1. Check your Skills – Chapter 12 BPS - 5TH ED.CHAPTER 12 1 FemaleMaleTotal Accidents181864578275 Homicide45728703327 Suicide34521522497 26201147914099  Calculate the probabilities for the following using this table: 1. A = the victim was male P(A) = 11479/14099 = 0.81 2. A = the victim was male, given B = the death was accidental P(A|B) = P(A and B)/P(B) = (6457/14099)/(8275/14099) = 6457/8275 =.78 3. That the death was accidental, given the victim was male P(B|A) = P(B and A)/P(A) = (6457/14099)/(11479/14099) = 6457/11479 =.56 4. Let A be the event that a victim of violent death was a woman and B the event that the death was a suicide. Find the proportion of suicides among violent deaths of womenevent P(B|A) = P(B and A)/P(A) = (345/14099)/(2620/14099) = 345/2620 =.13

2. EXERCISE 12.31  Getting into college. Ramon has applied to both Princeton and Stanford. He thinks the probability that Princeton will admit him is 0.4, the probability that Stanford will admit him is 0.5, and the probability that both will admit him is 0.2.probability  Make a Venn diagram. Then answer these questions.  (a) What is the probability that neither university admits Ramon?probability  (b) What is the probability that he gets into Stanford but not Princeton?probability  (c) Are admission to Princeton and admission to Stanford independent events? BPS - 5TH ED.CHAPTER 11 2

3. Sampling Distributions BPS - 5TH ED.CHAPTER 11 3

4. SAMPLING TERMINOLOGY  Parameter  fixed, unknown number that describes the population  Statistic  known value calculated from a sample  a statistic is often used to estimate a parameter  Variability  different samples from the same population may yield different values of the sample statistic  Sampling Distribution  tells what values a statistic takes and how often it takes those values in repeated sampling BPS - 5TH ED.CHAPTER 11 4

5. PARAMETER VS. STATISTIC BPS - 5TH ED.CHAPTER 11 5 A properly chosen sample of 1600 people across the United States was asked if they regularly watch a certain television program, and 24% said yes. The parameter of interest here is the true proportion of all people in the U.S. who watch the program, while the statistic is the value 24% obtained from the sample of 1600 people.

6. PARAMETER VS. STATISTIC BPS - 5TH ED.CHAPTER 11 6 u The mean of a population is denoted by µ – this is a parameter. u The mean of a sample is denoted by – this is a statistic. is used to estimate µ. u The true proportion of a population with a certain trait is denoted by p – this is a parameter. u The proportion of a sample with a certain trait is denoted by (“p-hat”) – this is a statistic. is used to estimate p.

7. THE LAW OF LARGE NUMBERS Consider sampling at random from a population with true mean µ. As the number of (independent) observations sampled, n increases, the mean of the sample gets closer and closer to the true mean of the population. BPS - 5TH ED.CHAPTER 11 7 As n gets larger and larger, approaches the value of µ

8. THE LAW OF LARGE NUMBERS GAMBLING  The “ house ” in a gambling operation is not gambling at all  the games are defined so that the gambler has a negative expected gain per play (the true mean gain after all possible plays is negative)  each play is independent of previous plays, so the law of large numbers guarantees that the average winnings of a large number of customers will be close the the (negative) true average BPS - 5TH ED.CHAPTER 11 8

9. SAMPLING DISTRIBUTION  The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size (n) from the same population  to describe a distribution we need to specify the shape, center, and spread  we will discuss the distribution of the sample mean (x-bar) in this chapter BPS - 5TH ED.CHAPTER 11 9

10. CASE STUDY BPS - 5TH ED.CHAPTER 11 10 Does This Wine Smell Bad? Dimethyl sulfide (DMS) is sometimes present in wine, causing “off-odors”. Winemakers want to know the odor threshold – the lowest concentration of DMS that the human nose can detect. Different people have different thresholds, and of interest is the mean threshold in the population of all adults.

11. CASE STUDY BPS - 5TH ED.CHAPTER 11 11 Suppose the mean threshold of all adults is  =25 micrograms of DMS per liter of wine, with a standard deviation of  =7 micrograms per liter and the threshold values follow a bell-shaped (normal) curve. Does This Wine Smell Bad?

12. WHERE SHOULD 95% OF ALL INDIVIDUAL THRESHOLD VALUES FALL?  mean plus or minus two standard deviations 25  2(7) = 11 25 + 2(7) = 39  95% should fall between 11 & 39  What about the mean (average) of a sample of n adults? What values would be expected? BPS - 5TH ED.CHAPTER 11 12

13. SAMPLING DISTRIBUTION  What about the mean (average) of a sample of n adults? What values would be expected? BPS - 5TH ED.CHAPTER 11 13 u Answer this by thinking: “What would happen if we took many samples of n subjects from this population?” (let’s say that n=10 subjects make up a sample) –take a large number of samples of n=10 subjects from the population –calculate the sample mean (x-bar) for each sample –make a histogram (or stemplot) of the values of x-bar –examine the graphical display for shape, center, spread

14. CASE STUDY BPS - 5TH ED.CHAPTER 11 14 Mean threshold of all adults is  =25 micrograms per liter, with a standard deviation of  =7 micrograms per liter and the threshold values follow a bell-shaped (normal) curve. Many (1000) repetitions of sampling n=10 adults from the population were simulated and the resulting histogram of the 1000 x-bar values is on the next slide. Does This Wine Smell Bad?

15. CASE STUDY BPS - 5TH ED.CHAPTER 11 15 Does This Wine Smell Bad?

16. MEAN AND STANDARD DEVIATION OF SAMPLE MEANS BPS - 5TH ED.CHAPTER 11 16 If numerous samples of size n are taken from a population with mean  and standard deviation , then the mean of the sampling distribution of is  (the population mean) and the standard deviation is: (  is the population s.d.)

17. MEAN AND STANDARD DEVIATION OF SAMPLE MEANS BPS - 5TH ED.CHAPTER 11 17  Since the mean of is , we say that is an unbiased estimator of  u Individual observations have standard deviation , but sample means from samples of size n have standard deviation. Averages are less variable than individual observations

18. SAMPLING DISTRIBUTION OF SAMPLE MEANS BPS - 5TH ED.CHAPTER 11 18 If individual observations have the N(µ,  ) distribution, then the sample mean of n independent observations has the N(µ,  / ) distribution. “If measurements in the population follow a Normal distribution, then so does the sample mean.”

19. CASE STUDY BPS - 5TH ED.CHAPTER 11 19 Mean threshold of all adults is  =25 with a standard deviation of  =7, and the threshold values follow a bell-shaped (normal) curve. Does This Wine Smell Bad? (Population distribution)

20. CENTRAL LIMIT THEOREM BPS - 5TH ED.CHAPTER 11 20 “No matter what distribution the population values follow, the sample mean will follow a Normal distribution if the sample size is large.” If a random sample of size n is selected from ANY population with mean  and standard deviation , then when n is large the sampling distribution of the sample mean is approximately Normal: is approximately N(µ,  / )

21. CENTRAL LIMIT THEOREM: SAMPLE SIZE BPS - 5TH ED.CHAPTER 11 21 u How large must n be for the CLT to hold? –depends on how far the population distribution is from Normal v the further from Normal, the larger the sample size needed v a sample size of 25 or 30 is typically large enough for any population distribution encountered in practice v recall: if the population is Normal, any sample size will work (n≥1)

22. CENTRAL LIMIT THEOREM: SAMPLE SIZE AND DISTRIBUTION OF X-BAR BPS - 5TH ED.CHAPTER 11 22 n=1 n=25 n=10 n=2