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Name _______________________________ AP STATISTICS CHAPTER 7: What is the difference between a parameter and a statistic? What does the symbol p̂ represent? Under what conditions is a statistic said to be UNBIASED? The SAMPLING DISTRIBUTION of a statistic is the taken by the statistic in all possible samples . SUMMARY/QUESTIONS TO ASK IN CLASS Name _______________________________ AP STATISTICS CHAPTER 7: What determines the spread (variability) of the sampling distribution? The methods we use for evaluating the variability of a sampling distribution in this course will apply if…. SUMMARY/QUESTIONS TO ASK IN CLASS Name _______________________________ AP STATISTICS CHAPTER 7: Recall that p̂ = THE SAMPLING DISTRIBUTION OF A SAMPLE PROPORTION Choose an SRS of size n from a population with population proportion p having some characteristic of interest. Let p̂ be the proportion of the sample having that characteristic. Then: 1) The mean of the sampling distribution is 2) The standard deviation of the sampling distribution is given by RULES OF THUMB (Conditions that need to be verified before we can proceed): 1) Use this formula for standard deviation when 2) Use the normal approximation when Example: A polling organization asks an SRS of 1500 first year college students whether they applied for admission to any other college. In fact, 35% of all first-year students applied to colleges besides the one they are attending. What is the probability that the random sample of 1500 students will give a result within 2 percentage points of the true value? 1. STATE: What is our goal? Re-state the problem using mathematical symbols. 2. PLAN: Summarize salient facts and conditions. 3. DO: Perform necessary calculations. 4. CONCLUDE: Answer the question, in context. SUMMARY/QUESTIONS TO ASK IN CLASS Name _______________________________ AP STATISTICS CHAPTER 7: Example 2: Suppose one student tossed a coin 200 times and found only 42% heads. Do you believe that this is likely to happen? Example 3: Assume that 30% of the students at HH wear contacts. In a sample of 100 students, what is the probability that more than 35% of them wear contacts? Example 4: About 11% of American adults are black. Therefore, the proportion of blacks in an SRS of 1500 adults should be close to .11. If a national sample contains only 9.2% black, should we suspect that the sampling procedure is somehow underrepresenting blacks? SUMMARY/QUESTIONS TO ASK IN CLASS Name _______________________________ AP STATISTICS CHAPTER 7: Sample Means: How do you know if a statistic is unbiased? General Properties of Sample Means: Suppose that x is the mean of an SRS of size n drawn from a large population with mean and standard deviation . Rule 1: Rule 2: When are we allowed to use Rule #2? Rule 3: EX) The army reports that the distribution of head circumference among soldiers is approximately normal with mean 22.8 inches and standard deviation of 1.1 inches. a) What is the probability that a randomly selected soldier’s head will have a circumference that is greater than 23.5 inches? b) What is the probability that a random sample of five soldiers will have an average head circumference that is greater than 23.5 inches? SUMMARY/QUESTIONS TO ASK IN CLASS Name _______________________________ AP STATISTICS CHAPTER 7: Example 2: Suppose a team of biologists has been studying the Pinedale children’s fishing pond. Let x represent the length of a single trout taken at random from the pond. This group of biologists has determined that the length has a normal distribution with mean of 10.2 inches and standard deviation of 1.4 inches. What is the probability that a single trout taken at random from the pond is between 8 and 12 inches long? What is the probability that the mean length of five trout taken at random is between 8 and 12 inches long? What sample mean would be at the 95th percentile? (Assume n = 5) Example 3 A soft-drink bottler claims that, on average, cans contain 12 oz of soda. Let x denote the actual volume of soda in a randomly selected can. Suppose that x is normally distributed with s = .16 oz. Sixteen cans are randomly selected and a mean of 12.1 oz is calculated. What is the probability that the mean of 16 cans will exceed 12.1 oz? SUMMARY/QUESTIONS TO ASK IN CLASS Name _______________________________ AP STATISTICS CHAPTER 7: IF THE POPULATION HAS ANY DISTRIBUTION, WE MAY EMPLOY THE . When n is the distribution. , the sample mean x has What are the conditions for using the CLT? Ex: A hot dog manufacturer asserts that one of its brands has an average fat content of 18 grams per hot dog. The standard deviation is 1. In a sample of 36 hot dogs, what is the probability that the mean fat content will be at least 18.4 grams? SUMMARY/QUESTIONS TO ASK IN CLASS Name _______________________________ AP STATISTICS CHAPTER 7: CHAPTER 9 – SUMMARY SAMPLE PROPORTIONS Standard deviation – use when the population is at least 10 times as large as the sample. s µp = p (1 - p) n Use the normal approximation when np 10 and n 1 p 10 p¾¾ ® N ( p, s µp ) Then µ SAMPLE MEANS Standard deviation - use when the population is at least 10 times as large as the sample. x n If the population is normally distributed, then x N ( , n ) Central limit theorem: Given any distribution, if n is large ( n 30 ), then x N ( , n ) SUMMARY/QUESTIONS TO ASK IN CLASS