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Chapter 5. Sampling Distributions for Counts and proportions * Reading assignments: Read the ch.5 of textbook. Now, suppose for a moment that we know the measurement of every member of the population. Then, if we knew the size of the sample we would draw, we could list every possible sample that could be selected from the population. Next, for each sample, we could calculate a numerical characteristic such as a mean or proportion. A table or density curve which represents the likelihood of drawing samples that have a certain calculated numerical characteristic is called a sampling distribution. The concept of a sampling distribution is very important in statistics, and in fact, this concept is the foundation for the rest of the course. To illustrate the idea of a sampling distribution, consider the following example. Example 1: Consider the following population {1,2,3,4} with identical probability 1/4. Suppose that a random sample of size 2 is to be selected without replacement from this population. There are 12 possible samples. Compute the sample mean for each ? of the 12 samples and construct the sampling distribution of X sample 1,2 2, 3,1 1,3 4,1 2,3 3,2 4,2 1,4 2,4 3,4 4,3 1 Sample mean 1.5 1.5 2 2 2.5 2.5 2.5 3 2.5 3 3.5 3.5 x 2 12 , if x 1.5, P X P X x 2 12 , if x 2, x 4 12 , if x 2.5, P X P X x 2 12 , if x 3, P X x 2 12 , if x 3.5. 1.5 1 6 2 1 6 2.5 1 3 3 1 6 3.5 1 6 2.5 . Then E X ? What is Var X 1 2 3 4 2.5 . The population mean, 4 What is the value of ? Notations : Population mean, variance, and standard deviation of random variable X: , 2 , Sample mean, variance, standard deviation of random variable X : X , S2 , S 2 Mean, variance, standard deviation of the sampling distribution of X : X , X , Population proportion, and sample proportion : , p 5.2. The sampling distribution of a sample mean, X X Let's consider a random sample of size n from a population having mean, has a mean X standard deviation . The sampling distribution of X 2 , and standard deviation X . Var X 2X n n and , Note: When the population distribution is normal, the sampling distribution of the sample mean is also normal. Example 2 : The inside diameter of a randomly selected piston ring is a random variable with mean value 12 cm and standard deviation 0.04 cm. (a) If X is the sample mean diameter for a random sample of n=16 rings, where is the sampling distribution of centered, and what is the standard X deviation of the X distribution? (b) Answer the questions posed in part (a) for a sample size n=64 rings. Example 3 : When sampling without replacement in example 1 is done, we have got X 1.5 2 2.5 3 3.5 2 12 2 12 4 12 2 12 2 12 probability Suppose now that a random sample of size 2 is to be selected with replacement from this population. There are 16 possible possible samples. Compute the sample mean for each of the 16 samples and construct the sampling distribution of X . X probability E X ? 1 1 16 1.5 2 16 2 3 16 2.5 4 16 3 3 16 3.5 2 16 4 1 16 Var X ? In what ways are the two sampling distributions similar? In what ways are they different? They are both symmetric around population mean. In the example 1, values are changing from 1.5 to 3.5 whereas, in example 3, they are changing from 1 to 4. Example 4 : The breaking strength of a rivet has a mean value of 10,000 psi and a standard deviation of 500 psi. (a) What is the probability that the sample mean breaking strength for a random sample of 40 rivets is between 9900 and 10,200? (b) If the sample size had been 15 rather than 40, could the probability requested in part (a) be calculated from the given information? Sampling distribution of a sample mean Case 1: If a population distribution follows N(µ, 2), then the sample mean independent observations has the N(µ,( / n)2) X of n Case 2: If a population distribution doesn't follow N(µ, 2), then we use the general theory(Central Limit Theorem, CLT) in statistics, which is that as the sample size increases, the distribution of X gets closer to a normal distribution. Central Limit Theorem (CLT) is well When the sample size is sufficiently large, the sampling distribution of X approximated by a normal curve, even when the population distribution is not itself normal. CLT can be safely applied when n is greater than 30. The standardized variable, Z X X X X has a standard normal distribution, i.e. N(0,1). n Example 5: A random sample is to be selected from a population with mean, =100 and standard deviation, =10. What are the mean and standard deviation of X when (a) n=9 ? (b) n=400 ? Example 6 : Suppose that the mean value of interpupillary distance for all adult males if =65 mm and the population standard deviation is =5 mm. (a) If the distribution of interpupillary distance if normal and a sample of n=25 adult males are selected, what is the probability that the sample average for these 25 will be between 64 and 67 mm? At least 68 mm? distance, X (b) Suppose that a sample of 100 adult males is obtained. Without assuming that interpupillary distance is normally distributed, what is the approximate probability that the sample average is between 64 and 67 mm? At least 68mm? 5.1.Sampling distributions for counts and proportions NOTE: Refer to the Binomial distribution in Ch 4 on class notes. The sampling distribution of p has a mean p and the standard deviation, p 1 n . When n is large and is not too near 0 or 1, the sampling 5 and n 1 5 then it is distribution of p is appropriately normal. If both n safe to use the normal approximation. Example 7: A certain chromosome defect occurs is only one out of 200 Caucasian adult males. A random sample of n=200 such individuals is obtained i.e. =1/200 and n=200. (a) What is the mean value of the sample proportion p, and what is the standard deviation of the sample proportion? (b) Does p have approximately a normal distribution in this case? Example 8: Newsweek (Nov. 23, 1992) reported that 40% of all U.S. employees participate in "self-insurance" health plans ( =0.4). (a) In a random sample of 100 employees, what is the approximate probability that at least half of those in the sample participate in such a plan? (b) Suppose you were told that at least 60 of 100 employees in a sample from Texas participated in such a lan. Would you continue to believe that =0.4?