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Check your Skills – Chapter 12 Calculate the probabilities for the following using this table: 1. A = the victim was male Accidents Homicide Suicide Female 1818 457 345 2620 Male 6457 2870 2152 11479 Total 8275 3327 2497 14099 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 women P(B|A) = P(B and A)/P(A) = (345/14099)/(2620/14099) = 345/2620 = .13 CHAPTER 12 BPS - 5TH ED. 1 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. Make a Venn diagram. Then answer these questions. (a) What is the probability that neither university admits Ramon? (b) What is the probability that he gets into Stanford but not Princeton? (c) Are admission to Princeton and admission to Stanford independent events? CHAPTER 11 BPS - 5TH ED. 2 CHAPTER 11 Sampling Distributions CHAPTER 11 BPS - 5TH ED. 3 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 CHAPTER 11 BPS - 5TH ED. 4 PARAMETER VS. STATISTIC 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. CHAPTER 11 BPS - 5TH ED. 5 PARAMETER VS. STATISTIC mean of a population is denoted by µ – this is a parameter. The mean of a sample is denoted by x – this is a statistic. x is used to estimate µ. The The true proportion of a population with a certain trait is denoted by p – this is a parameter. The proportion of a sample with a certain trait is denoted by p̂ (“p-hat”) – this is a statistic. p̂ is used to estimate p. CHAPTER 11 BPS - 5TH ED. 6 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. As n gets larger and larger, x approaches the value of µ CHAPTER 11 BPS - 5TH ED. 7 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 CHAPTER 11 BPS - 5TH ED. 8 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 CHAPTER 11 BPS - 5TH ED. 9 CASE STUDY 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. CHAPTER 11 BPS - 5TH ED. 10 CASE STUDY Does This Wine Smell Bad? 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. CHAPTER 11 BPS - 5TH ED. 11 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? CHAPTER 11 BPS - 5TH ED. 12 SAMPLING DISTRIBUTION What about the mean (average) of a sample of n adults? What values would be expected? 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 CHAPTER 11 BPS - 5TH ED. 13 CASE STUDY Does This Wine Smell Bad? 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. CHAPTER 11 BPS - 5TH ED. 14 CASE STUDY Does This Wine Smell Bad? CHAPTER 11 BPS - 5TH ED. 15 MEAN AND STANDARD DEVIATION OF SAMPLE MEANS If numerous samples of size n are taken from a population with mean and standard deviation , then the mean of the sampling distribution of X is (the population mean) and the standard deviation is: n ( is the population s.d.) CHAPTER 11 BPS - 5TH ED. 16 MEAN AND STANDARD DEVIATION OF SAMPLE MEANS the mean of X is , we say that X is an unbiased estimator of Since observations have standard deviation , but sample means X from samples of size n have standard deviation . Individual n Averages are less variable than individual observations CHAPTER 11 BPS - 5TH ED. 17 SAMPLING DISTRIBUTION OF SAMPLE MEANS If individual observations have the N(µ, ) distribution, then the sample mean X of n independent observations has the N(µ, / n ) distribution. “If measurements in the population follow a Normal distribution, then so does the sample mean.” CHAPTER 11 BPS - 5TH ED. 18 CASE STUDY Does This Wine Smell Bad? Mean threshold of all adults is =25 with a standard deviation of =7, and the threshold values follow a bell-shaped (normal) curve. CHAPTER 11 (Population distribution) BPS - 5TH ED. 19 CENTRAL LIMIT THEOREM 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 X is approximately Normal: X is approximately N(µ, / n ) “No matter what distribution the population values follow, the sample mean will follow a Normal distribution if the sample size is large.” CHAPTER 11 BPS - 5TH ED. 20 CENTRAL LIMIT THEOREM: SAMPLE SIZE How large must n be for the CLT to hold? – depends on how far the population distribution is from Normal the further from Normal, the larger the sample size needed a sample size of 25 or 30 is typically large enough for any population distribution encountered in practice recall: if the population is Normal, any sample size will work (n≥1) CHAPTER 11 BPS - 5TH ED. 21 CENTRAL LIMIT THEOREM: SAMPLE SIZE AND DISTRIBUTION OF X-BAR CHAPTER 11 n=1 n=2 n=10 n=25 BPS - 5TH ED. 22