this year, what is the expected profit or loss of this policy for the insurance company? Ruomeng Lan (TAMU) Math 166 October 15, / 12

10 Examples For a $2 scratch-off lottery ticket, there is a 3% chance of winning $20 and a 1% chance of winning $100. What is the expected profit or loss of purchasing one of these lotto tickets? Is it fair? Remark: A lotto or a gambling game is fair if the expected profit/loss is zero. Ruomeng Lan (TAMU) Math 166 October 15, / 12

11 Expected Value of Binomial Experiments Recall: We have the following probability distribution table for the experiment of flipping a fair coin twice Number of Heads Probability 1/4 1/2 1/4 What is the expected number of occurrences of head? Ruomeng Lan (TAMU) Math 166 October 15, / 12

12 Expected Value of Binomial Experiments Expected Value of Binomial Experiments: The expected value of the binomial distribution with n trials and probability of success p in a single trial and q = 1 p is E(X ) = np. Example: A unfair coin has the property that the probability of flipping a head is 40%. If we flip the coin 15 times, what is the expect number of occurrences of a head? Ruomeng Lan (TAMU) Math 166 October 15, / 12