HONORS STATISTICS. Mrs. Garrett Block 2 & 3

Transcription

1 HONORS STATISTICS Mrs. Garrett Block 2 & 3 Tuesday December 4,

2 Daily Agenda 1. Welcome to class 2. Please find folder and take your seat. 3. Review OTL C7#1 4. Notes and practice 7.2 day 1 5. Folders and gradesheets 6. Collect Folders 2

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4 OTL C7#1 DUE ON Tuesday December 4th Pg 470: 7.4 and 7.5 (remember class discussion) Pg : 7.12, 7.13, 7.14 make histogram Pg 480: 7.20 ASKIP 7.14 start READING NOTES C7-2 pages

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8 Review Example: Find the mean and standard deviation of the following data set

9 7.2 Means of Random Variables The MEAN of a discrete random variable is called the EXPECTED VALUE It is the balance point of the probability distribution. It is NON RESISTANT. WHAT DOES EXPECTED VALUE MEAN??? my answer will be "yes" 9

10 7.2 Variances of Random Variables 10

11 Class examples page , 7.26, X = the grade point average of students in a particular class X P(X) draw the histogram of the probability distribution below probability Grade point average Find the expected value Find the variance and standard deviation 11

12 7.26 page 486: The Tri State Pick 3 lottery game offers a choice of several bets. You choose a 3 digit number. The lottery commission announces the winning three digit number, chosen randomly at the end of each day. The "box" pays $83.33 if the number you choose has the same digits as the winning number, in any order. Find the expected payoff for a $1 bet on the box. (Assume a number with three different digits is chosen) let X = expected payoff on a $1 bet (not your total take!) X P(X) 12

13 Basically the more data you find or the larger your survey is... the closer your sample results will be to the true value of the mean. 13

14 Chapter 7 Section 2 Introduction to means and variance rules 14

15 OTL C7#2 DUE ON WEDNESDAY DECEMBER 5 Pg 486: 7.25, 7.27, 7.29 Pg 491: 7.32 ASKIP 7.25 show your arithmetic Finish READING NOTES C7-2 pages Continue READING NOTES C7-2 pages

16 Owned and rented housing How do rented housing units differ from units occupied by their owners? (7.4, and 7.5) Find the mean number of rooms for both distributions. μx (owned) = 1(0.003) + 2(0.002) + 3(0.023) + 4(0.104) + 5(0.210) + 6(0.224) + 7(0.197) + 8(0.149) + 9(0.053) + 10(0.035) = Remember the distribution histogram? The histogram appears to be centered around 6 rooms too μx (rented) = 1(0.008) + 2(0.027) + 3(0.287) + 4(0.363) + 5(0.164) + 6(0.093) + 7(0.039) + 8(0.013) + 9(0.003) + 10(0.003) = Remember the distribution histogram? The histogram appears to be centered around 4 rooms too. The means reflect the centers of the histograms > owned homes have on average more rooms than rented units. And more swimming pools too. 16

17 7.27 KENO a) Find the probability distribution of the payoff X of the game. b) Find the probability distribution of the payoff X of the game. μx = 0(0.75) + 3(0.25) = 0.75 c) In the long run, how much does the casino keep from each dollar bet? The casino keeps 25 cents from each dollar bet (in the long run) This means on average you lose 25 cents of each dollar that you bet... 17

18 7.29 Households and Families Find the mean and standard deviation of both household size and family size. Combine these with your description from Exercise 7.12 & 7.13 to give a comparison of the two distributions. μx (household) = 1(0.25) + 2(0.32) + 3(0.17) + 4(0.15) + 5(0.07) + 6(0.03) + 7(0.01) = 2.6 people per household on average. μx (family) = 1( (0.42) + 3(0.23) + 4(0.21) + 5(0.09) + 6(0.03) + 7(0.02) = 3.14 people per household on average. Family Household probability probability Both distributions are skewed to the right. Families tend to have more people per unit. They have a larger expected value also < 3.14 Number of people in a household or family σx 2 (household) = (1-2.6) 2 (0.25) + (2-2.6) 2 (0.32) + (3-2.6) 2 (0.17) + (4-2.6) 2 (0.15) + (5-2.6) 2 (0.07) + (6-2.6) 2 (0.03) + (7-2.6) 2 (0.01) = σx(household) = 2.02 = people per unit σx 2 (family) = (1-3.14) 2 (0.0) + (2-3.14) 2 (0.42) + (3-3.14) 2 (0.23) + (4-3.14) 2 (0.21) + (5-3.14) 2 (0.09) + (6-3.14) 2 (0.03) + (7-3.14) 2 (0.02) = 1.56 σx(family) = 1.56 = people per unit Family units appear to be less variable than household units. This is apparent in the histograms of the distributions. 18

19 7.32 Is red hot? a) A gambler knows that red and black are equally likely to occur on each spin of a roulette wheel. He observes five consecutive reds and bets heavily on red at the next spin. Asked why, he says that "red is hot" and that the run of reds is likely to continue. Explain to the gambler what is wrong with this reasoning. The wheel is not affected by what has occurred on the last spin. It has no memory, outcomes are independent. Reds and blacks will continue to be equally likely on each remaining spin. b) After hearing you explain why red and black remain equally probable after five reds on the roulette wheel, the gambler moves to a poker game. He is dealt five straight red cards. He remembers what you said and assumes that the next card dealt in the same hand is equally likely to be red or black. Is the gamble right or wrong? Why? He is WRONG. Unlike the wheel, cards are affected by each card that is dealt. Removing a card changes the number of each color that is left in the deck. If five red cards are dealt then there are less red cards remaining in the deck and the probability of a red will decrease. 19